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Stark conjectures for CM elliptic curves over number fields
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Journal of Number Theory 103 (2003) 163–196
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Stark conjectures for CM elliptic curves over number fields Jeffrey Stopple Mathematics Department, University of California at Santa Barbara, Santa Barbara, CA 93106, USA Received 1 December 2001; revised 10 February 2003 Communicated by H. Stark
Abstract We present an elliptic curve analog of the Stark conjecture for the value of the L-function at s ¼ 0: Although implied by the general Beilinson conjectures, the approach here is very concrete. Several cases are proved. r 2003 Elsevier Inc. All rights reserved. Keywords: Stark conjectures; Beilinson conjectures; Special values of L-functions
1. Introduction In [1], Bloch constructs symbols in K2 ðEÞ for a CM elliptic curve E defined over Q; corresponding to divisors supported on torsion points of the curve. This construction, and the special properties of such curves, allowed him to prove the Bloch–Beilinson conjecture for such curves. In [2], Deninger extends Bloch’s results, for certain elliptic curves ‘of Shimura type’ or ‘type (S)’, described below. To explain the Bloch–Beilinson conjectures in the context of a curve E defined over a number field F in a simple format, we tensor K2 ðEF Þ with Q to get a vector space QK2 ðEF Þ: Let xi be a basis, and let Fj denote the embeddings of F into C: There are regulator maps for each embedding, described in Section 2 below (along with some facts about K-theory). Roughly speaking, the Beilinson conjectures consists of two parts.
E-mail address: [email protected]. 0022-314X/03/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0022-314X(03)00112-4
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1. ‘Dimension conjecture’: dim QK2 ðEF Þ ¼ ½F : Q; 2. ‘L-value conjecture’: Lð0; EÞð½F :QÞ EQ det½regðxi ÞFj : Note that for a CM elliptic curve, the functional equation gives the order of the zero at s ¼ 0 as ½F : Q: This follows from our knowledge of the Gamma factors for a Hecke character L-function. This is, of course, in strong contrast to the case of s ¼ 1: What is known by the work of Bloch and Deninger is that there are at least ½F : Q linearly independent symbols xi which make the L-value conjecture hold. On the other hand, nothing at all is known about upper bounds on the dimension. Remarkably, there is not even a single example where the dimension is known to be finite. For CM curves over number fields, the hypotheses under which one can prove anything are somewhat restrictive. For simplicity assume E has complex multiplication by the ring of integers OK of the complex quadratic field K; and E is defined over an extension F of K: Shimura showed [15, Theorem 7.44] that the following conditions are equivalent: 1. F ðEtors Þ is contained in K ab : 2. F is abelian over K and the Hecke character c corresponding to E on the ideals of F factors through the norm map from F to K: We will say that E is of type (S) if either of these hold. This condition is closely related to one considered by Gross [6]. He calls a CM curve defined over a Galois extension F of Q a ‘Q-curve’ if it is isogenous over F to all its Galois conjugates. Similarly, one defines K-curve via isogeny with all GalðF =KÞ conjugates. If E is type (S) then it is a K-curve, since then the Hecke character c is clearly Galois invariant and this is an isogeny invariant [6, Proposition (9.1.3)]. Conversely, suppose F is abelian over K and E is a K-curve defined over F : If O K ¼ f71g and the 2-Sylow subgroup of GalðF =KÞ is cyclic, then E is type (S) [2, Proposition 2]. The key fact for results about special values of L-functions is that if E is of type (S), then Lðs; EÞ factors as a product of L functions of Hecke characters of K: In [12], results on the conjecture of Birch and Swinnerton–Dyer were obtained this way. In Section 3 below we prove a negative result: if the curve E is not of Shimura type, then the Bloch construction gives only symbols with regulator equal 0. The problem is caused by the Galois action on the torsion points. In Section 4 we try to use the Galois action on K2 ðEF Þ to an advantage, by developing an elliptic curve analog of the Stark conjecture. The relevant L-functions are Artin–Hecke L-functions. Although we would like to work more generally, we restrict attention to CM curves. One reason is that for elliptic curves over number fields the continuation of the L-function to s ¼ 0 is still only conjectural. More
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significant is that CM curves have at worst additive bad reduction at any prime. The K-theory of E becomes more complicated if there is split multiplicative bad reduction. In Section 5 assuming the Dimension conjecture (1) and the L-value conjecture (2) above, we prove the analog Stark’s result [16] for rational characters. In Section 6 assuming the Dimension conjecture and that E is type (S), we prove the elliptic curve analog of the Stark conjecture for an abelian extension of the complex quadratic field K: In particular, taking a trivial character of the Galois group, we have re-derived the result of Deninger in [2] that (2) above holds. This is not an independent proof; the results rely on the same facts about curves of type (S) from [4] that [2] uses. However, it is a very classical proof. The two main ideas are an extension of the Frobenius determinant theorem, and a distribution relation for values of Kronecker–Eisenstein series on isogenous curves. In Section 7 we prove a result for a general CM elliptic curve E (i.e., not necessarily type (S)) defined over an abelian extension F of K: There exists an extension M of F ; and a character w of GalðM=F Þ; such that the elliptic Stark conjecture is true for E and w:
2. Analytic prerequisites 2.1. The Bloch–Wigner dilogarithm The regulator on K2 of curves over number fields is a map into cohomology. Here, however, we will follow the philosophy of [9] where one finds the advice ‘‘In general, the more concrete one is able to make the [Borel] regulator map, the more explicit the information one is able to extract from it.’’ So we will use Bloch’s original, function theoretic approach to the regulator as in [1]. Recall that the classical Euler dilogarithm is defined by N X zn jzjo1 n2 n¼1 Z z logð1 tÞ dt zAC\½1; NÞ
¼ t 0
Li2 ðzÞ ¼
after analytic continuation. The Bloch–Wigner dilogarithm DðzÞ ¼ ImðLi2 ðzÞÞ þ log jzj argð1 zÞ is well defined independent of path used to continue Li2 and arg. For a torus C=L; L ¼ ½o1 ; o2 corresponding to a point t in H; we have the q-symmetrized, or elliptic, dilogarithm X Dq ðzÞ ¼ Dðzqn Þ; q ¼ expð2pitÞ; zAC =qZ DC=L: nAZ
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We define another real valued function JðzÞ ¼ log jzj log j1 zj; and Jq ðzÞ ¼
N X
n
Jðzq Þ
n¼0
N X n¼1
2 jzj 2 Jðz q Þ þ log jqj B3 log ; 3 jqj
1 n
where B3 ðtÞ is the third Bernoulli polynomial B3 ðtÞ ¼ t3 3=2t2 þ 1=2t: Together these functions make the regulator function: Rq ðzÞ ¼ Dq ðzÞ iJq ðzÞ: (This normalization differs by i from the one usually taken.) Recall that Weil [18] defines the Kronecker–Eisenstein–Lerch series for x; x0 points in C=L; and ReðsÞ4a=2 þ 1 as Ka ðx; x0 ; s; LÞ ¼
X
/x0 ; oSðx þ wÞa jx þ wj 2s :
Here /x0 ; oS ¼ expððx0 w x0 wÞ=AðLÞÞ; where AðLÞ ¼ o1 o1 ImðtÞ=p; so that o1 o2 o1 o2 ¼ 2pidAðLÞ with AðLÞ40 and d ¼ 71 chosen so that o2 =o1 ¼ dt with t in H: In this way AðLÞ is independent of choice of generators for the lattice, while Að½1; tÞ ¼ ImðtÞ=p: And means sum over wAL; with wax if xAL: We have the functional equation GðsÞKa ðx; x0 ; s; LÞ ¼ AðLÞaþ1 2s Gða þ 1 sÞKa ðx0 ; x; a þ 1 sÞ/x0 ; xS
ð1Þ
For the special case a ¼ 1; x ¼ 0; s ¼ 2; we will use the more concise notation K2;1 ðu; LÞ ¼ K1 ð0; u; 2; LÞ ¼
X
/u; oS
w jwj4
:
Observe that the behavior under homothety is simple: A2 ðcLÞK2;1 ðcu; cLÞ ¼ cA2 ðLÞK2;1 ðu; LÞ: Bloch showed in [1] that for L ¼ ½1; t Rq ðexpð2piuÞÞ ¼ pA2 ð½1; tÞK2;1 ðu; ½1; tÞ:
ð2Þ
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It is worth observing how this function depends on the choice of the lattice basis. Suppose we have a basis o1 ; o2 and t ¼ o2 =o1 : Let * 1 ¼ do1 þ co2 ; o * 2 ¼ bo1 þ ao2 o and t* ¼ ðat þ bÞ=ðct þ dÞ: Then o1 A2 ð½1; tÞK2;1 ðu=o1 ; ½1; tÞ ¼ A2 ð½o1 ; o2 ÞK2;1 ðu; ½o1 ; o2 Þ * 1; o * 2 ÞK2;1 ðu; ½o * 1; o * 2 Þ ¼ A2 ð½o * 1 ; ½1; t* Þ: * 1 A2 ð½1; t* ÞK2;1 ðu=o ¼o
ð3Þ
This is a problem in general, as there is no canonical choice for the lattice basis. 2.2. Curves over number fields Now let E be an elliptic curve with invariant differential o defined over a number field F ; and with complex multiplication by K: Let M be an extension of F ; and P a point in EðMÞ: For each embedding F : M+C we get an elliptic curve EF over C corresponding to a lattice L: If F restricts to a real embedding of F ; the real period gives a canonical choice for o1 : Extend to a lattice basis with any complex o2 such that t ¼ o2 =o1 is in the upper half plane, let q ¼ expð2pitÞ and u in C=L corresponding to FðPÞ: Then Rq ðexpð2piu=o1 ÞÞ ¼ pA2 ð½1; tÞK2;1 ðu=o1 ; ½1; tÞ is well defined. On the other hand, if F restricts to a complex embedding of F ; we choose any basis o1 ; o2 for L; and define t and q as before. If F0 is the embedding which differs from F by complex conjugation, we can certainly choose basis o1 0 ¼ o1 ; o2 0 ¼ o2 for L0 ¼ L; so t0 ¼ t and q0 ¼ q: Lemma 1. Suppose the embeddings F; F0 differ by complex conjugation. Then Rq ðexpð2piu=o1 ÞÞ ¼ Rq0 ðexpð2piu0 =o1 0 ÞÞ Proof. We have ImðtÞ2 K2;1 ðu=o1 ; LÞ ¼ Imðt0 Þ2 K2;1 ðu0 =o1 0 ; L% 0 Þ ¼ Imðt0 Þ2 K2;1 ðu0 =o1 0 ; L0 Þ: So (4) follows from (2).
&
ð4Þ
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Remark. Of course, this still depends on the choice of the basis. If we change t to ðat þ bÞ=ðct þ dÞ; then (3) implies that Rq changes by ct þ d: To motivate what follows, we will summarize some relevant facts from K-theory. For a commutative ring R; recall that K0 ðRÞ is just the Grothendieck group, with generators ½M for each projective R-module M; and relations ½M þ ½M 0 ¼ ½M"M 0 : In particular for a field k; K0 ðkÞ ¼ Z: We will say no more about K1 than the fact that for fields, K1 ðkÞ ¼ k : For K2 ðkÞ; the Matsumoto relations give that K2 ðkÞ ¼ k #k =f f #1 f j f a0; 1g: The class of f #g; denoted f f ; gg; is called a symbol. For a curve E over a field F ; we have the function field F ðEÞ; and a map a K1 ðF ðEÞÞK0 ðF Þ ¼ DivðEÞ; PAEðF Þ
f/
X
ordP f ðPÞ:
The kernel F ¼ K1 ðF Þ of this map is relatively uninteresting; the cokernel PicðEÞ is important. For functions f ; g in F ðEÞ ; and fixed PAE; the tame symbol at P is defined by TP ð f ; gÞ ¼ ð 1ÞordP g ordP f
f ordP g ðPÞ; gordP f
and is trivial on tensors f #1 f ; thus is a function on symbols f f ; gg: In analogy to the divisor map above we have ‘
TP
K2 ðF ðEÞÞ !
a
K1 ðF Þ:
PAEðF Þ
Here the cokernel is mysterious. The kernel is, modulo torsion, our object of study K2 ðEÞ: When F is a number field we get a group K2 ðEF Þ for each embedding F of F into C: Associated to a symbol f f ; gg in K2 ðEF Þ we have the divisors divð f Þ; divðgÞ of the elliptic functions f ; g: Let X X ordQ ð f ÞordQ0 ðgÞðQ Q0 Þ ¼ say aP ðPÞ Q;Q0
P
be their twisted convolution divð f Þ divðgÞ: Let uP be the point on C=L corresponding to P on E: The regulator associated to the symbol f f ; gg and the
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embedding F is defined to be regðf f ; ggÞF ¼
X
aP Rq ðexpð2piuP =o1 ÞÞ=p
P
¼
X
aP A2 ð½1; tÞK2;1 ðuP =o1 ; ½1; tÞ;
ð5Þ
P
where the dependence of each of the parameters q; t; o1 ; and uP on the embedding F is suppressed. One can show [1] that this is a Steinberg function, i.e. trivial on the relations that define K2 ; hence the notation in (5) makes sense. Now suppose that the number field F has degree n over Q; and the extension M is Galois over F with Galois group G: Let S ¼ HomðM; CÞ; and let M PC be the complex vector space with basis S: A typical element in MC is written F zF F: Complex conjugation acts on both C and S; and we define Minkowski space MR to be the points such that zF ¼ zF : This Euclidean space is canonically isomorphic to RS [8, chapter I, Section 5]. We define a regulator map l : K2 ðEM Þ-MR ; x/
X
regðxÞF F:
Relation (4) of Lemma 1 shows that l actually maps to MR ; not just MC : The action of G on S on the left is the opposite action on the field: for g an element of G; g 1 FðxÞ ¼ Fðg xÞ; which we are writing Fðxg Þ: So the F coefficient of g lðxÞ; which is regðxÞg 1 F ; is equal to regðxg Þf ; the F coefficient of lðxg Þ: Thus the map l is a G module homomorphism. Remark. Of course, this l still depends on the choice of lattice basis at each embedding. Suppose as before that F; F0 are related by complex conjugation and we -
have chosen t and t0 ¼ t: If we have a vector of symbols x ; then by the remark after Lemma 1, we compute that changing t to ðat þ bÞ=ðct þ dÞ changes the vectors -
regð x ÞF -
regð x ÞF0
-
into ðct þ dÞregð x ÞF ; -
into ðct þ dÞregð x ÞF0 :
This changes the determinant of any matrix in which these vectors appear, by jct þ dj2 : Since our curve has complex multiplication, t is in K and this factor is in Q : Thus modulo Q ; our determinants will be independent of choice of lattice basis.
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2.3. Isogenies between curves Suppose we have elliptic curves E and E 0 defined over a number field F ; and an F isogeny f : E-E 0 : We identify the isogeny with a scalar fAC such that for the corresponding lattices, fLCL0 and f : C=L-C=L0 ; z/fz: The isogeny f gives a contravariant map on the function fields of the two curves f f ¼ f 3f: This respects the Matsumoto relations so we get a map on K2 of the function fields. It is easy to show the tame symbol satisfies TP ðf f ; f gÞ ¼ TfðPÞ ð f ; gÞ so we get a map f : K2 ðE 0 Þ-K2 ðEÞ; f f ; gg/ff f ; f gg: Theorem 2. We have the distribution relation f K2;1 ðfðxÞ; L0 Þ ¼ deg f
X
K2;1 ðx t; LÞ:
tAker f
As a consequence, the regulator map l of Section 2.2 satisfies lðf f f ; ggÞ ¼ FðfÞlðf f ; ggÞ; i.e. we have a commutative diagram K2 ðE 0 Þ
f
!
kl
lk FR
K2 ðEÞ
FðfÞ
!
