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On the Orthogonal of Cyclotomic Units in Positive Characteristic
Journal of Number Theory 79, 258283 (1999) Article ID jnth.1999.2428, available online at http:www.idealibrary.com on
On the Orthogonal of Cyclotomic Units in Positive Characteristic Bruno Angles Laboratoire de Mathematiques Emile Picard, C.N.R.S. U.M. 5580, U.F.R. M.I.G., Universite Toulouse III, 118 Route de Narbonne, 31062 Toulouse Cedex 4, France E-mail: anglespicard.ups-tlse.fr Communicated by D. Goss Received October 20, 1998
INTRODUCTION Let P be a monic irreducible polynomial in F q[T ] of degree d. Let FF q(T ) be a subfield of the maximal real subfield of the P th cyclotomic function field. In this paper, we study the orthogonal of the cyclotomic units of F for the local symbol at P attached to the Carlitz module (see [SC, AN]). Let l=[F : F q(T )] and let r l(P) be the number of BernoulliCarlitz numbers B(k) such that k#0((q d &1)l), 0
THE ORTHOGONAL OF CYCLOTOMIC UNITS
259
Let v P be the normalized P-adic valuation on Q P , then v P extends to a valuation on 0 which is also denoted by v P . Note that v P (P)=1. We view Q as contained in Q P . Let FQ P be a finite extension, F/0. We set: O F , the valuation ring of F; U F , the group of units of F ; PF , the maximal ideal of O F ; n for n 1, U (n) F =1+P F ;
e F is the ramification index of FQ P and f F is the inertia degree of FQ P ; let ? # O F , we say that ? is a prime of F if v P (?)=1e F . Let A be an unitary commutative ring; we denote the set of invertible elements of A by A*. Let F be a field and let LF be a finite extension. Then N LF is the norm map from L* to F* and Tr LF is the trace map from L to F. Let kQ be a finite extension. We set: O k , the integral closure of Z in k ; E k =O * k ; h k , the class number of the ideal class group of O k .
1. PRELIMINARIES In this section, we recall the definitions of the BernoulliCarlitz numbers and of the symbol attached to the P th cyclotomic function field. Let Z P Ä EndG a , b [ [b] C , be the local Carlitz module, i.e., it is a ring homomorphism such that [T ] C =TX+X q and for all b # Z P , [b] C = bX+higher terms (see [AN, Sect. 4], for more details). Let 4 P =[: # 0, [P] C (:)=0]. Then 4 P is a Z P -module via the Carlitz module and we have an isomorphism of Z P -modules, 4 P & Z P PZ P . Let * P be a fixed element in 4 P , * P {0. Set K=Q P (* P ). Recall that KQ P is an abelian extension, KQ P is totally ramified, * P is a prime of K and we have the isomorphism Gal(KQ P ) & (Z P PZ P )*,
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where _ maps to the class of A # Z P such that _(* P )=[A] C (* P ). We set G=Gal(KQ P ). For A # Z P "PZ P , we denote the element _ # G such that _(* P )=[A] C (* P ) by _ A . The limit lim n(1P n )[P n ] C exists in Q P[[X ]] [AN, Proposition 1.5] i and is denoted by Log C . Set L 0 =1 and for i 1, L i =(T q &T ) L i&1 . Then, by [AN, Proposition 4.1], we have (&1) i q X i. Li i 0
Log C (X )= :
Let Exp C be the unique element in Q P[[X ]] such that Log C b Exp C =Exp C b Log C =X. i
Set D 0 =1 and for i 1, D i =(T q &T ) D qi&1 . We have 1 qi X . Di i 0
Exp C (X )= :
Now, we recall the definition of the BernoulliCarlitz numbers B(k). Let k 0 be an integer and write k=a 0 +a 1 q+ } } } +a N q N where 0 a i ai q&1, for i=0, ..., N. Set 1 k => N i=0 D i . The BernoulliCarlitz numbers are defined by X B(k) k X . =: Exp C (X ) k 0 1 k We have B(0)=1 and if k0(q&1), B(k)=0. Furthermore, for k 1, we have 1k B(k+1&q i ). D 1 i k+1&q i i 1
B(k)=& :
We refer the reader to [Ge, GO] for more informations on Bernoulli Carlitz numbers. d Let L(X )=PX+X q be the basic Lubin-Tate polynomial, then L gives rise to a ring homomorphism (see [AN, Sect. 1]): Z P Ä EndG a , b [ [b] L . d Note that [P] L =PX+W q and for all ` # F P , [`] L =`X. We set Log L = n n lim n(1P )[P ] L , such a limit exists in Q P[[X ]]. Let Exp L be the unique element in Q P[[X ]] such that Exp L b Log L =Log L b Exp L =X.
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1.1. Lemma. (i) (ii)
Set f P =Exp C b Log L and g P =Exp L b Log C .
f P and g P are elements of Z P[[X ]]. For all b in Z P , we have f P b [b] L =[b] C b f P , g P b [b] C =[b] L b g P .
Proof.
This comes from [AN, Proposition 1.1 and Proposition 1.6].
K
We set * L = g P (* P ). Then * L {0 and [P] L (* L )=0.