FR
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Proof. We will first need a distribution relation for the isogeny given by multiplication by d on C=L (found in [7, Lemme 2.4.2]). Fix x in C=L and x0 Ad 1 L; then X
d 2þa 2s Ka ðx0 ; dx; s; LÞ ¼
/ dx0 ; x þ tSKa ð0; x þ t; s; LÞ:
tAd 1 L=L
This is easy to prove as the left side is just d2
X
/o0 ; dxS
o0
ðdo0 þ dx0 Þa jdo0 þ dx0 j2s
while the right is just X
X
oa dx0
t
/o; x þ tS
ðo þ dx0 Þa jo þ dx0 j2s
:
The result follows from the orthogonality relation X
/o; tS ¼
tAd 1 L=L
d2
in case o ¼ do0 ;
0
otherwise:
Now suppose L; L0 ; and f are as above, so fLCL0 : By the fundamental theorem of elementary divisors (see, for example, [15, Lemma 3.11]), we can choose bases fo1 ; o2 g of L and fo1 0 ; o2 0 g of L0 so that fo1 ¼ d1 o1 0 ;
fo2 ¼ d2 o2 0
for some integers d1 ; d2 ; so ½L0 : fL ¼ d1 d2 ¼ deg f: Note that jfj2 AðLÞ ¼ d1 d2 AðL0 Þ; a fact we will use to compare the pairing /; S0 corresponding to L0 with /; S: We compute Ka ð0; fx; s; L0 Þ ¼
X o0 a0AL0 a
¼
/o0 ; fxS0
f
X
jfj2s
oa0Af 1 L0
o% 0 a jo0 j2s
/o; dxS
oa joj2s
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a
¼
f
X
/t; dxSKa ðt; dx; s; LÞ
2s
jfj
tAker f
a
¼
X
f
deg f2s a 2 jfj2s
/½deg ft; tSKa ð0; x t; s; LÞ
tAker f tAker ½deg f
by the distribution relation for the isogeny ½deg f above. We use the orthogonality relation, for each tAker ½deg f X
/½deg ft; tS ¼
tAker f
deg f
if tAker f;
0
if tAker ½deg f\ker f
to deduce Ka ð0; fx; s; L0 Þ ¼
f
a
jfj
2s
deg f2s a 1
X
Ka ð0; x t; s; LÞ:
tAker f
The distribution relation follows from taking s ¼ 2; a ¼ 1; and ff ¼ deg f: For the commutative diagram, notice that if for each Q we fix a P in f 1 ðQÞ; divðf f Þ ¼
X
X
Q
TAker f
ordQ ð f ÞðP TÞ;
and similarly with divðf gÞ: Thus divðf f Þ divðf gÞ ¼ degðfÞ f ðdivð f Þ divðgÞÞ; and the result follows from the distribution relation.
&
Remark. Suppose the curve E has complex multiplication by OK : The theorem says the image of K2 ðEÞ in FR is an OK module. If the ‘Dimension conjecture’ (1) is true, this is a lattice in FR with complex multiplication.
3. Torsion divisorial support As mentioned in the introduction, a weak form of the Beilinson conjectures is known for CM elliptic curves of type (S). What can be done more generally? Throughout this section we assume that E is defined over an abelian extension F of K; since the most interesting case is the smallest extension of K over which E is defined, the Hilbert class field F ¼ Kð jðEÞÞ:
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Recall that F ðEtors Þ is always an abelian extension of F ; and that K ab is equal to Kð jðEÞ; xðEtors ÞÞ; where x ¼ xðPÞ is Weber’s function for E: Thus if F is abelian over K; so is F ðxðEtors ÞÞ: Remark. If the class number of K is one, Kð jðEÞÞ ¼ K: Then F abelian over K implies F DKðxðEtors ÞÞDKðEtors Þ so F ðEtors ÞDKðEtors ÞDK ab and so E is of type (S). We assume for the rest of this section that the class number of K is greater than one. We would like to know when regðxÞF is equal to zero. In the paper [13], Schappacher considers this question for curves over Q: Concerning the d torsion, he remarks in (5.6) that since the function x-K1 ðx0 ; x; sÞ is odd, if a divisor a is fixed under a- a; then K1 ð0; a; sÞ ¼ 0: Similar considerations appear in the thesis of Ross [11], from which we will borrow the idea of ‘torsion divisorial support’: Let L be any extension of F ; and let i : F +L denote inclusion. This gives rise to two maps in K theory, i : K2 ðEF Þ-K2 ðEL Þ; induced by base extension, and i : K2 ðEL Þ-K2 ðEF Þ; induced by restriction of scalars. The map i 3i acts by multiplication by ½L : F ; and i 3i is the norm map Y i 3i : f f ; ggf f ; ggs : sAGalðL=F Þ
If we fix an embedding F : L+C; and let f f ; ggAK2 ðEL Þ; we see X regði f f ; ggÞF ¼ regði 3i f f ; ggÞFjF ¼ regðf f ; ggs ÞFjF :
ð6Þ
sAGalðL=F Þ
One sees immediately that the divisor corresponding to i f f ; gg is invariant under GalðL=F Þ: Ross then makes a definition similar to Definition 3. Let N an ideal of OK : Then a symbol in K2 ðEF Þ is said to have N Q torsion divisorial support if it is of the form i j f fj ; gj g with all fj ; gj defined over F ðEN Þ; and such that the divisors divð fj Þ; divðgj Þ are supported on EN :
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If there is a s in GalðF ðEN Þ=F Þ such that Ps ¼ P for all N torsion points P; then we see the regulator is zero on any element x with N torsion divisorial support. In this context the following lemma will be useful. Lemma 4. For an ideal N of OK not dividing 2OK ; the following are equivalent: 1. There exists an N torsion point P of E such that 8sAGalðF ðEN Þ=F Þ;
Ps a P:
2. For all ideals A in OF ; cðAÞc 1 mod N; where c is the Hecke character of E 3. F ðEN Þ ¼ F ðxðEN ÞÞ: Proof. If s is the Artin symbol of an ideal A; then Ps ¼ cðAÞP: Thus 1 and 2 are equivalent. Since E is defined over F ; it is isomorphic over F to an equation of the form y2 ¼ x3 þ Ax þ B: For a point P ¼ ðx; yÞ; we have Ps ¼ P if and only if xs ¼ x and ys ¼ y: If F ðEN Þ ¼ F ðxðEN ÞÞ then clearly such a Galois action cannot happen. Conversely, suppose P is N torsion such that for all s; Ps a P; with N minimal ˜ s ¼ xðPÞ: ˜ Choose a for P: Let sAGalðF ðEN Þ=F ðxðEN ÞÞÞ: So for all P˜ in EN ; xðPÞ s ˜ From the prime Q of OF so that s is the Artin symbol for Q; then P˜ ¼ cðQÞP: Weierstrass equation, if yðPÞs a yðPÞ; it must equal yðPÞ: So cðQÞP ¼ P; thus ˜ cðQÞP˜ ¼ P˜ and P˜ s ¼ P; ˜ so s is trivial. This cðQÞ 1 modulo N: Then for all P; shows 133: & Definition 5. E is of type (R) if there exists an ideal N of OK not dividing 2OK such that any of the equivalent conditions above hold. Remark. This is a necessary condition for the regulator of a symbol with N torsion divisorial support to be nonzero. Lemma 6. If E is of type (S), then it is of type (R) Proof. This is Lemma 4.7 of [4], where they show that (S) implies that for any N divisible by both the conductor of E and the conductor of F over K; F ðEN Þ ¼ F ðxðEN ÞÞ; and is in fact the ray class field of K modulo N: & Theorem 7. E is of type (R) if and only if it is of type (S).
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Proof. Recall we are assuming the class number of K is greater than 1, and thus 0 O K ¼ f71g: By Robert [10, Corollaire 2] we know there exists an elliptic curve E 0 defined over F which is of type (S). The proof constructs a Hecke character c which has the relevant property. By Theorem 9.1.3 of [6], we may assume that E 0 and E have the same j invariant. Thus E 0 is a model of E and so c ¼ wc0 for some quadratic Dirichlet character w associated to an extension M=F : We will show that M is abelian over K: Let N be an ideal of OK such that the Hecke character c of E is never 1 modulo N: Since E 0 is of type (S) it is of type (R) by Lemma 6, and we may assume there exists an ideal N0 divisible by N such that F ðxðE 0 N0 ÞÞ is equal to F ðE 0 N0 Þ: Let Q a prime ideal of OF which splits completely in F ðxðE 0 N0 ÞÞ: We will show that Q splits in M: Since Q splits completely, the corresponding Frobenius automorphism s is trivial, so Ps ¼ P for all N0 torsion P on E 0 ; thus c0 ðQÞ 1 mod N0 : This means cðQÞ ¼ wðQÞc0 ðQÞ 71 mod N0 because wðQÞ ¼ 71: Thus cðQÞ 71 mod N as N divides N0 : By hypothesis on N; we must have cðQÞ 1 mod N; and therefore wðQÞ ¼ 1: Thus Q splits in M: This then is enough to say that MCF ðxðE 0 N0 ÞÞ; a ray class field of K: Thus M is abelian over K: By Lemme 1 of [10], we see that E is of type (S). The point is that E 0 is of type (S) so F ðE 0 tors Þ is contained in K ab : With M also contained in K ab ; we get F ðEtors Þ is contained in K ab : & Remark. If the curve E is not of Shimura type, then any symbol with torsion divisorial support has regulator equal 0.
4. Stark conjectures The result of the previous section presents two alternatives. One could consider instead the construction of symbols in K2 ðEÞ based on points of infinite order, as in [5]. Or one can try to used the Galois action to an advantage. This possibility is suggested in [9, p. 187]: ‘‘Because of the compatibility with the action of correspondences and Tate twists, the Beilinson regulators admit a ‘motivic’ formulation. This generalizes Stark’s conjectures on the factoring of the regulator according to the Galois action, relating the eigenpieces of unit groups to the values of Artin L-functions at s ¼ 0y’’ In this section we begin to work out the analog of Stark’s conjectures for K2 of an elliptic curve with complex multiplication. The approach here is as concrete and down to earth as possible.
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4.1. Notation For any finite group C and class functions w1 ; w2 on C; we let /w1 ; w2 SC denote the scalar product on C /w1 ; w2 SC ¼
1 X w ðtÞw2 ðtÞ: xC tAC 1
For any field k and abelian group A; we let kA denote k#A: Groups always act on the left, even if written as instead of s a: Suppose E is an elliptic curve over a number field F with complex multiplication pffiffiffiffi by OK ; where K ¼ Qð DÞ: There are two cases: 1. KDF : There exists a Hecke character c for F such that Lðs; EÞ ¼ Lðs; cÞLðs; cÞ: 2. KD / F ; let H ¼ F K: There exists a Hecke character c for H such that Lðs; EÞ ¼ Lðs; cÞ: We take a field M Galois over F which also contains K: Let G ¼ GalðM=F Þ; N ¼ GalðM=HÞ; and ½F : Q ¼ n: Fix an embedding F1 of M (and thus also H; K) into C; such that F1 ð jðEÞÞ ¼ jðOK Þ: Let SK ¼ HomK ðM; CÞ; so S ¼ SK ,SK : Recall the spaces MC ; MR of Section 2.2. Let MQ be the field M itself, viewed as xSK copies of the field K: The details of the Q½G embedding inside MR are in 4.4 below. The headings of the following subsections indicate which part of [17] we are imitating. 4.2. L-Functions We take a finite-dimensional complex vector space V and a representation r : G-GLðV Þ with character w: Let V denote the contragredient representation of V : We define ( Lðs; c#wÞLðs; c#wÞ in case 1; Lðs; E#wÞ ¼ Lðs; c#ResN wÞ in case 2 in terms of (products of) Artin–Hecke L functions. (Viewing the standard L function of the elliptic curve as coming from a Galois representation into cohomology, this is just the L function of the tensor product representation.) We see immediately
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this is well behaved with respect to direct sums: Lðs; E#ðw1 "w2 ÞÞ ¼ Lðs; E#w1 ÞLðs; E#w2 Þ:
ð7Þ
For induction, we need the following fact about Artin–Hecke L functions: if F 0 lies between F and M; fixed by G 0 ; and w0 is the character of a representation of G 0 ; then 0 0 Lðs; c#IndG G0 ðw ÞÞ ¼ Lðs; ðc3NormF 0 =F Þ#w Þ:
ð8Þ
One shows then that Lðs; EF 0 #w0 Þ ¼ Lðs; EF #Indðw0 ÞÞ:
ð9Þ
(There are three cases to check, depending on whether both F 0 and F contain K; neither do, or only F 0 does.) Proposition 8. Let n ¼ ½F : Q: Then Lðs; E#wÞ has a zero at s ¼ 0 of order n dimðV Þ: Proof. Consider first case 2. Via Brauer induction there exist one dimensional characters wi on subgroups Gi of N and integers ni such that X ResN ðwÞ ¼ ni IndGi ðwi Þ: i
Thus Lðs; c#ResN ðwÞÞ ¼
Y
Lðs; c#IndGi ðwi ÞÞni
i
¼
Y
Lðs; ðc3NormM Gi =H Þ#wi Þni :
i
Each of the L-functions Lðs; ðc3NormM Gi =H Þ#wi Þ has a zero at s ¼ 0 of order ½M Gi : K so the product has a zero of order X X ni ½M Gi : K ¼ ni ½M Gi : H½H : K i
i
¼ ½H : K
X
ni dimðIndGi ðwi ÞÞ
i
¼ ½F : QdimðV Þ: In case 1 similarly each of Lðs; c#wÞ and Lðs; c#wÞ have a zero at s ¼ 0 of order dimðV Þ½F : K; so the product has a zero of order n dimðV Þ: &
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Remark. To get an appropriate regulator determinant, we want an automorphism of a vector space whose dimension is equal to the order of the zero. Proposition 9. n dim ðV Þ ¼ dim HomG ðV ; MC Þ Proof. The representation of G in MC ¼ MQ #C is just n copies of the regular representation IndG e ð1Þ of G; where e denotes the identity element of G: We have n dimðV Þ ¼ n/Rese ðwÞ; 1Se ¼ /Rese ðwÞ; n 1Se ¼ /w; n IndG e ð1ÞSG by Frobenius reciprocity. Since the character of the right regular representation takes rational integer values, it is real. So the above is equal to G /wn IndG e ð1Þ; 1SG ¼ dimðV #MC Þ ¼ dim HomG ðV ; MC Þ
by duality.
&
4.3. Stark regulator Proposition 10. We suppose from now on the dimension conjecture of Section 1; specifically, that for all L; F DLDM; we have dim QK2 ðEL Þ ¼ ½L : Q: Then QK2 ðEM ÞDMQ as Q½G modules. Proof. Via the corollary on p. 104 of [14], we need only show that for all subgroups C of G C dimðQK2 ðEM ÞC Þ ¼ dimðMQ Þ:
Let L be the fixed field of C: Via Galois descent for K groups tensored with Q; QK2 ðEM ÞC DQK2 ðEL Þ: By our assumption the dimension is ½L : Q: On the other hand, C ResC IndG e ð1Þ ¼ ½G : CInde ð1Þ
by the Induction-Restriction theorem [14, p. 58]. This representation contains the trivial representation ½G : C ¼ ½L : F times. In MQ we have n ¼ ½F : Q copies of this representation, so C dimðMQ Þ ¼ ½L : Q:
&
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Remark. One would like to try to get by with the weaker assumption dim QK2 ðEL ÞX½L : Q; since this is already known for curves of type (S), by the work of Deninger [2]. One might hope to then prove that MQ +QK2 ðEM Þ as Q½G modules. But the inequality on the dimensions is not strong enough to prove this. It might, for example, be true that QK2 ðEL Þ ¼ QK2 ðEM Þ for every L; which would say QK2 ðEM Þ is trivial as a Q½G module. Now, assuming the Dimension conjecture, let f : MQ -QK2 ðEM Þ a Q½G isomorphism. Recalling the map l defined in Section 2.2, we see l3f is a G automorphism of MR ; and we use the same notation when extending scalars to MC : Via functoriality, this defines an automorphism ðl3f ÞV : ðl3f ÞV
HomG ðV ; MC Þ ! HomG ðV ; MC Þ; A/l3f 3A: We define RðE; wÞ to be the determinant of ðl3f ÞV : (This determinant actually depends on the choice of map f ; which is suppressed from the notation.) Let cðE; wÞ be the coefficient of the first term in the Taylor expansion of Lðs; E#wÞ at s ¼ 0; and define AðE; wÞ ¼
RðE; wÞ : cðE; wÞ
Elliptic Stark Conjecture. AðE; wÞ belongs to QðwÞ; and for s in GalðQðwÞ=QÞ we have AðE; wÞs ¼ AðE; ws Þ:
4.4. Towards Stark’s version From our assumption on the dimensions of the K-groups in the previous subsection we deduced the existence of a Q½G isomorphism f from MQ to QK2 ðEM Þ; which implies there exists a set M of n ‘Minkowski symbols’ each of whose G conjugates generate a xG dimensional Q vector space, in which G acts by the regular representation. (These symbols collectively play a role analogous to that of the Minkowski unit in the unit group.) Conversely, a set M of n symbols in distinct G orbits will let us define a Q½G isomorphism fM : In this subsection we deduce a more explicit formula for RðE; wÞ; by considering an explicit isomorphism fM : We must first specify the Q½G embedding of M inside MR : Consider first case 1. We choose representatives for the G orbits in S by taking F1 as in Section 4.1,
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F2 ; y; Fn=2 any representatives of the G orbits in SK ; along with their complex conjugates. Let m in M generate a normal basis of M as an F vector space, and f fi g any basis of F as a K vector space. A typical element of M is then written n=2 X X sAG
ci ðsÞfi ms
i¼1
with all the ci ðsÞ in K; which we identify with n=2 X X sAG
ci ðsÞsFi þ ci ðsÞsFi
i¼1
in MR : To get a basis over Q we let fi7
pffiffiffiffi ð17 DÞ fi ¼ 2
for i ¼ 1; y; n=2: Case 2 is more complicated as F is not a K vector space. We write n ¼ r þ 2s with r and s the number of real embeddings, resp. pairs of complex conjugate embeddings of F : We choose representatives F1 as above, F2 ; y; Fr in SK so that Fi jF is a real embedding for ipr: Thus Fi is equal g 1 i Fi for some gi in G: For roips; Fi jF is a complex embedding, so if we fix once and for all a g in G\N; we can actually define Fiþs ¼ gFi in this case; then g 1 Fiþs ¼ Fi : Let m in M generate a normal basis of M over F : Let f fi g a basis of F as a Q vector space. Thus a typical element of M can be written n X X sAG
ci ðsÞfi ms
i¼1
with the ci ðsÞ in Q: For ipr we identify the element fi ms in M with pffiffiffiffi pffiffiffiffi ð1 þ DÞsFi þ ð1 DÞsFi : When roips; we identify the element fi ms in M with pffiffiffiffi pffiffiffiffi sFi þ sFi þ DsFiþs DsFiþs ; and fiþs ms with pffiffiffiffi pffiffiffiffi DsFi DsFi þ sFiþs þ sFiþs : We then extend Q-linearly to get MQ inside MR : The determinant of ðl3fM ÞV in HomG ðV ; MC Þ is equal to the determinant of 1#l3fM in ðV #MC ÞG : We’re now ready to compute this determinant.
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Consider first case 1. We arbitrarily label the n symbols as xi;þ and xi; for i ¼ 1; y; n=2: We take the isomorphism fM ð fi7 ms Þ ¼ xsi;7 ;
i ¼ 1; y; n=2; sAG:
For tAG let regM ðtÞ denote the n n matrix of 2 2 blocks " # regðxti;þ ÞFj regðxti;þ ÞFj regðxti; ÞFj
regðxti; ÞFj
for i; j ¼ 1; y; n=2: Then Proposition 11. In case 1 we have ! X pffiffiffiffi dimðV Þn=2 det rðtÞ#regM ðtÞ ; RðE; w; fM Þ ¼ ð DÞ
ð10Þ
tAG
where as usual r is the representation of G with character w: Proof. A typical vector in ðV #MC ÞG looks like n=2 X X sAG
rðsÞvi #sFi þ rðsÞvi 0 #sFi
i¼1
with arbitrary vectors vi ; vi 0 in V : This space inherits a natural inner product from the one /; S on V ; namely the inner product of a typical vector as above with another, formed of vectors wi ; wi 0 is just n=2 X
/vi ; wi S þ /vi 0 ; wi 0 S:
i¼1
We let ep for p ¼ 1; y; dimðV Þ an orthonormal basis of V : We choose a basis for ðV #MC ÞG of the form X rðsÞep #ð fi7 ms Þ; v7 p;i ¼ sAG
for p ¼ 1; y; dimðV Þ; and i ¼ 1; y; n=2: We are identifying fi7 ms with its image in MR as in Section 4.1. This lets us compute X rðsÞep #lðxsi;7 Þ 1#l3fM ðv7 p;i Þ ¼ sAG
¼
X
sAG
rðsÞep #
X FAS
regðxi;7 ÞF sF:
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Write each F as t 1 Fk or t 1 Fk and change variables s/st to get ( ) X X t t regðxi;7 ÞFk rðtÞep #sFk þ regðxi;7 ÞFk rðtÞep #sFk : sAG
tAG
To get matrix coefficients we compute an inner product * pffiffiffiffi + X ð17 DÞ t 7 7 eq /1#l3fM ðvp;i Þ; vq; j S ¼ regðxi;7 ÞFj rðtÞep ; 2 tAG * pffiffiffiffi + X ð17 DÞ t eq ; þ regðxi;7 ÞFj rðtÞep ; 2 tAG where the choices of 7 on different sides of the inner product are of course independent. Taking all four possible choices of the 7 gives us a 2 2 block: pffiffiffi # " #" pffiffiffi 1 D 1þ D regðxti;þ ÞFj regðxti;þ ÞFj X 2 2 /rðtÞep ; eq S pffiffiffi pffiffiffi : 1þ D 1 D regðxti; ÞFj regðxti; ÞFj tAG 2
2
The determinant of the matrix with these (doubly indexed) coefficients is our regulator RðE; w; fM Þ: & Case 2 seems, at first, simpler. We have symbols xi for i ¼ 1; y; n in distinct G orbits. We take the isomorphism fM ð fi ms Þ ¼ xsi ;
i ¼ 1; y; n; sAG:
Now a typical vector in ðV #MC ÞG looks like n X X sAG
rðsÞvi #sFi
i¼1
with arbitrary vectors vi in V : The inner product of a typical vector as above with another formed of vectors wi is just n X
/vi ; wi S:
i¼1
We let ep for p ¼ 1; y; dimðV Þ an orthonormal basis of V : We choose a basis for ðV #MC ÞG of the form X rðsÞep #ð fi ms Þ; vp;i ¼ sAG
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for p ¼ 1; y; dimðV Þ; and i ¼ 1; y; n: This lets us compute 1#l3fM ðvp;i Þ ¼
X
rðsÞep #lðxsi Þ
sAG
¼
X
rðsÞep #
¼
sAG
regðxi ÞF sF
FAS
sAG
( X X
X
)
regðxti ÞFk rðtÞep
#sFk
tAG
after writing each F as t 1 Fk and changing variables s/st just as before. In order to compute matrix coefficients as inner products, we must rewrite the typical basis vector vq; j with jpr as vq; j ¼
X
rðsÞeq #ðð1 þ
pffiffiffiffi pffiffiffiffi DÞsFj þ ð1 DÞsFj Þ
rðsÞeq #ðð1 þ
pffiffiffiffi pffiffiffiffi DÞsFj þ ð1 DÞsg 1 j Fj Þ
sAG
¼
X
sAG
¼
X
rðsÞðð1 þ
pffiffiffiffi pffiffiffiffi DÞeq þ ð1 DÞrðgj Þeq Þ#sFj
sAG
after using the relation Fj ¼ g 1 j Fj of 4.1 above, and a change of variables. We then see that for jpr /1#l3fM ðvp;i Þ; vq; j S X pffiffiffiffi pffiffiffiffi regðxti ÞFj /rðtÞep ; ð1 þ DÞeq þ ð1 DÞrðgj Þeq S ¼ tAG
¼ ð1
pffiffiffiffi X DÞ regðxti ÞFj /rðtÞep ; eq S tAG
pffiffiffiffi X þ ð1 þ DÞ regðxti ÞFj /rðtÞep ; rðgj Þeq S: tAG
In the second sum above we use that r acts by isometries /rðtÞep ; rðgj Þeq S ¼ /rðg 1 j tÞep ; eq S and change the variables t/gj t: Then gt
regðxi j ÞFj ¼ regðxti Þg 1 Fj ¼ regðxti ÞFj j
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so finally /1#l3fM ðvp;i Þ; vq; j S X pffiffiffiffi pffiffiffiffi fð1 DÞregðxti ÞFj þ ð1 þ DÞregðxti ÞFj g/rðtÞep ; eq S ¼ tAG
for jpr: Similarly if rojpr þ s; we rewrite vq; j as X pffiffiffiffi rðsÞðeq Drðg 1 Þeq Þ#sFj vq; j ¼ sAG
pffiffiffiffi þ rðsÞð Deq þ rðgÞeq Þ#sFjþs Þ
using the relation between Fj and Fjþs above and a change of variables. Using the same change of variables as above, we see that X /1#l3fM ðvp;i Þ; vq; j S ¼ /rðtÞep ; eq S tAG
ðregðxti ÞFj þ regðxti ÞFj
pffiffiffiffi pffiffiffiffi Dregðxti ÞFjs þ Dregðxti ÞFjþs Þ:
5. Rational characters For the regulator determinant we have RðE; w1 "w2 Þ ¼ RðE; w1 ÞRðE; w2 Þ: Induction properties, as well, follow from [17, p. 29]: If w is the character of a representation of GalðM=F 0 Þ; then RðEF 0 ; wÞ ¼ RðEF ; IndðwÞÞ: For example, with F 0 ¼ M we get RðEM Þ ¼ RðEF ; IndG e ð1ÞÞ ¼
Y
RðEF ; wÞdimðwÞ :
wAGˆ
So we have factored the regulator determinant into pieces. By the usual properties of L-functions, the same holds for the cðE; wÞ and thus for the ratios AðE; wÞ: Remark. If we take the trivial representation 1 of G; then comparing (6) and (10) we see we have recovered the map i : K2 ðEM Þ-K2 ðEF Þ; and the determinant RðEF Þ is one piece of the determinant RðEM Þ: As in Section 3, if E is not of type (S), the regulator map is zero on symbols with torsion divisorial support. The other terms in the product are not, however, a priori zero on symbols with torsion divisorial
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support. For example, take F ¼ Kð jðEÞÞ the Hilbert class field, and M ¼ F ðEQ Þ; where pOK ¼ QQ is a split prime, and take r an odd character of GalðM=F Þ ¼ F p: This is a simple observation, but, as mentioned at the beginning of this section, it is the motivation for looking at this analog of Stark’s conjecture. Theorem 12. Suppose E is an elliptic curve defined over F ; and M is a Galois extension, with w a character of a representation of GalðM=F Þ taking rational values. Then there exists integers m; ni and intermediate fields Fi such that Lðs; EF #wÞm ¼
Y
Lðs; EFi Þni :
i
If we assume the dimension conjecture and the L-value conjecture of Section 1, we get that AðE; wÞm ¼
Y
AðEFi Þni AQ:
i
Proof. We will treat the case when F does not contain K; the other case is easier. Let m be the exponent of G ¼ GalðM=F Þ: By standard facts about representations (e.g. [14, p. 103]) there are subgroups Ci of G and integers ni such that mw¼
X
ni IndG Ci ð1Þ;
m ResN ðwÞ ¼
i
X
ni ResN IndG Ci ð1Þ:
i
So Lðs; E#mwÞ ¼
Y
ni Lðs; c#ResN IndG Ci ð1ÞÞ :
i
We want to use the Induction-Restriction theorem [14, p. 58] on each term, so we need for each i a decomposition of G into double cosets G¼
[
NgCi :
g
Fix an element dAG; deN; since ½G : N ¼ 2; there are at most two double cosets NeCi and NdCi : There are two cases: 1. Ci 5N; Then there is only a single double coset NeCi ¼ NdCi : Let C˜ i ¼ Ci -N; Fi the fixed field of Ci and F˜i the fixed field of C˜ i : The Induction-Restriction theorem says N ResN IndG Ci ð1Þ ¼ IndC˜ i ð1Þ:
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So N Lðs; c#ResN IndG Ci ð1ÞÞ ¼ Lðs; c#IndC˜ i ð1ÞÞ ˜
¼ Lðs; c3NormFHi Þ ¼ Lðs; EFi Þ: 2. Ci oN: The two double cosets are distinct. Let Di ¼ dCi d 1 ; also a subgroup of N: Let Fi the fixed field of Ci ; and Li the fixed field of Di : The InductionRestriction theorem says N N ResN IndG Ci ð1Þ ¼ IndCi ð1Þ"IndDi ð1Þ:
So N N Lðs; c#ResN IndG Ci ð1ÞÞ ¼ Lðs; c#IndC˜ i ð1ÞÞLðs; c#IndD˜ i ð1ÞÞ
¼ Lðs; c3NormFHi ÞLðs; c3NormLHi Þ: Since E is defined over the subfield F of H; and d generates GalðH=F Þ; we get from [6, Theorem 10.1.3] that dc ¼ c: This gives ¼ Lðs; c3NormFHi ÞLðs; c3NormFHi Þ ¼ Lðs; EFi Þ: Assuming the L-value conjecture of Section 1, there exist rational numbers AðEFi Þ ¼
RðEFi Þ : cðEFi Þ
Then the induction and direct sum properties imply the following weak form of the elliptic Stark conjecture for rational characters: Y AðE; wÞm ¼ AðEFi Þni AQ: & i
6. Abelian over complex quadratic This section is devoted to the proof of the following Theorem 13. Suppose the field M has abelian Galois group G over the complex quadratic field K; and that the curve E is type (S). Then the elliptic Stark conjecture is true.
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To minimize notation, we will consider the case where E is defined over F ¼ Kð jðEÞÞ: This makes F the Hilbert class field of K; and n ¼ ½F : Q ¼ 2h where h is the class number of OK : For abelian Galois groups we may as well assume the character w satisfies dimðwÞ ¼ 1: By pulling the representation w back to a larger Galois group, we can assume M is the ray class field modulo G; where G is principal and is divisible by the conductors of w and the Hecke character c: Furthermore, since E is type (S), c ¼ f3NormF =K for some Hecke character f of K: We begin by considering partial L-functions, which have only the Euler factors prime to G: By abuse of notation the dependence on G is suppressed. Then we have Lðs; c#wÞ ¼ Lðs; f3NormF =K #wÞ ¼ Lðs; f#IndGG ðwÞÞ ¼
h Y
Lðs; f#wi Þ
i¼1
¼
h X Y
wi ðgÞLðs; f; gÞ;
i¼1 gAG
where IndGG ðwÞ decomposes as "wi and we have the partial L-functions associated to an Artin symbol g ¼ ½ ; M=K in G X fðCÞNðCÞ s : Lðs; f; gÞ ¼ CCOK ½C; M=K ¼ g This gives the hth derivative at s ¼ 0 as LðhÞ ð0; c#wÞ ¼
h X Y
wi ðgÞL0 ð0; f; gÞ:
ð11Þ
i¼1 gAG
In the next subsection we will give a generalization of what is usually called the Frobenius determinant relation (actually due to Dedekind.) 6.1. Generalized Dedekind determinant Suppose G is a finite abelian group, GoG; and w : G-C is a character. Write p ¼ IndGG ðwÞ ¼ "wi for the induced representation. Fix once and for all a set S of coset representatives for G\G: Let W be the usual space the induced representation acts in: W ¼ fF : G-CjF ðsxÞ ¼ wðsÞF ðxÞ; 8sAG; xAGg;
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where G acts by multiplication: pðgÞF ðxÞ ¼ F ðxgÞ: Let f : G-C be any function, and define the operator on W pð f Þ ¼
X
f ðgÞpðgÞ:
gAG
The characters wi form a basis of W ; and so do the characteristic functions of cosets FGg ; where FGg ðs0 g0 Þ :¼
wðs0 Þ
if g ¼ g0 ;
0
otherwise:
The Dedekind determinant relation computes the determinant of pð f Þ with respect to these two canonical bases of W : Lemma 14. det pð f Þ ¼
½G:G Y
X
i¼1
gAG
"
¼ det
X
f ðgÞwi ðgÞ # 0 1
wðtÞf ðtg g Þ
tAG
: g;g0 AS
Proof. The functions wi are eigenvectors of each pðgÞ; with eigenvalue wi ðgÞ; thus also eigenvectors of pð f Þ with eigenvalue X
f ðgÞwi ðgÞ;
gAG
and so the first formula for the determinant is clear. Relative to our set of coset representatives S; define a function (a factor set) S S-G; ðg; g0 Þ/gðg; g0 Þ; so that gðg; g0 Þg ¼ g0 g00 ;
where Gg ¼ Gg0 Gg00
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in the quotient G\G: An explicit computation shows that X X wðsgðg; g0 ÞÞf ðsg00 ÞFGg0 ; pð f ÞFGg ¼ g00 AS sAG
with g; g0 ; g00 related as above. A change of variables give the matrix coefficients of pð f Þ as in the lemma. & Remark. The case when G is the trivial subgroup, and p is the regular representation of G is the usual Frobenius determinant relation. Taking L0 ð0; f; gÞ for the function f ðgÞ in the Dedekind determinant, and using (11) realizes the L-function value as a determinant: " # X ðhÞ 0 0 1 L ð0; c#wÞ ¼ det wðtÞL ð0; f; tg g Þ : ð12Þ tAG
g;g0
This application of the Dedekind determinant has long been a key ingredient for special values of L-functions. For example, results on the conjecture of Birch and Swinnerton were obtained in [3,4]. 6.2. Partial L-functions and Kronecker series In this subsection we present, for completeness, a calculation of the derivative at s ¼ 0 of a partial L-function as the value at s ¼ 2 of a Kronecker series. Since F1 ð jðEÞÞ ¼ jðOK Þ; we have, for the lattice L corresponding to EF1 L ¼ OOK for some OAC : Further, for A an ideal of OK ; we have E ½A;M=K is defined over F ; corresponding to a lattice LA ¼ hðAÞOA 1 for some hðAÞ: The isogeny between E and E ½A;M=K is just multiplication by hðAÞ: Our hypothesis that E is type (S) implies that E is isogenous over F to all its Galois conjugates, which gives that hðAÞAF for all A: By composing isogenies one sees that h is a crossed homomorphism: hðA0 AÞ ¼ hðA0 Þ½A;M=K hðAÞ: For more details on curves of type (S) (used throughout this section) see [4]. Choose a set of representative AAA for the ideal class group of K; all prime to our fixed G: Choose also a fixed set of representatives BAB so the Artin symbols ½B; M=K give every element of G ¼ GalðM=F Þ: The ideals in B are principal as F is the Hilbert class field, and we have that B ¼ ðfðBÞÞ: A given element gAG is then of
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the form ½AB; M=K for some A and some B: We get all ideals in this class by summing over a in A 1 G since then fðBÞ þ a fðBÞ mod G and ½AðfðBÞ þ aÞ; M=K ¼ ½AðfðBÞÞ; M=K: So X
Lðs; f; ½AB; M=KÞ ¼
aAA 1 G
fðAÞ fððfðBÞ þ aÞÞ : NðAÞs NðfðBÞ þ aÞs
Now, in general, we have fððlÞÞ ¼ ffin ðlÞl; where ffin ðlÞAK only depends on l mod G: Since B ¼ ðfðBÞÞ we get that ffin ðfðBÞÞ ¼ 1; and fððfðBÞ þ aÞÞ ¼ ffin ðfðBÞ þ aÞfðBÞ þ a ¼ fðBÞ þ a: Thus Lðs; f; ½AB; M=KÞ ¼
fðAÞ NðAÞs
X
fðBÞ þ a
aAA 1 G
jfðBÞ þ aj2s
:
ð13Þ
Let nAOK so that ðn=OÞ ¼ G 1 ; i.e., n is a G torsion point on C=L: We see that hðAÞn
X
fðBÞ þ a
aAA 1 G
jfðBÞ þ aj2s
2s
jhðAÞnj
¼ K1 ðfðBÞhðAÞn; 0; s; LA Þ
ð14Þ
since aAA 1 G exactly when o ¼ hðAÞnaAhðAÞOA 1 ¼ LA : We combine Eqs. (13) and (14), multiply by GðsÞ and use the functional equation (1) to see that GðsÞLðs; f; ½AB; M=KÞ ¼
fðAÞ jhðAÞnj2s AðLA Þ2 2s Gð2 sÞ NðAÞs hðAÞn K1 ð0; fðBÞhðAÞn; 2 s; LA Þ:
Thus the partial L-function derivative at s ¼ 0 is computed by the Kronecker series value at s ¼ 2; which we are denoting K2;1 :
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Lemma 15. L0 ð0; f; ½AB; M=KÞ ¼
fðAÞ hðAÞn
A2 ðLA ÞK2;1 ðfðBÞhðAÞn; LA Þ:
6.3 For convenience we now number our representatives Ai AA for the ideal class group of K; and choose them so that Ai ¼ Ai if jðAi Þ is real (1pipr), and Aiþs ¼ Ai if jðAi Þ is complex (roips). Consider the matrix on the right-hand side of Eq. (12). In terms of our representatives, we have " # X 0
1 wð½B; M=KÞL ð0; f; ½Ai Aj B; M=KÞ : det B
i; j
To use the relation between these partial L-function derivatives and Kronecker series above, we need to change A 1 i to Ai : This induces a permutation of the ideal classes as well as an extra term from the representatives of the principal ideals in each row. The permutation of the rows only changes the determinant by 71; but the extra principal ideal needs to be absorbed from each row by a change of variables in the sum, which alters the determinant. This gives the L-function value as " # X 0 aðwÞdet wð½B; M=KÞL ð0; f; ½Ai Aj B; M=KÞ ; B
i; j
where aðwÞ is the product of all terms introduced by these change of variables. Note that, as the conjecture will require, aðwÞAQðwÞ;
aðws Þ ¼ aðwÞs
8sAGalðQðwÞ=QÞ:
We use Lemma 15 (with Ai Aj instead of A) on each entry of the matrix. Factor fðAi Þ from row i for each i; and similarly fðAj Þ from each column. Note that Q 2 Q 2 i fðAi Þ is in K ; since i Ai is principal. In fact in the ‘other half’ of the L function (coming from f instead of f) we will see the complex conjugate of these terms. So modulo Q we can ignore them. We have so far shown that LðhÞ ð0; c#wÞEQðwÞ " # X A2 ðLAi Aj Þ det wð½B; M=KÞ K2;1 ðfðBÞhðAi Aj Þn; LAi Aj Þ : hðAi Aj Þn B i; j We can now apply the distribution relation of Proposition 2 to the isogeny hðAi Þ½Aj ;F =K : E ½Aj ;F =K -E ½Ai Aj ;F =K
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of degree NðAi Þ: We need to use the fact that in general AðLA Þ ¼ AðhðAÞOA 1 Þ ¼
pffiffiffiffiffiffiffi jhðAÞOj2 jDj ; 2pNðAÞ
where D is the discriminant of K: So A2 ðLAi Aj Þ ¼
jhðAi Þ½Aj ;F =K j4 NðAi Þ2
A2 ðLAj Þ:
In summary, we have got LðhÞ ð0; c#wÞEQðwÞ 2 6 6 6 det6 6 4
X BAB tAker hðAi Þ
3 7 7 A2 ðLAj Þ 7 wð½B; M=KÞ K2;1 ðfðBÞhðAj Þn t; LAj Þ7 : 7 hðAj Þn 5 i; j
By class field theory, the elements Aj of the class group correspond via our fixed embedding F1 to the other embeddings Fj which have the same restriction to K: That is C=LAj ¼ E ½Aj ;M=K ¼ EFj : We use the homothety property to factor a scalar hðAj ÞO out of the K2;1 in each column, to convert LAj to A 1 j : If we take our representatives to be integral ideals not divisible by any rational integer but 1, it is easy to see that A 1 j has a lattice basis of the form ½1; tj as required. The divisor associated to the sum over the torsion points in the kernel of hðAi Þ comes from a symbol xi in QK2 ðEM Þ by the theorem of Bloch [1]. Multiplication by fðBÞ on the torsion points of one of these curves acts by Galois automorphism ½B; M=K: Recall these fix F ; and since our isogenies hðAi Þ are defined over F we can re-write the sum over BAB as a sum over tAG: Thus we see " # X wðtÞregðxti ÞFj : LðhÞ ð0; c#wÞEQðwÞ det O=n tAG
i; j
% we get the same formula but with the Applying this construction to c instead of c; conjugate embeddings Fj instead of Fj : However, we have so far only constructed h symbols in K2 which we can relate to the L-value. To get 2h symbols, we take advantage of the fact that K2 ðEM Þ is an OK module. More specifically, the point is that we can vary the ideal G; changing the matrix of partial L functions and thus also
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the matrix of symbols, by a constant in K: For any ideal P; one sees that for any g in G; the partial L-functions satisfy LG ðs; f; gÞ ¼ LGP ðs; f; gÞ þ fðPÞNðPÞ s LG ðs; f; g ½P; M=K 1 Þ; where we now, of course, need to keep track of the ideal in the notation for the partial L-function. Thus if P is principal X
wðtÞLGP 0 ð0; f; gtÞ
tAG
¼ ð1 fðPÞwð½P; M=KÞÞ
X
wðtÞLG 0 ð0; f; gtÞ:
tAG
Choose two principal prime ideals Pþ and P ; and let G7 ¼ GP7 : Let p7 ¼ 1 fðP7 Þwð½P7 ; M=KÞ: When we change G to Gþ or G ; the matrix on the right-hand side of (12) changes by the scalar pþ or p : Following the calculation of the previous section to the end, we see the same is true for the matrix of regulators of symbols supported on the torsion. That is, let " R¼
X
# wðtÞregðxti ÞFj
tAG
i; j
the matrix corresponding to symbols on the G torsion, then " 7
p R¼
X
# wðtÞregðxti;7 ÞFj
tAG
; i; j
where the symbols xi;7 come from the G7 torsion. We have shown above that detðRÞdetðRÞEQðwÞ LðhÞ ð0; c#wÞLðhÞ ð0; c#wÞ ¼ Lð2hÞ ð0; E#wÞ: By Proposition 11, the elliptic Stark conjecture will follow from the linear algebra pffiffiffiffi Lemma 16. For an h h matrix R; and scalars pþ ; p in Qð DÞ; "
pþ R det
p R
pþ R p R
#
¼ k detðRÞdetðRÞ;
kA
Q if h is even; pffiffiffiffi Q D if h is odd:
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Proof. Write the matrix as pþ I h p Ih
pþ Ih p Ih
!
R 0
0 : R
It follows from the Laplace expansion theorem that the determinant of the first pffiffiffiffi matrix is ðpþ p p pþ Þh ; and pþ p p pþ is in Q D: & 6.4. The simple zero In the Stark conjectures, the case when the L-function has a simple zero gets special attention. We observe here that if ½F : QdimðV Þ ¼ 1; then F ¼ Q: Since E is defined over F ; we must necessarily have the class number hðKÞ ¼ 1: Since dimðV Þ ¼ 1; w factors through an abelian extension GalðM=QÞ; and M contains K by hypothesis. By the remark at the beginning of Section 3, EL is of type (S) for any intermediate field L with QDLDM: Assuming the Dimension conjecture, the results in the previous subsection give the elliptic Stark conjecture in this case.
7. Curves not type (S) The positive results so far have all been for curves of type (S), even though we introduced the elliptic Stark conjecture to study the general case. In this section we remedy this defect with the following. Theorem 17. If F is abelian over K; and E is any elliptic curve over F with complex multiplication by OK ; then there is a Galois extension M of F ; and a character w of GalðM=F Þ such that the elliptic Stark conjecture holds for Lðs; E#wÞ: Remark. This theorem does not assume the ‘Dimension conjecture’ of Section 1. Proof. By Robert [10, Corollaire 2] we know there exists an elliptic curve E 0 defined over F which is of type (S). By Theorem 9.1.3 of [6], we may assume that E 0 and E have the same j invariant. Thus E 0 is a model of E and we can write Weierstrass equations E: y2 ¼ 4x3 g2 x g3 ; E 0 : y2 ¼ 4x3 d 2 g2 x d 3 g3 pffiffiffi with d in F : The curves E and E 0 become isomorphic over M ¼ F ð d Þ via f : E-E 0 ; ðx; yÞ/ðx0 ¼ dx; y0 ¼ d 3=2 yÞ:
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We get a map on functions fields and a map on K-groups f : QK2 ðEF 0 Þ-QK2 ðEF Þ; as in Section 2. The inclusions i : F ðEÞ-MðEÞ and i0 : F ðE 0 Þ-MðEÞ also give maps i : QK2 ðEF Þ-QK2 ðEM Þ; i0 : QK2 ðEF 0 Þ-QK2 ðEM Þ: Note that the triangle formed by these three maps does not commute. In fact MðEÞ ¼ MðE 0 Þ is a Galois extension of F ðxÞ ¼ F ðx0 Þ; with Galois group the Klein four group. The quadratic subfields are MðxÞ; F ðEÞ; and F ðE 0 Þ: The subgroups of order two which fix these fields are generated by ½ 1; t; and t0 ; where 8 8 0 -x0 ; -x; > > pffiffiffi > pffiffiffi : pffiffiffi : pffiffiffi d - d; d - d: Of course, t0 ¼ t3½ 1; and both t and t0 restrict to the same nontrivial automorphism of M over F : For f in F ðE 0 Þ; we see ðf f Þt ¼ f f ;
f t ¼ f 3½ 1 ¼ ½ 1 f :
Thus for symbols x in QK2 ðEF 0 Þ; ði f xÞt ¼ i f x;
ði0 xÞt ¼ ½ 1 i0 x:
Since E 0 is type (S) there exist n ¼ ½F : Q symbols xi such that the ‘L-value conjecture’ of Section 1 is true. In the notation of Section 4, this says that AðEF 0 Þ ¼ RðEF 0 Þ=cðEF 0 Þ is rational. Let w be the nontrivial character of GalðM=F Þ: Since Lðs; EF ÞLðs; EF 0 Þ ¼ Lðs; EM Þ ¼ Lðs; EF ÞLðs; EF #wÞ we see that cðEF 0 Þ ¼ cðEF ; wÞ: It remains to relate RðEF 0 Þ to RðEF ; wÞ: For notational convenience we suppress the inclusions i and i0 : We see that for symbols xi as above, f xi þ xi t f xi xi ; ¼ 2 2 so the n symbols ðf xi þ xi Þ=2 satisfy the requirement of Section 4 to be a set M of ‘Minkowski symbols’. Fix any embedding Fj of F into C: In the formula for RðEF ; w; fM Þ in Proposition 11, the i; j entry in the sum over the Galois group
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simplifies as f xi þ xi f xi þ xi t
reg ¼ regðxi ÞFj : reg 2 2 Fj Fj Thus RðEF 0 Þ ¼ RðE; wÞ and so AðEF ; wÞ ¼ RðEF ; wÞ=cðEF ; wÞ ¼ RðEF 0 Þ=cðEF 0 Þ is rational.
&
Acknowledgments The author thanks Fernando Rodriguez Villegas and Dinakar Ramakrishnan for helpful conversations, and Bill Jacob for teaching the K-theory.
References [1] S. Bloch, Higher Regulators, Algebraic K-theory, and Zeta Functions of Elliptic Curves, Lecture Notes, UC Irvine, 1977. [2] C. Deninger, Higher regulators and Hecke L-series of imaginary quadratic fields I, Invent Math. 69 (1989) 1–69. [3] E. de Shalit, Iwasawa Theory of Elliptic Curves with Complex Multiplication, Academic Press, New York, 1987. [4] C. Goldstein, N. Schappacher, Se´ries d’Eisenstein et fonctions L de courbes elliptiques a` multiplication complexe, J. Reine Angew. Math. 327 (1981) 184–218. [5] A. Goncharov, A. Levin, Zagier’s conjecture on LðE; 2Þ; Invent. Math. 132 (1998) 393–432. [6] B. Gross, Arithmetic on Elliptic Curves with Complex Multiplication, Lecture Notes, Vol. 776, Springer, Berlin. [7] J. Mestre, N. Schappacher, Se´ries de Kronecker et fonctions L des puissances syme´triques de courbes elliptiques sur Q; in Arithmetic Algebraic Geometry (Texel, 1989), Birkha¨user, Boston, 1991, pp. 209–245. [8] J. Neukirch, Algebraic Number Theory, Springer, Berlin, 1999. [9] D. Ramakrishnan, Regulators, algebraic cycles, and values of L-functions, in: M.R. Stein, R.K. Dennis (Eds.), Algebraic K theory and Algebraic Number Theory, Contemporary Mathematics, Vol. 83, American Mathematical Society, Providence, RI, 1987. [10] G. Robert, Sur le corps de de´finition de certaines courbes elliptiques a multiplications complexes, in Se´minaire de The´orie des Nombres 1983–84, Birkha¨user, Basel, 235–253. [11] R. Ross, K2 of elliptic and Fermat curves, Ph.D. Thesis, Rutgers, 1990. [12] K. Rubin, Elliptic curves with complex multiplication and the conjecture of Birch and SwinnertonDyer, Invent. Math. 64 (3) (1981) 455–470. [13] N. Schappacher, Les conjectures de Beilinson pour les courbes elliptiques, Aste´risque 198–200 (1991) 305–317. [14] J.P. Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics, Vol. 42, Springer, Berlin. [15] G. Shimura, Arithmetic Theory of Automorphic Functions, Princeton University Press, Princeton, NJ, 1971. [16] H.M. Stark, L-functions at s ¼ 1; II. Artin L-functions with rational characters, Adv. in Math. 17 (1975) 60–92. [17] J. Tate, Les Conjectures de Stark sur les Fonctions L d’Artin en s ¼ 0; Birkha¨user, Basel, 1984. [18] A. Weil, Elliptic Functions according to Eisenstein and Kronecker, Springer, Berlin, 1976.
Journal of Number Theory 103 (2003) 163–196
http://www.elsevier.com/locate/jnt
Stark conjectures for CM elliptic curves over number fields Jeffrey Stopple Mathematics Department, University of California at Santa Barbara, Santa Barbara, CA 93106, USA Received 1 December 2001; revised 10 February 2003 Communicated by H. Stark
Abstract We present an elliptic curve analog of the Stark conjecture for the value of the L-function at s ¼ 0: Although implied by the general Beilinson conjectures, the approach here is very concrete. Several cases are proved. r 2003 Elsevier Inc. All rights reserved. Keywords: Stark conjectures; Beilinson conjectures; Special values of L-functions
1. Introduction In [1], Bloch constructs symbols in K2 ðEÞ for a CM elliptic curve E defined over Q; corresponding to divisors supported on torsion points of the curve. This construction, and the special properties of such curves, allowed him to prove the Bloch–Beilinson conjecture for such curves. In [2], Deninger extends Bloch’s results, for certain elliptic curves ‘of Shimura type’ or ‘type (S)’, described below. To explain the Bloch–Beilinson conjectures in the context of a curve E defined over a number field F in a simple format, we tensor K2 ðEF Þ with Q to get a vector space QK2 ðEF Þ: Let xi be a basis, and let Fj denote the embeddings of F into C: There are regulator maps for each embedding, described in Section 2 below (along with some facts about K-theory). Roughly speaking, the Beilinson conjectures consists of two parts.
E-mail address: [email protected]. 0022-314X/03/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0022-314X(03)00112-4
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1. ‘Dimension conjecture’: dim QK2 ðEF Þ ¼ ½F : Q; 2. ‘L-value conjecture’: Lð0; EÞð½F :QÞ EQ det½regðxi ÞFj : Note that for a CM elliptic curve, the functional equation gives the order of the zero at s ¼ 0 as ½F : Q: This follows from our knowledge of the Gamma factors for a Hecke character L-function. This is, of course, in strong contrast to the case of s ¼ 1: What is known by the work of Bloch and Deninger is that there are at least ½F : Q linearly independent symbols xi which make the L-value conjecture hold. On the other hand, nothing at all is known about upper bounds on the dimension. Remarkably, there is not even a single example where the dimension is known to be finite. For CM curves over number fields, the hypotheses under which one can prove anything are somewhat restrictive. For simplicity assume E has complex multiplication by the ring of integers OK of the complex quadratic field K; and E is defined over an extension F of K: Shimura showed [15, Theorem 7.44] that the following conditions are equivalent: 1. F ðEtors Þ is contained in K ab : 2. F is abelian over K and the Hecke character c corresponding to E on the ideals of F factors through the norm map from F to K: We will say that E is of type (S) if either of these hold. This condition is closely related to one considered by Gross [6]. He calls a CM curve defined over a Galois extension F of Q a ‘Q-curve’ if it is isogenous over F to all its Galois conjugates. Similarly, one defines K-curve via isogeny with all GalðF =KÞ conjugates. If E is type (S) then it is a K-curve, since then the Hecke character c is clearly Galois invariant and this is an isogeny invariant [6, Proposition (9.1.3)]. Conversely, suppose F is abelian over K and E is a K-curve defined over F : If O K ¼ f71g and the 2-Sylow subgroup of GalðF =KÞ is cyclic, then E is type (S) [2, Proposition 2]. The key fact for results about special values of L-functions is that if E is of type (S), then Lðs; EÞ factors as a product of L functions of Hecke characters of K: In [12], results on the conjecture of Birch and Swinnerton–Dyer were obtained this way. In Section 3 below we prove a negative result: if the curve E is not of Shimura type, then the Bloch construction gives only symbols with regulator equal 0. The problem is caused by the Galois action on the torsion points. In Section 4 we try to use the Galois action on K2 ðEF Þ to an advantage, by developing an elliptic curve analog of the Stark conjecture. The relevant L-functions are Artin–Hecke L-functions. Although we would like to work more generally, we restrict attention to CM curves. One reason is that for elliptic curves over number fields the continuation of the L-function to s ¼ 0 is still only conjectural. More
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significant is that CM curves have at worst additive bad reduction at any prime. The K-theory of E becomes more complicated if there is split multiplicative bad reduction. In Section 5 assuming the Dimension conjecture (1) and the L-value conjecture (2) above, we prove the analog Stark’s result [16] for rational characters. In Section 6 assuming the Dimension conjecture and that E is type (S), we prove the elliptic curve analog of the Stark conjecture for an abelian extension of the complex quadratic field K: In particular, taking a trivial character of the Galois group, we have re-derived the result of Deninger in [2] that (2) above holds. This is not an independent proof; the results rely on the same facts about curves of type (S) from [4] that [2] uses. However, it is a very classical proof. The two main ideas are an extension of the Frobenius determinant theorem, and a distribution relation for values of Kronecker–Eisenstein series on isogenous curves. In Section 7 we prove a result for a general CM elliptic curve E (i.e., not necessarily type (S)) defined over an abelian extension F of K: There exists an extension M of F ; and a character w of GalðM=F Þ; such that the elliptic Stark conjecture is true for E and w:
2. Analytic prerequisites 2.1. The Bloch–Wigner dilogarithm The regulator on K2 of curves over number fields is a map into cohomology. Here, however, we will follow the philosophy of [9] where one finds the advice ‘‘In general, the more concrete one is able to make the [Borel] regulator map, the more explicit the information one is able to extract from it.’’ So we will use Bloch’s original, function theoretic approach to the regulator as in [1]. Recall that the classical Euler dilogarithm is defined by N X zn jzjo1 n2 n¼1 Z z logð1 tÞ dt zAC\½1; NÞ
¼ t 0
Li2 ðzÞ ¼
after analytic continuation. The Bloch–Wigner dilogarithm DðzÞ ¼ ImðLi2 ðzÞÞ þ log jzj argð1 zÞ is well defined independent of path used to continue Li2 and arg. For a torus C=L; L ¼ ½o1 ; o2 corresponding to a point t in H; we have the q-symmetrized, or elliptic, dilogarithm X Dq ðzÞ ¼ Dðzqn Þ; q ¼ expð2pitÞ; zAC =qZ DC=L: nAZ
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We define another real valued function JðzÞ ¼ log jzj log j1 zj; and Jq ðzÞ ¼
N X
n
Jðzq Þ
n¼0
N X n¼1
2 jzj 2 Jðz q Þ þ log jqj B3 log ; 3 jqj
1 n
where B3 ðtÞ is the third Bernoulli polynomial B3 ðtÞ ¼ t3 3=2t2 þ 1=2t: Together these functions make the regulator function: Rq ðzÞ ¼ Dq ðzÞ iJq ðzÞ: (This normalization differs by i from the one usually taken.) Recall that Weil [18] defines the Kronecker–Eisenstein–Lerch series for x; x0 points in C=L; and ReðsÞ4a=2 þ 1 as Ka ðx; x0 ; s; LÞ ¼
X
/x0 ; oSðx þ wÞa jx þ wj 2s :
Here /x0 ; oS ¼ expððx0 w x0 wÞ=AðLÞÞ; where AðLÞ ¼ o1 o1 ImðtÞ=p; so that o1 o2 o1 o2 ¼ 2pidAðLÞ with AðLÞ40 and d ¼ 71 chosen so that o2 =o1 ¼ dt with t in H: In this way AðLÞ is independent of choice of generators for the lattice, while Að½1; tÞ ¼ ImðtÞ=p: And means sum over wAL; with wax if xAL: We have the functional equation GðsÞKa ðx; x0 ; s; LÞ ¼ AðLÞaþ1 2s Gða þ 1 sÞKa ðx0 ; x; a þ 1 sÞ/x0 ; xS
ð1Þ
For the special case a ¼ 1; x ¼ 0; s ¼ 2; we will use the more concise notation K2;1 ðu; LÞ ¼ K1 ð0; u; 2; LÞ ¼
X
/u; oS
w jwj4
:
Observe that the behavior under homothety is simple: A2 ðcLÞK2;1 ðcu; cLÞ ¼ cA2 ðLÞK2;1 ðu; LÞ: Bloch showed in [1] that for L ¼ ½1; t Rq ðexpð2piuÞÞ ¼ pA2 ð½1; tÞK2;1 ðu; ½1; tÞ:
ð2Þ
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It is worth observing how this function depends on the choice of the lattice basis. Suppose we have a basis o1 ; o2 and t ¼ o2 =o1 : Let * 1 ¼ do1 þ co2 ; o * 2 ¼ bo1 þ ao2 o and t* ¼ ðat þ bÞ=ðct þ dÞ: Then o1 A2 ð½1; tÞK2;1 ðu=o1 ; ½1; tÞ ¼ A2 ð½o1 ; o2 ÞK2;1 ðu; ½o1 ; o2 Þ * 1; o * 2 ÞK2;1 ðu; ½o * 1; o * 2 Þ ¼ A2 ð½o * 1 ; ½1; t* Þ: * 1 A2 ð½1; t* ÞK2;1 ðu=o ¼o
ð3Þ
This is a problem in general, as there is no canonical choice for the lattice basis. 2.2. Curves over number fields Now let E be an elliptic curve with invariant differential o defined over a number field F ; and with complex multiplication by K: Let M be an extension of F ; and P a point in EðMÞ: For each embedding F : M+C we get an elliptic curve EF over C corresponding to a lattice L: If F restricts to a real embedding of F ; the real period gives a canonical choice for o1 : Extend to a lattice basis with any complex o2 such that t ¼ o2 =o1 is in the upper half plane, let q ¼ expð2pitÞ and u in C=L corresponding to FðPÞ: Then Rq ðexpð2piu=o1 ÞÞ ¼ pA2 ð½1; tÞK2;1 ðu=o1 ; ½1; tÞ is well defined. On the other hand, if F restricts to a complex embedding of F ; we choose any basis o1 ; o2 for L; and define t and q as before. If F0 is the embedding which differs from F by complex conjugation, we can certainly choose basis o1 0 ¼ o1 ; o2 0 ¼ o2 for L0 ¼ L; so t0 ¼ t and q0 ¼ q: Lemma 1. Suppose the embeddings F; F0 differ by complex conjugation. Then Rq ðexpð2piu=o1 ÞÞ ¼ Rq0 ðexpð2piu0 =o1 0 ÞÞ Proof. We have ImðtÞ2 K2;1 ðu=o1 ; LÞ ¼ Imðt0 Þ2 K2;1 ðu0 =o1 0 ; L% 0 Þ ¼ Imðt0 Þ2 K2;1 ðu0 =o1 0 ; L0 Þ: So (4) follows from (2).
&
ð4Þ
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Remark. Of course, this still depends on the choice of the basis. If we change t to ðat þ bÞ=ðct þ dÞ; then (3) implies that Rq changes by ct þ d: To motivate what follows, we will summarize some relevant facts from K-theory. For a commutative ring R; recall that K0 ðRÞ is just the Grothendieck group, with generators ½M for each projective R-module M; and relations ½M þ ½M 0 ¼ ½M"M 0 : In particular for a field k; K0 ðkÞ ¼ Z: We will say no more about K1 than the fact that for fields, K1 ðkÞ ¼ k : For K2 ðkÞ; the Matsumoto relations give that K2 ðkÞ ¼ k #k =f f #1 f j f a0; 1g: The class of f #g; denoted f f ; gg; is called a symbol. For a curve E over a field F ; we have the function field F ðEÞ; and a map a K1 ðF ðEÞÞK0 ðF Þ ¼ DivðEÞ; PAEðF Þ
f/
X
ordP f ðPÞ:
The kernel F ¼ K1 ðF Þ of this map is relatively uninteresting; the cokernel PicðEÞ is important. For functions f ; g in F ðEÞ ; and fixed PAE; the tame symbol at P is defined by TP ð f ; gÞ ¼ ð 1ÞordP g ordP f
f ordP g ðPÞ; gordP f
and is trivial on tensors f #1 f ; thus is a function on symbols f f ; gg: In analogy to the divisor map above we have ‘
TP
K2 ðF ðEÞÞ !
a
K1 ðF Þ:
PAEðF Þ
Here the cokernel is mysterious. The kernel is, modulo torsion, our object of study K2 ðEÞ: When F is a number field we get a group K2 ðEF Þ for each embedding F of F into C: Associated to a symbol f f ; gg in K2 ðEF Þ we have the divisors divð f Þ; divðgÞ of the elliptic functions f ; g: Let X X ordQ ð f ÞordQ0 ðgÞðQ Q0 Þ ¼ say aP ðPÞ Q;Q0
P
be their twisted convolution divð f Þ divðgÞ: Let uP be the point on C=L corresponding to P on E: The regulator associated to the symbol f f ; gg and the
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embedding F is defined to be regðf f ; ggÞF ¼
X
aP Rq ðexpð2piuP =o1 ÞÞ=p
P
¼
X
aP A2 ð½1; tÞK2;1 ðuP =o1 ; ½1; tÞ;
ð5Þ
P
where the dependence of each of the parameters q; t; o1 ; and uP on the embedding F is suppressed. One can show [1] that this is a Steinberg function, i.e. trivial on the relations that define K2 ; hence the notation in (5) makes sense. Now suppose that the number field F has degree n over Q; and the extension M is Galois over F with Galois group G: Let S ¼ HomðM; CÞ; and let M PC be the complex vector space with basis S: A typical element in MC is written F zF F: Complex conjugation acts on both C and S; and we define Minkowski space MR to be the points such that zF ¼ zF : This Euclidean space is canonically isomorphic to RS [8, chapter I, Section 5]. We define a regulator map l : K2 ðEM Þ-MR ; x/
X
regðxÞF F:
Relation (4) of Lemma 1 shows that l actually maps to MR ; not just MC : The action of G on S on the left is the opposite action on the field: for g an element of G; g 1 FðxÞ ¼ Fðg xÞ; which we are writing Fðxg Þ: So the F coefficient of g lðxÞ; which is regðxÞg 1 F ; is equal to regðxg Þf ; the F coefficient of lðxg Þ: Thus the map l is a G module homomorphism. Remark. Of course, this l still depends on the choice of lattice basis at each embedding. Suppose as before that F; F0 are related by complex conjugation and we -
have chosen t and t0 ¼ t: If we have a vector of symbols x ; then by the remark after Lemma 1, we compute that changing t to ðat þ bÞ=ðct þ dÞ changes the vectors -
regð x ÞF -
regð x ÞF0
-
into ðct þ dÞregð x ÞF ; -
into ðct þ dÞregð x ÞF0 :
This changes the determinant of any matrix in which these vectors appear, by jct þ dj2 : Since our curve has complex multiplication, t is in K and this factor is in Q : Thus modulo Q ; our determinants will be independent of choice of lattice basis.
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2.3. Isogenies between curves Suppose we have elliptic curves E and E 0 defined over a number field F ; and an F isogeny f : E-E 0 : We identify the isogeny with a scalar fAC such that for the corresponding lattices, fLCL0 and f : C=L-C=L0 ; z/fz: The isogeny f gives a contravariant map on the function fields of the two curves f f ¼ f 3f: This respects the Matsumoto relations so we get a map on K2 of the function fields. It is easy to show the tame symbol satisfies TP ðf f ; f gÞ ¼ TfðPÞ ð f ; gÞ so we get a map f : K2 ðE 0 Þ-K2 ðEÞ; f f ; gg/ff f ; f gg: Theorem 2. We have the distribution relation f K2;1 ðfðxÞ; L0 Þ ¼ deg f
X
K2;1 ðx t; LÞ:
tAker f
As a consequence, the regulator map l of Section 2.2 satisfies lðf f f ; ggÞ ¼ FðfÞlðf f ; ggÞ; i.e. we have a commutative diagram K2 ðE 0 Þ
f
!
kl
lk FR
K2 ðEÞ
FðfÞ
!
FR
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Proof. We will first need a distribution relation for the isogeny given by multiplication by d on C=L (found in [7, Lemme 2.4.2]). Fix x in C=L and x0 Ad 1 L; then X
d 2þa 2s Ka ðx0 ; dx; s; LÞ ¼
/ dx0 ; x þ tSKa ð0; x þ t; s; LÞ:
tAd 1 L=L
This is easy to prove as the left side is just d2
X
/o0 ; dxS
o0
ðdo0 þ dx0 Þa jdo0 þ dx0 j2s
while the right is just X
X
oa dx0
t
/o; x þ tS
ðo þ dx0 Þa jo þ dx0 j2s
:
The result follows from the orthogonality relation X
/o; tS ¼
tAd 1 L=L
d2
in case o ¼ do0 ;
0
otherwise:
Now suppose L; L0 ; and f are as above, so fLCL0 : By the fundamental theorem of elementary divisors (see, for example, [15, Lemma 3.11]), we can choose bases fo1 ; o2 g of L and fo1 0 ; o2 0 g of L0 so that fo1 ¼ d1 o1 0 ;
fo2 ¼ d2 o2 0
for some integers d1 ; d2 ; so ½L0 : fL ¼ d1 d2 ¼ deg f: Note that jfj2 AðLÞ ¼ d1 d2 AðL0 Þ; a fact we will use to compare the pairing /; S0 corresponding to L0 with /; S: We compute Ka ð0; fx; s; L0 Þ ¼
X o0 a0AL0 a
¼
/o0 ; fxS0
f
X
jfj2s
oa0Af 1 L0
o% 0 a jo0 j2s
/o; dxS
oa joj2s
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a
¼
f
X
/t; dxSKa ðt; dx; s; LÞ
2s
jfj
tAker f
a
¼
X
f
deg f2s a 2 jfj2s
/½deg ft; tSKa ð0; x t; s; LÞ
tAker f tAker ½deg f
by the distribution relation for the isogeny ½deg f above. We use the orthogonality relation, for each tAker ½deg f X
/½deg ft; tS ¼
tAker f
deg f
if tAker f;
0
if tAker ½deg f\ker f
to deduce Ka ð0; fx; s; L0 Þ ¼
f
a
jfj
2s
deg f2s a 1
X
Ka ð0; x t; s; LÞ:
tAker f
The distribution relation follows from taking s ¼ 2; a ¼ 1; and ff ¼ deg f: For the commutative diagram, notice that if for each Q we fix a P in f 1 ðQÞ; divðf f Þ ¼
X
X
Q
TAker f
ordQ ð f ÞðP TÞ;
and similarly with divðf gÞ: Thus divðf f Þ divðf gÞ ¼ degðfÞ f ðdivð f Þ divðgÞÞ; and the result follows from the distribution relation.
&
Remark. Suppose the curve E has complex multiplication by OK : The theorem says the image of K2 ðEÞ in FR is an OK module. If the ‘Dimension conjecture’ (1) is true, this is a lattice in FR with complex multiplication.
3. Torsion divisorial support As mentioned in the introduction, a weak form of the Beilinson conjectures is known for CM elliptic curves of type (S). What can be done more generally? Throughout this section we assume that E is defined over an abelian extension F of K; since the most interesting case is the smallest extension of K over which E is defined, the Hilbert class field F ¼ Kð jðEÞÞ:
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Recall that F ðEtors Þ is always an abelian extension of F ; and that K ab is equal to Kð jðEÞ; xðEtors ÞÞ; where x ¼ xðPÞ is Weber’s function for E: Thus if F is abelian over K; so is F ðxðEtors ÞÞ: Remark. If the class number of K is one, Kð jðEÞÞ ¼ K: Then F abelian over K implies F DKðxðEtors ÞÞDKðEtors Þ so F ðEtors ÞDKðEtors ÞDK ab and so E is of type (S). We assume for the rest of this section that the class number of K is greater than one. We would like to know when regðxÞF is equal to zero. In the paper [13], Schappacher considers this question for curves over Q: Concerning the d torsion, he remarks in (5.6) that since the function x-K1 ðx0 ; x; sÞ is odd, if a divisor a is fixed under a- a; then K1 ð0; a; sÞ ¼ 0: Similar considerations appear in the thesis of Ross [11], from which we will borrow the idea of ‘torsion divisorial support’: Let L be any extension of F ; and let i : F +L denote inclusion. This gives rise to two maps in K theory, i : K2 ðEF Þ-K2 ðEL Þ; induced by base extension, and i : K2 ðEL Þ-K2 ðEF Þ; induced by restriction of scalars. The map i 3i acts by multiplication by ½L : F ; and i 3i is the norm map Y i 3i : f f ; ggf f ; ggs : sAGalðL=F Þ
If we fix an embedding F : L+C; and let f f ; ggAK2 ðEL Þ; we see X regði f f ; ggÞF ¼ regði 3i f f ; ggÞFjF ¼ regðf f ; ggs ÞFjF :
ð6Þ
sAGalðL=F Þ
One sees immediately that the divisor corresponding to i f f ; gg is invariant under GalðL=F Þ: Ross then makes a definition similar to Definition 3. Let N an ideal of OK : Then a symbol in K2 ðEF Þ is said to have N Q torsion divisorial support if it is of the form i j f fj ; gj g with all fj ; gj defined over F ðEN Þ; and such that the divisors divð fj Þ; divðgj Þ are supported on EN :
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If there is a s in GalðF ðEN Þ=F Þ such that Ps ¼ P for all N torsion points P; then we see the regulator is zero on any element x with N torsion divisorial support. In this context the following lemma will be useful. Lemma 4. For an ideal N of OK not dividing 2OK ; the following are equivalent: 1. There exists an N torsion point P of E such that 8sAGalðF ðEN Þ=F Þ;
Ps a P:
2. For all ideals A in OF ; cðAÞc 1 mod N; where c is the Hecke character of E 3. F ðEN Þ ¼ F ðxðEN ÞÞ: Proof. If s is the Artin symbol of an ideal A; then Ps ¼ cðAÞP: Thus 1 and 2 are equivalent. Since E is defined over F ; it is isomorphic over F to an equation of the form y2 ¼ x3 þ Ax þ B: For a point P ¼ ðx; yÞ; we have Ps ¼ P if and only if xs ¼ x and ys ¼ y: If F ðEN Þ ¼ F ðxðEN ÞÞ then clearly such a Galois action cannot happen. Conversely, suppose P is N torsion such that for all s; Ps a P; with N minimal ˜ s ¼ xðPÞ: ˜ Choose a for P: Let sAGalðF ðEN Þ=F ðxðEN ÞÞÞ: So for all P˜ in EN ; xðPÞ s ˜ From the prime Q of OF so that s is the Artin symbol for Q; then P˜ ¼ cðQÞP: Weierstrass equation, if yðPÞs a yðPÞ; it must equal yðPÞ: So cðQÞP ¼ P; thus ˜ cðQÞP˜ ¼ P˜ and P˜ s ¼ P; ˜ so s is trivial. This cðQÞ 1 modulo N: Then for all P; shows 133: & Definition 5. E is of type (R) if there exists an ideal N of OK not dividing 2OK such that any of the equivalent conditions above hold. Remark. This is a necessary condition for the regulator of a symbol with N torsion divisorial support to be nonzero. Lemma 6. If E is of type (S), then it is of type (R) Proof. This is Lemma 4.7 of [4], where they show that (S) implies that for any N divisible by both the conductor of E and the conductor of F over K; F ðEN Þ ¼ F ðxðEN ÞÞ; and is in fact the ray class field of K modulo N: & Theorem 7. E is of type (R) if and only if it is of type (S).
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Proof. Recall we are assuming the class number of K is greater than 1, and thus 0 O K ¼ f71g: By Robert [10, Corollaire 2] we know there exists an elliptic curve E 0 defined over F which is of type (S). The proof constructs a Hecke character c which has the relevant property. By Theorem 9.1.3 of [6], we may assume that E 0 and E have the same j invariant. Thus E 0 is a model of E and so c ¼ wc0 for some quadratic Dirichlet character w associated to an extension M=F : We will show that M is abelian over K: Let N be an ideal of OK such that the Hecke character c of E is never 1 modulo N: Since E 0 is of type (S) it is of type (R) by Lemma 6, and we may assume there exists an ideal N0 divisible by N such that F ðxðE 0 N0 ÞÞ is equal to F ðE 0 N0 Þ: Let Q a prime ideal of OF which splits completely in F ðxðE 0 N0 ÞÞ: We will show that Q splits in M: Since Q splits completely, the corresponding Frobenius automorphism s is trivial, so Ps ¼ P for all N0 torsion P on E 0 ; thus c0 ðQÞ 1 mod N0 : This means cðQÞ ¼ wðQÞc0 ðQÞ 71 mod N0 because wðQÞ ¼ 71: Thus cðQÞ 71 mod N as N divides N0 : By hypothesis on N; we must have cðQÞ 1 mod N; and therefore wðQÞ ¼ 1: Thus Q splits in M: This then is enough to say that MCF ðxðE 0 N0 ÞÞ; a ray class field of K: Thus M is abelian over K: By Lemme 1 of [10], we see that E is of type (S). The point is that E 0 is of type (S) so F ðE 0 tors Þ is contained in K ab : With M also contained in K ab ; we get F ðEtors Þ is contained in K ab : & Remark. If the curve E is not of Shimura type, then any symbol with torsion divisorial support has regulator equal 0.
4. Stark conjectures The result of the previous section presents two alternatives. One could consider instead the construction of symbols in K2 ðEÞ based on points of infinite order, as in [5]. Or one can try to used the Galois action to an advantage. This possibility is suggested in [9, p. 187]: ‘‘Because of the compatibility with the action of correspondences and Tate twists, the Beilinson regulators admit a ‘motivic’ formulation. This generalizes Stark’s conjectures on the factoring of the regulator according to the Galois action, relating the eigenpieces of unit groups to the values of Artin L-functions at s ¼ 0y’’ In this section we begin to work out the analog of Stark’s conjectures for K2 of an elliptic curve with complex multiplication. The approach here is as concrete and down to earth as possible.
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4.1. Notation For any finite group C and class functions w1 ; w2 on C; we let /w1 ; w2 SC denote the scalar product on C /w1 ; w2 SC ¼
1 X w ðtÞw2 ðtÞ: xC tAC 1
For any field k and abelian group A; we let kA denote k#A: Groups always act on the left, even if written as instead of s a: Suppose E is an elliptic curve over a number field F with complex multiplication pffiffiffiffi by OK ; where K ¼ Qð DÞ: There are two cases: 1. KDF : There exists a Hecke character c for F such that Lðs; EÞ ¼ Lðs; cÞLðs; cÞ: 2. KD / F ; let H ¼ F K: There exists a Hecke character c for H such that Lðs; EÞ ¼ Lðs; cÞ: We take a field M Galois over F which also contains K: Let G ¼ GalðM=F Þ; N ¼ GalðM=HÞ; and ½F : Q ¼ n: Fix an embedding F1 of M (and thus also H; K) into C; such that F1 ð jðEÞÞ ¼ jðOK Þ: Let SK ¼ HomK ðM; CÞ; so S ¼ SK ,SK : Recall the spaces MC ; MR of Section 2.2. Let MQ be the field M itself, viewed as xSK copies of the field K: The details of the Q½G embedding inside MR are in 4.4 below. The headings of the following subsections indicate which part of [17] we are imitating. 4.2. L-Functions We take a finite-dimensional complex vector space V and a representation r : G-GLðV Þ with character w: Let V denote the contragredient representation of V : We define ( Lðs; c#wÞLðs; c#wÞ in case 1; Lðs; E#wÞ ¼ Lðs; c#ResN wÞ in case 2 in terms of (products of) Artin–Hecke L functions. (Viewing the standard L function of the elliptic curve as coming from a Galois representation into cohomology, this is just the L function of the tensor product representation.) We see immediately
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this is well behaved with respect to direct sums: Lðs; E#ðw1 "w2 ÞÞ ¼ Lðs; E#w1 ÞLðs; E#w2 Þ:
ð7Þ
For induction, we need the following fact about Artin–Hecke L functions: if F 0 lies between F and M; fixed by G 0 ; and w0 is the character of a representation of G 0 ; then 0 0 Lðs; c#IndG G0 ðw ÞÞ ¼ Lðs; ðc3NormF 0 =F Þ#w Þ:
ð8Þ
One shows then that Lðs; EF 0 #w0 Þ ¼ Lðs; EF #Indðw0 ÞÞ:
ð9Þ
(There are three cases to check, depending on whether both F 0 and F contain K; neither do, or only F 0 does.) Proposition 8. Let n ¼ ½F : Q: Then Lðs; E#wÞ has a zero at s ¼ 0 of order n dimðV Þ: Proof. Consider first case 2. Via Brauer induction there exist one dimensional characters wi on subgroups Gi of N and integers ni such that X ResN ðwÞ ¼ ni IndGi ðwi Þ: i
Thus Lðs; c#ResN ðwÞÞ ¼
Y
Lðs; c#IndGi ðwi ÞÞni
i
¼
Y
Lðs; ðc3NormM Gi =H Þ#wi Þni :
i
Each of the L-functions Lðs; ðc3NormM Gi =H Þ#wi Þ has a zero at s ¼ 0 of order ½M Gi : K so the product has a zero of order X X ni ½M Gi : K ¼ ni ½M Gi : H½H : K i
i
¼ ½H : K
X
ni dimðIndGi ðwi ÞÞ
i
¼ ½F : QdimðV Þ: In case 1 similarly each of Lðs; c#wÞ and Lðs; c#wÞ have a zero at s ¼ 0 of order dimðV Þ½F : K; so the product has a zero of order n dimðV Þ: &
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Remark. To get an appropriate regulator determinant, we want an automorphism of a vector space whose dimension is equal to the order of the zero. Proposition 9. n dim ðV Þ ¼ dim HomG ðV ; MC Þ Proof. The representation of G in MC ¼ MQ #C is just n copies of the regular representation IndG e ð1Þ of G; where e denotes the identity element of G: We have n dimðV Þ ¼ n/Rese ðwÞ; 1Se ¼ /Rese ðwÞ; n 1Se ¼ /w; n IndG e ð1ÞSG by Frobenius reciprocity. Since the character of the right regular representation takes rational integer values, it is real. So the above is equal to G /wn IndG e ð1Þ; 1SG ¼ dimðV #MC Þ ¼ dim HomG ðV ; MC Þ
by duality.
&
4.3. Stark regulator Proposition 10. We suppose from now on the dimension conjecture of Section 1; specifically, that for all L; F DLDM; we have dim QK2 ðEL Þ ¼ ½L : Q: Then QK2 ðEM ÞDMQ as Q½G modules. Proof. Via the corollary on p. 104 of [14], we need only show that for all subgroups C of G C dimðQK2 ðEM ÞC Þ ¼ dimðMQ Þ:
Let L be the fixed field of C: Via Galois descent for K groups tensored with Q; QK2 ðEM ÞC DQK2 ðEL Þ: By our assumption the dimension is ½L : Q: On the other hand, C ResC IndG e ð1Þ ¼ ½G : CInde ð1Þ
by the Induction-Restriction theorem [14, p. 58]. This representation contains the trivial representation ½G : C ¼ ½L : F times. In MQ we have n ¼ ½F : Q copies of this representation, so C dimðMQ Þ ¼ ½L : Q:
&
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Remark. One would like to try to get by with the weaker assumption dim QK2 ðEL ÞX½L : Q; since this is already known for curves of type (S), by the work of Deninger [2]. One might hope to then prove that MQ +QK2 ðEM Þ as Q½G modules. But the inequality on the dimensions is not strong enough to prove this. It might, for example, be true that QK2 ðEL Þ ¼ QK2 ðEM Þ for every L; which would say QK2 ðEM Þ is trivial as a Q½G module. Now, assuming the Dimension conjecture, let f : MQ -QK2 ðEM Þ a Q½G isomorphism. Recalling the map l defined in Section 2.2, we see l3f is a G automorphism of MR ; and we use the same notation when extending scalars to MC : Via functoriality, this defines an automorphism ðl3f ÞV : ðl3f ÞV
HomG ðV ; MC Þ ! HomG ðV ; MC Þ; A/l3f 3A: We define RðE; wÞ to be the determinant of ðl3f ÞV : (This determinant actually depends on the choice of map f ; which is suppressed from the notation.) Let cðE; wÞ be the coefficient of the first term in the Taylor expansion of Lðs; E#wÞ at s ¼ 0; and define AðE; wÞ ¼
RðE; wÞ : cðE; wÞ
Elliptic Stark Conjecture. AðE; wÞ belongs to QðwÞ; and for s in GalðQðwÞ=QÞ we have AðE; wÞs ¼ AðE; ws Þ:
4.4. Towards Stark’s version From our assumption on the dimensions of the K-groups in the previous subsection we deduced the existence of a Q½G isomorphism f from MQ to QK2 ðEM Þ; which implies there exists a set M of n ‘Minkowski symbols’ each of whose G conjugates generate a xG dimensional Q vector space, in which G acts by the regular representation. (These symbols collectively play a role analogous to that of the Minkowski unit in the unit group.) Conversely, a set M of n symbols in distinct G orbits will let us define a Q½G isomorphism fM : In this subsection we deduce a more explicit formula for RðE; wÞ; by considering an explicit isomorphism fM : We must first specify the Q½G embedding of M inside MR : Consider first case 1. We choose representatives for the G orbits in S by taking F1 as in Section 4.1,
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F2 ; y; Fn=2 any representatives of the G orbits in SK ; along with their complex conjugates. Let m in M generate a normal basis of M as an F vector space, and f fi g any basis of F as a K vector space. A typical element of M is then written n=2 X X sAG
ci ðsÞfi ms
i¼1
with all the ci ðsÞ in K; which we identify with n=2 X X sAG
ci ðsÞsFi þ ci ðsÞsFi
i¼1
in MR : To get a basis over Q we let fi7
pffiffiffiffi ð17 DÞ fi ¼ 2
for i ¼ 1; y; n=2: Case 2 is more complicated as F is not a K vector space. We write n ¼ r þ 2s with r and s the number of real embeddings, resp. pairs of complex conjugate embeddings of F : We choose representatives F1 as above, F2 ; y; Fr in SK so that Fi jF is a real embedding for ipr: Thus Fi is equal g 1 i Fi for some gi in G: For roips; Fi jF is a complex embedding, so if we fix once and for all a g in G\N; we can actually define Fiþs ¼ gFi in this case; then g 1 Fiþs ¼ Fi : Let m in M generate a normal basis of M over F : Let f fi g a basis of F as a Q vector space. Thus a typical element of M can be written n X X sAG
ci ðsÞfi ms
i¼1
with the ci ðsÞ in Q: For ipr we identify the element fi ms in M with pffiffiffiffi pffiffiffiffi ð1 þ DÞsFi þ ð1 DÞsFi : When roips; we identify the element fi ms in M with pffiffiffiffi pffiffiffiffi sFi þ sFi þ DsFiþs DsFiþs ; and fiþs ms with pffiffiffiffi pffiffiffiffi DsFi DsFi þ sFiþs þ sFiþs : We then extend Q-linearly to get MQ inside MR : The determinant of ðl3fM ÞV in HomG ðV ; MC Þ is equal to the determinant of 1#l3fM in ðV #MC ÞG : We’re now ready to compute this determinant.
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Consider first case 1. We arbitrarily label the n symbols as xi;þ and xi; for i ¼ 1; y; n=2: We take the isomorphism fM ð fi7 ms Þ ¼ xsi;7 ;
i ¼ 1; y; n=2; sAG:
For tAG let regM ðtÞ denote the n n matrix of 2 2 blocks " # regðxti;þ ÞFj regðxti;þ ÞFj regðxti; ÞFj
regðxti; ÞFj
for i; j ¼ 1; y; n=2: Then Proposition 11. In case 1 we have ! X pffiffiffiffi dimðV Þn=2 det rðtÞ#regM ðtÞ ; RðE; w; fM Þ ¼ ð DÞ
ð10Þ
tAG
where as usual r is the representation of G with character w: Proof. A typical vector in ðV #MC ÞG looks like n=2 X X sAG
rðsÞvi #sFi þ rðsÞvi 0 #sFi
i¼1
with arbitrary vectors vi ; vi 0 in V : This space inherits a natural inner product from the one /; S on V ; namely the inner product of a typical vector as above with another, formed of vectors wi ; wi 0 is just n=2 X
/vi ; wi S þ /vi 0 ; wi 0 S:
i¼1
We let ep for p ¼ 1; y; dimðV Þ an orthonormal basis of V : We choose a basis for ðV #MC ÞG of the form X rðsÞep #ð fi7 ms Þ; v7 p;i ¼ sAG
for p ¼ 1; y; dimðV Þ; and i ¼ 1; y; n=2: We are identifying fi7 ms with its image in MR as in Section 4.1. This lets us compute X rðsÞep #lðxsi;7 Þ 1#l3fM ðv7 p;i Þ ¼ sAG
¼
X
sAG
rðsÞep #
X FAS
regðxi;7 ÞF sF:
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Write each F as t 1 Fk or t 1 Fk and change variables s/st to get ( ) X X t t regðxi;7 ÞFk rðtÞep #sFk þ regðxi;7 ÞFk rðtÞep #sFk : sAG
tAG
To get matrix coefficients we compute an inner product * pffiffiffiffi + X ð17 DÞ t 7 7 eq /1#l3fM ðvp;i Þ; vq; j S ¼ regðxi;7 ÞFj rðtÞep ; 2 tAG * pffiffiffiffi + X ð17 DÞ t eq ; þ regðxi;7 ÞFj rðtÞep ; 2 tAG where the choices of 7 on different sides of the inner product are of course independent. Taking all four possible choices of the 7 gives us a 2 2 block: pffiffiffi # " #" pffiffiffi 1 D 1þ D regðxti;þ ÞFj regðxti;þ ÞFj X 2 2 /rðtÞep ; eq S pffiffiffi pffiffiffi : 1þ D 1 D regðxti; ÞFj regðxti; ÞFj tAG 2
2
The determinant of the matrix with these (doubly indexed) coefficients is our regulator RðE; w; fM Þ: & Case 2 seems, at first, simpler. We have symbols xi for i ¼ 1; y; n in distinct G orbits. We take the isomorphism fM ð fi ms Þ ¼ xsi ;
i ¼ 1; y; n; sAG:
Now a typical vector in ðV #MC ÞG looks like n X X sAG
rðsÞvi #sFi
i¼1
with arbitrary vectors vi in V : The inner product of a typical vector as above with another formed of vectors wi is just n X
/vi ; wi S:
i¼1
We let ep for p ¼ 1; y; dimðV Þ an orthonormal basis of V : We choose a basis for ðV #MC ÞG of the form X rðsÞep #ð fi ms Þ; vp;i ¼ sAG
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for p ¼ 1; y; dimðV Þ; and i ¼ 1; y; n: This lets us compute 1#l3fM ðvp;i Þ ¼
X
rðsÞep #lðxsi Þ
sAG
¼
X
rðsÞep #
¼
sAG
regðxi ÞF sF
FAS
sAG
( X X
X
)
regðxti ÞFk rðtÞep
#sFk
tAG
after writing each F as t 1 Fk and changing variables s/st just as before. In order to compute matrix coefficients as inner products, we must rewrite the typical basis vector vq; j with jpr as vq; j ¼
X
rðsÞeq #ðð1 þ
pffiffiffiffi pffiffiffiffi DÞsFj þ ð1 DÞsFj Þ
rðsÞeq #ðð1 þ
pffiffiffiffi pffiffiffiffi DÞsFj þ ð1 DÞsg 1 j Fj Þ
sAG
¼
X
sAG
¼
X
rðsÞðð1 þ
pffiffiffiffi pffiffiffiffi DÞeq þ ð1 DÞrðgj Þeq Þ#sFj
sAG
after using the relation Fj ¼ g 1 j Fj of 4.1 above, and a change of variables. We then see that for jpr /1#l3fM ðvp;i Þ; vq; j S X pffiffiffiffi pffiffiffiffi regðxti ÞFj /rðtÞep ; ð1 þ DÞeq þ ð1 DÞrðgj Þeq S ¼ tAG
¼ ð1
pffiffiffiffi X DÞ regðxti ÞFj /rðtÞep ; eq S tAG
pffiffiffiffi X þ ð1 þ DÞ regðxti ÞFj /rðtÞep ; rðgj Þeq S: tAG
In the second sum above we use that r acts by isometries /rðtÞep ; rðgj Þeq S ¼ /rðg 1 j tÞep ; eq S and change the variables t/gj t: Then gt
regðxi j ÞFj ¼ regðxti Þg 1 Fj ¼ regðxti ÞFj j
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so finally /1#l3fM ðvp;i Þ; vq; j S X pffiffiffiffi pffiffiffiffi fð1 DÞregðxti ÞFj þ ð1 þ DÞregðxti ÞFj g/rðtÞep ; eq S ¼ tAG
for jpr: Similarly if rojpr þ s; we rewrite vq; j as X pffiffiffiffi rðsÞðeq Drðg 1 Þeq Þ#sFj vq; j ¼ sAG
pffiffiffiffi þ rðsÞð Deq þ rðgÞeq Þ#sFjþs Þ
using the relation between Fj and Fjþs above and a change of variables. Using the same change of variables as above, we see that X /1#l3fM ðvp;i Þ; vq; j S ¼ /rðtÞep ; eq S tAG
ðregðxti ÞFj þ regðxti ÞFj
pffiffiffiffi pffiffiffiffi Dregðxti ÞFjs þ Dregðxti ÞFjþs Þ:
5. Rational characters For the regulator determinant we have RðE; w1 "w2 Þ ¼ RðE; w1 ÞRðE; w2 Þ: Induction properties, as well, follow from [17, p. 29]: If w is the character of a representation of GalðM=F 0 Þ; then RðEF 0 ; wÞ ¼ RðEF ; IndðwÞÞ: For example, with F 0 ¼ M we get RðEM Þ ¼ RðEF ; IndG e ð1ÞÞ ¼
Y
RðEF ; wÞdimðwÞ :
wAGˆ
So we have factored the regulator determinant into pieces. By the usual properties of L-functions, the same holds for the cðE; wÞ and thus for the ratios AðE; wÞ: Remark. If we take the trivial representation 1 of G; then comparing (6) and (10) we see we have recovered the map i : K2 ðEM Þ-K2 ðEF Þ; and the determinant RðEF Þ is one piece of the determinant RðEM Þ: As in Section 3, if E is not of type (S), the regulator map is zero on symbols with torsion divisorial support. The other terms in the product are not, however, a priori zero on symbols with torsion divisorial
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support. For example, take F ¼ Kð jðEÞÞ the Hilbert class field, and M ¼ F ðEQ Þ; where pOK ¼ QQ is a split prime, and take r an odd character of GalðM=F Þ ¼ F p: This is a simple observation, but, as mentioned at the beginning of this section, it is the motivation for looking at this analog of Stark’s conjecture. Theorem 12. Suppose E is an elliptic curve defined over F ; and M is a Galois extension, with w a character of a representation of GalðM=F Þ taking rational values. Then there exists integers m; ni and intermediate fields Fi such that Lðs; EF #wÞm ¼
Y
Lðs; EFi Þni :
i
If we assume the dimension conjecture and the L-value conjecture of Section 1, we get that AðE; wÞm ¼
Y
AðEFi Þni AQ:
i
Proof. We will treat the case when F does not contain K; the other case is easier. Let m be the exponent of G ¼ GalðM=F Þ: By standard facts about representations (e.g. [14, p. 103]) there are subgroups Ci of G and integers ni such that mw¼
X
ni IndG Ci ð1Þ;
m ResN ðwÞ ¼
i
X
ni ResN IndG Ci ð1Þ:
i
So Lðs; E#mwÞ ¼
Y
ni Lðs; c#ResN IndG Ci ð1ÞÞ :
i
We want to use the Induction-Restriction theorem [14, p. 58] on each term, so we need for each i a decomposition of G into double cosets G¼
[
NgCi :
g
Fix an element dAG; deN; since ½G : N ¼ 2; there are at most two double cosets NeCi and NdCi : There are two cases: 1. Ci 5N; Then there is only a single double coset NeCi ¼ NdCi : Let C˜ i ¼ Ci -N; Fi the fixed field of Ci and F˜i the fixed field of C˜ i : The Induction-Restriction theorem says N ResN IndG Ci ð1Þ ¼ IndC˜ i ð1Þ:
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So N Lðs; c#ResN IndG Ci ð1ÞÞ ¼ Lðs; c#IndC˜ i ð1ÞÞ ˜
¼ Lðs; c3NormFHi Þ ¼ Lðs; EFi Þ: 2. Ci oN: The two double cosets are distinct. Let Di ¼ dCi d 1 ; also a subgroup of N: Let Fi the fixed field of Ci ; and Li the fixed field of Di : The InductionRestriction theorem says N N ResN IndG Ci ð1Þ ¼ IndCi ð1Þ"IndDi ð1Þ:
So N N Lðs; c#ResN IndG Ci ð1ÞÞ ¼ Lðs; c#IndC˜ i ð1ÞÞLðs; c#IndD˜ i ð1ÞÞ
¼ Lðs; c3NormFHi ÞLðs; c3NormLHi Þ: Since E is defined over the subfield F of H; and d generates GalðH=F Þ; we get from [6, Theorem 10.1.3] that dc ¼ c: This gives ¼ Lðs; c3NormFHi ÞLðs; c3NormFHi Þ ¼ Lðs; EFi Þ: Assuming the L-value conjecture of Section 1, there exist rational numbers AðEFi Þ ¼
RðEFi Þ : cðEFi Þ
Then the induction and direct sum properties imply the following weak form of the elliptic Stark conjecture for rational characters: Y AðE; wÞm ¼ AðEFi Þni AQ: & i
6. Abelian over complex quadratic This section is devoted to the proof of the following Theorem 13. Suppose the field M has abelian Galois group G over the complex quadratic field K; and that the curve E is type (S). Then the elliptic Stark conjecture is true.
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To minimize notation, we will consider the case where E is defined over F ¼ Kð jðEÞÞ: This makes F the Hilbert class field of K; and n ¼ ½F : Q ¼ 2h where h is the class number of OK : For abelian Galois groups we may as well assume the character w satisfies dimðwÞ ¼ 1: By pulling the representation w back to a larger Galois group, we can assume M is the ray class field modulo G; where G is principal and is divisible by the conductors of w and the Hecke character c: Furthermore, since E is type (S), c ¼ f3NormF =K for some Hecke character f of K: We begin by considering partial L-functions, which have only the Euler factors prime to G: By abuse of notation the dependence on G is suppressed. Then we have Lðs; c#wÞ ¼ Lðs; f3NormF =K #wÞ ¼ Lðs; f#IndGG ðwÞÞ ¼
h Y
Lðs; f#wi Þ
i¼1
¼
h X Y
wi ðgÞLðs; f; gÞ;
i¼1 gAG
where IndGG ðwÞ decomposes as "wi and we have the partial L-functions associated to an Artin symbol g ¼ ½ ; M=K in G X fðCÞNðCÞ s : Lðs; f; gÞ ¼ CCOK ½C; M=K ¼ g This gives the hth derivative at s ¼ 0 as LðhÞ ð0; c#wÞ ¼
h X Y
wi ðgÞL0 ð0; f; gÞ:
ð11Þ
i¼1 gAG
In the next subsection we will give a generalization of what is usually called the Frobenius determinant relation (actually due to Dedekind.) 6.1. Generalized Dedekind determinant Suppose G is a finite abelian group, GoG; and w : G-C is a character. Write p ¼ IndGG ðwÞ ¼ "wi for the induced representation. Fix once and for all a set S of coset representatives for G\G: Let W be the usual space the induced representation acts in: W ¼ fF : G-CjF ðsxÞ ¼ wðsÞF ðxÞ; 8sAG; xAGg;
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where G acts by multiplication: pðgÞF ðxÞ ¼ F ðxgÞ: Let f : G-C be any function, and define the operator on W pð f Þ ¼
X
f ðgÞpðgÞ:
gAG
The characters wi form a basis of W ; and so do the characteristic functions of cosets FGg ; where FGg ðs0 g0 Þ :¼
wðs0 Þ
if g ¼ g0 ;
0
otherwise:
The Dedekind determinant relation computes the determinant of pð f Þ with respect to these two canonical bases of W : Lemma 14. det pð f Þ ¼
½G:G Y
X
i¼1
gAG
"
¼ det
X
f ðgÞwi ðgÞ # 0 1
wðtÞf ðtg g Þ
tAG
: g;g0 AS
Proof. The functions wi are eigenvectors of each pðgÞ; with eigenvalue wi ðgÞ; thus also eigenvectors of pð f Þ with eigenvalue X
f ðgÞwi ðgÞ;
gAG
and so the first formula for the determinant is clear. Relative to our set of coset representatives S; define a function (a factor set) S S-G; ðg; g0 Þ/gðg; g0 Þ; so that gðg; g0 Þg ¼ g0 g00 ;
where Gg ¼ Gg0 Gg00
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in the quotient G\G: An explicit computation shows that X X wðsgðg; g0 ÞÞf ðsg00 ÞFGg0 ; pð f ÞFGg ¼ g00 AS sAG
with g; g0 ; g00 related as above. A change of variables give the matrix coefficients of pð f Þ as in the lemma. & Remark. The case when G is the trivial subgroup, and p is the regular representation of G is the usual Frobenius determinant relation. Taking L0 ð0; f; gÞ for the function f ðgÞ in the Dedekind determinant, and using (11) realizes the L-function value as a determinant: " # X ðhÞ 0 0 1 L ð0; c#wÞ ¼ det wðtÞL ð0; f; tg g Þ : ð12Þ tAG
g;g0
This application of the Dedekind determinant has long been a key ingredient for special values of L-functions. For example, results on the conjecture of Birch and Swinnerton were obtained in [3,4]. 6.2. Partial L-functions and Kronecker series In this subsection we present, for completeness, a calculation of the derivative at s ¼ 0 of a partial L-function as the value at s ¼ 2 of a Kronecker series. Since F1 ð jðEÞÞ ¼ jðOK Þ; we have, for the lattice L corresponding to EF1 L ¼ OOK for some OAC : Further, for A an ideal of OK ; we have E ½A;M=K is defined over F ; corresponding to a lattice LA ¼ hðAÞOA 1 for some hðAÞ: The isogeny between E and E ½A;M=K is just multiplication by hðAÞ: Our hypothesis that E is type (S) implies that E is isogenous over F to all its Galois conjugates, which gives that hðAÞAF for all A: By composing isogenies one sees that h is a crossed homomorphism: hðA0 AÞ ¼ hðA0 Þ½A;M=K hðAÞ: For more details on curves of type (S) (used throughout this section) see [4]. Choose a set of representative AAA for the ideal class group of K; all prime to our fixed G: Choose also a fixed set of representatives BAB so the Artin symbols ½B; M=K give every element of G ¼ GalðM=F Þ: The ideals in B are principal as F is the Hilbert class field, and we have that B ¼ ðfðBÞÞ: A given element gAG is then of
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the form ½AB; M=K for some A and some B: We get all ideals in this class by summing over a in A 1 G since then fðBÞ þ a fðBÞ mod G and ½AðfðBÞ þ aÞ; M=K ¼ ½AðfðBÞÞ; M=K: So X
Lðs; f; ½AB; M=KÞ ¼
aAA 1 G
fðAÞ fððfðBÞ þ aÞÞ : NðAÞs NðfðBÞ þ aÞs
Now, in general, we have fððlÞÞ ¼ ffin ðlÞl; where ffin ðlÞAK only depends on l mod G: Since B ¼ ðfðBÞÞ we get that ffin ðfðBÞÞ ¼ 1; and fððfðBÞ þ aÞÞ ¼ ffin ðfðBÞ þ aÞfðBÞ þ a ¼ fðBÞ þ a: Thus Lðs; f; ½AB; M=KÞ ¼
fðAÞ NðAÞs
X
fðBÞ þ a
aAA 1 G
jfðBÞ þ aj2s
:
ð13Þ
Let nAOK so that ðn=OÞ ¼ G 1 ; i.e., n is a G torsion point on C=L: We see that hðAÞn
X
fðBÞ þ a
aAA 1 G
jfðBÞ þ aj2s
2s
jhðAÞnj
¼ K1 ðfðBÞhðAÞn; 0; s; LA Þ
ð14Þ
since aAA 1 G exactly when o ¼ hðAÞnaAhðAÞOA 1 ¼ LA : We combine Eqs. (13) and (14), multiply by GðsÞ and use the functional equation (1) to see that GðsÞLðs; f; ½AB; M=KÞ ¼
fðAÞ jhðAÞnj2s AðLA Þ2 2s Gð2 sÞ NðAÞs hðAÞn K1 ð0; fðBÞhðAÞn; 2 s; LA Þ:
Thus the partial L-function derivative at s ¼ 0 is computed by the Kronecker series value at s ¼ 2; which we are denoting K2;1 :
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Lemma 15. L0 ð0; f; ½AB; M=KÞ ¼
fðAÞ hðAÞn
A2 ðLA ÞK2;1 ðfðBÞhðAÞn; LA Þ:
6.3 For convenience we now number our representatives Ai AA for the ideal class group of K; and choose them so that Ai ¼ Ai if jðAi Þ is real (1pipr), and Aiþs ¼ Ai if jðAi Þ is complex (roips). Consider the matrix on the right-hand side of Eq. (12). In terms of our representatives, we have " # X 0
1 wð½B; M=KÞL ð0; f; ½Ai Aj B; M=KÞ : det B
i; j
To use the relation between these partial L-function derivatives and Kronecker series above, we need to change A 1 i to Ai : This induces a permutation of the ideal classes as well as an extra term from the representatives of the principal ideals in each row. The permutation of the rows only changes the determinant by 71; but the extra principal ideal needs to be absorbed from each row by a change of variables in the sum, which alters the determinant. This gives the L-function value as " # X 0 aðwÞdet wð½B; M=KÞL ð0; f; ½Ai Aj B; M=KÞ ; B
i; j
where aðwÞ is the product of all terms introduced by these change of variables. Note that, as the conjecture will require, aðwÞAQðwÞ;
aðws Þ ¼ aðwÞs
8sAGalðQðwÞ=QÞ:
We use Lemma 15 (with Ai Aj instead of A) on each entry of the matrix. Factor fðAi Þ from row i for each i; and similarly fðAj Þ from each column. Note that Q 2 Q 2 i fðAi Þ is in K ; since i Ai is principal. In fact in the ‘other half’ of the L function (coming from f instead of f) we will see the complex conjugate of these terms. So modulo Q we can ignore them. We have so far shown that LðhÞ ð0; c#wÞEQðwÞ " # X A2 ðLAi Aj Þ det wð½B; M=KÞ K2;1 ðfðBÞhðAi Aj Þn; LAi Aj Þ : hðAi Aj Þn B i; j We can now apply the distribution relation of Proposition 2 to the isogeny hðAi Þ½Aj ;F =K : E ½Aj ;F =K -E ½Ai Aj ;F =K
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of degree NðAi Þ: We need to use the fact that in general AðLA Þ ¼ AðhðAÞOA 1 Þ ¼
pffiffiffiffiffiffiffi jhðAÞOj2 jDj ; 2pNðAÞ
where D is the discriminant of K: So A2 ðLAi Aj Þ ¼
jhðAi Þ½Aj ;F =K j4 NðAi Þ2
A2 ðLAj Þ:
In summary, we have got LðhÞ ð0; c#wÞEQðwÞ 2 6 6 6 det6 6 4
X BAB tAker hðAi Þ
3 7 7 A2 ðLAj Þ 7 wð½B; M=KÞ K2;1 ðfðBÞhðAj Þn t; LAj Þ7 : 7 hðAj Þn 5 i; j
By class field theory, the elements Aj of the class group correspond via our fixed embedding F1 to the other embeddings Fj which have the same restriction to K: That is C=LAj ¼ E ½Aj ;M=K ¼ EFj : We use the homothety property to factor a scalar hðAj ÞO out of the K2;1 in each column, to convert LAj to A 1 j : If we take our representatives to be integral ideals not divisible by any rational integer but 1, it is easy to see that A 1 j has a lattice basis of the form ½1; tj as required. The divisor associated to the sum over the torsion points in the kernel of hðAi Þ comes from a symbol xi in QK2 ðEM Þ by the theorem of Bloch [1]. Multiplication by fðBÞ on the torsion points of one of these curves acts by Galois automorphism ½B; M=K: Recall these fix F ; and since our isogenies hðAi Þ are defined over F we can re-write the sum over BAB as a sum over tAG: Thus we see " # X wðtÞregðxti ÞFj : LðhÞ ð0; c#wÞEQðwÞ det O=n tAG
i; j
% we get the same formula but with the Applying this construction to c instead of c; conjugate embeddings Fj instead of Fj : However, we have so far only constructed h symbols in K2 which we can relate to the L-value. To get 2h symbols, we take advantage of the fact that K2 ðEM Þ is an OK module. More specifically, the point is that we can vary the ideal G; changing the matrix of partial L functions and thus also
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the matrix of symbols, by a constant in K: For any ideal P; one sees that for any g in G; the partial L-functions satisfy LG ðs; f; gÞ ¼ LGP ðs; f; gÞ þ fðPÞNðPÞ s LG ðs; f; g ½P; M=K 1 Þ; where we now, of course, need to keep track of the ideal in the notation for the partial L-function. Thus if P is principal X
wðtÞLGP 0 ð0; f; gtÞ
tAG
¼ ð1 fðPÞwð½P; M=KÞÞ
X
wðtÞLG 0 ð0; f; gtÞ:
tAG
Choose two principal prime ideals Pþ and P ; and let G7 ¼ GP7 : Let p7 ¼ 1 fðP7 Þwð½P7 ; M=KÞ: When we change G to Gþ or G ; the matrix on the right-hand side of (12) changes by the scalar pþ or p : Following the calculation of the previous section to the end, we see the same is true for the matrix of regulators of symbols supported on the torsion. That is, let " R¼
X
# wðtÞregðxti ÞFj
tAG
i; j
the matrix corresponding to symbols on the G torsion, then " 7
p R¼
X
# wðtÞregðxti;7 ÞFj
tAG
; i; j
where the symbols xi;7 come from the G7 torsion. We have shown above that detðRÞdetðRÞEQðwÞ LðhÞ ð0; c#wÞLðhÞ ð0; c#wÞ ¼ Lð2hÞ ð0; E#wÞ: By Proposition 11, the elliptic Stark conjecture will follow from the linear algebra pffiffiffiffi Lemma 16. For an h h matrix R; and scalars pþ ; p in Qð DÞ; "
pþ R det
p R
pþ R p R
#
¼ k detðRÞdetðRÞ;
kA
Q if h is even; pffiffiffiffi Q D if h is odd:
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Proof. Write the matrix as pþ I h p Ih
pþ Ih p Ih
!
R 0
0 : R
It follows from the Laplace expansion theorem that the determinant of the first pffiffiffiffi matrix is ðpþ p p pþ Þh ; and pþ p p pþ is in Q D: & 6.4. The simple zero In the Stark conjectures, the case when the L-function has a simple zero gets special attention. We observe here that if ½F : QdimðV Þ ¼ 1; then F ¼ Q: Since E is defined over F ; we must necessarily have the class number hðKÞ ¼ 1: Since dimðV Þ ¼ 1; w factors through an abelian extension GalðM=QÞ; and M contains K by hypothesis. By the remark at the beginning of Section 3, EL is of type (S) for any intermediate field L with QDLDM: Assuming the Dimension conjecture, the results in the previous subsection give the elliptic Stark conjecture in this case.
7. Curves not type (S) The positive results so far have all been for curves of type (S), even though we introduced the elliptic Stark conjecture to study the general case. In this section we remedy this defect with the following. Theorem 17. If F is abelian over K; and E is any elliptic curve over F with complex multiplication by OK ; then there is a Galois extension M of F ; and a character w of GalðM=F Þ such that the elliptic Stark conjecture holds for Lðs; E#wÞ: Remark. This theorem does not assume the ‘Dimension conjecture’ of Section 1. Proof. By Robert [10, Corollaire 2] we know there exists an elliptic curve E 0 defined over F which is of type (S). By Theorem 9.1.3 of [6], we may assume that E 0 and E have the same j invariant. Thus E 0 is a model of E and we can write Weierstrass equations E: y2 ¼ 4x3 g2 x g3 ; E 0 : y2 ¼ 4x3 d 2 g2 x d 3 g3 pffiffiffi with d in F : The curves E and E 0 become isomorphic over M ¼ F ð d Þ via f : E-E 0 ; ðx; yÞ/ðx0 ¼ dx; y0 ¼ d 3=2 yÞ:
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We get a map on functions fields and a map on K-groups f : QK2 ðEF 0 Þ-QK2 ðEF Þ; as in Section 2. The inclusions i : F ðEÞ-MðEÞ and i0 : F ðE 0 Þ-MðEÞ also give maps i : QK2 ðEF Þ-QK2 ðEM Þ; i0 : QK2 ðEF 0 Þ-QK2 ðEM Þ: Note that the triangle formed by these three maps does not commute. In fact MðEÞ ¼ MðE 0 Þ is a Galois extension of F ðxÞ ¼ F ðx0 Þ; with Galois group the Klein four group. The quadratic subfields are MðxÞ; F ðEÞ; and F ðE 0 Þ: The subgroups of order two which fix these fields are generated by ½ 1; t; and t0 ; where 8 8 0 -x0 ; -x; > >
f t ¼ f 3½ 1 ¼ ½ 1 f :
Thus for symbols x in QK2 ðEF 0 Þ; ði f xÞt ¼ i f x;
ði0 xÞt ¼ ½ 1 i0 x:
Since E 0 is type (S) there exist n ¼ ½F : Q symbols xi such that the ‘L-value conjecture’ of Section 1 is true. In the notation of Section 4, this says that AðEF 0 Þ ¼ RðEF 0 Þ=cðEF 0 Þ is rational. Let w be the nontrivial character of GalðM=F Þ: Since Lðs; EF ÞLðs; EF 0 Þ ¼ Lðs; EM Þ ¼ Lðs; EF ÞLðs; EF #wÞ we see that cðEF 0 Þ ¼ cðEF ; wÞ: It remains to relate RðEF 0 Þ to RðEF ; wÞ: For notational convenience we suppress the inclusions i and i0 : We see that for symbols xi as above, f xi þ xi t f xi xi ; ¼ 2 2 so the n symbols ðf xi þ xi Þ=2 satisfy the requirement of Section 4 to be a set M of ‘Minkowski symbols’. Fix any embedding Fj of F into C: In the formula for RðEF ; w; fM Þ in Proposition 11, the i; j entry in the sum over the Galois group
ARTICLE IN PRESS J. Stopple / Journal of Number Theory 103 (2003) 163–196
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simplifies as f xi þ xi f xi þ xi t
reg ¼ regðxi ÞFj : reg 2 2 Fj Fj Thus RðEF 0 Þ ¼ RðE; wÞ and so AðEF ; wÞ ¼ RðEF ; wÞ=cðEF ; wÞ ¼ RðEF 0 Þ=cðEF 0 Þ is rational.
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Acknowledgments The author thanks Fernando Rodriguez Villegas and Dinakar Ramakrishnan for helpful conversations, and Bill Jacob for teaching the K-theory.
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