(1.2)
Set 4 L =[`* L , ` # F P ]. Note that K=Q P (* L ). 1.3. Lemma. Proof. have
i
d
i q q * L # d&1 i=0 ((&1) L i ) * P (P K ). i
Write g P (X )= i 0 A i X q , with A 0 =1 and A i # Z P for all i. We Log L b g P =Log C .
But by [AN, Proposition 2.2], we have d
Log L #X
mod X q Q P[[X ]].
g P #Log C
mod X q Q P[[X ]].
Thus d
Therefore, for 0 i d&1, we get Ai =
(&1) i . Li
K
Let z # PK and let t 1 , t 2 in 0 such that [P] C (t 1 )=z, [P] L (t 2 )=z. Set E 1 =K(t 1 ) and E 2 =K(t 2 ). Then E 1 and E 2 are abelian extensions of K. Let ( } , E 1 K ) and ( } , E 2 K ) be the local Artin maps associated to these abelian extensions. Let : # K*; we set ( z, :) C =(:, E 1 K )(t 1 )&t 1 # 4 P ,
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and ( z, :) L =(:, E 2 K )(t 2 )&t 2 # 4 L . In this paper, we are interested by the symbol ( } , } ) C which was introduced by Schultheis in [SC]. Let's give the basic properties of this symbol. 1.4. Proposition.
(i)
Let z 1 , z 2 in PK , a 1 , a 2 in Z P and : # K*,
( [a 1 ] C (z 1 )+[a 2 ] C (z 2 ), :) C =[a 1 ] C (( z 1 , :) C )+[a 2 ] C (( z 2 , :) C ). (ii)
Let z in PK and : 1 , : 2 in K*, ( z, : 1 : 2 ) C =( z, : 1 ) C +( z, : 2 ) C .
ab (iii) Let z # PK , : # k* and _ # Gal(Q ab P Q P ), where Q P /0 is the maximal abelian extension of Q P . Then
_(( z, :) C )=( _(z), _(:)) C . (iv)
Let z # PK , z{0. We have ( z, z) C =0.
(v)
Let z # PK and : # K*. Then ( z, :) C = f P (( g P (z), :) L ).
Proof.
See [AN, Sect. 1]. K
1.5. Lemma. d z # P qK . Proof.
Let z # PK . Then (z, :) C =0 for all : in U K if and only if
Let t # 0 such that [P] C (t)=z. Write E=K(t). We have ( z, :) C =0 : # N EK (E*).
Therefore (\: # U K , ( z, :) C =0) (U K /N EK (E*)). But U K /N EK (E*) if and only if EK is unramified. By [AN, Proposition 2.1], d EK is unramified if and only if z # P qK . K
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263
2. THE CARLITZKUMMER MORPHISMS In this section, we give the definition of the CarlitzKummer morphisms and we prove the basic properties of these morphisms. The CarlitzKummer morphisms play a central ro^le in the next sections. Recall that K=Q P (* P )=Q P (* L ). Let : # K*. Write :=* rL : 1 , where r # Z and : 1 # U K . Since KQ P is totally ramified, there exists g 1(X ) # Z P[[X ]] such that : 1 = g 1(* L ). Write g(X )=X rg 1(X ). Let g$(X ) be the derivative of g(X ), we set D(:)=
g$(* L ) g(* L )
d
mod P qK &2 .
Note that D(:) is independent of the choice of g 1(X ). By [LA, Chap. 9, q d &2 Sect. 2], D: K* Ä P &1 is a morphism of groups and D(U K )/ K P K q d &2 O K P K . 2.1. Theorem. Let z # P 2K and : # U K . Then (z, :) C = Proof.
_
1 Tr KQP (g P (z) D(:)) P
&
(* P ). C
By [AN, Theorem 2.9], if z # P 2K , we have (z, :) L =
_
1 Tr KQP (Log L(z) D(:)) P
&
(* L ). L
Let z # P 2K . By [AN, Proposition 2.2], we have Log L( g P (z))#g P (z)
d
mod P dK .
d
d
q &2 Note that Tr KQP (P qK )/P 2Z P . Furthermore, if : # U K , D(:) # O K P K . Therefore, if : # U K , we get
( g P (z), :) L =
_
1 Tr KQP (g P (z) D(:)) P
&
(* L ). L
Note that f P (* L )=* P . It remains to apply Lemma 1.1 and Proposition 1.4. K
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Now we define the k th CarlitzKummer morphism . k , this morphism is a little bit different from the one defined in Okada's work [OK]. Let k be an integer, 1 k q d &2. Let : # U K ; write q d &3
d
D(:)# : a j * Lj
mod P qK &2 ,
j=0
where a j # Z P for j=0, ..., q d &3 (we can do so because O K =Z P[* L ]). Then . k(:) # F P is defined by . k(:)#a k&1(P). 2.2. Proposition. Let k be an integer, 1 k q d &2. Then . k : U K Ä F P is well-defined and is a morphism of groups. Furthermore, . k is surjective. d
Proof. Note that PO K =P qK &1 . Thus . k is well-defined. Since D is a morphism of groups, . k is a morphism of groups too. Let n 1 and let ; # F P . We have d
D(1&;* nL )# &n : ; l* ln&1 L
mod P qK &2 .
l 1
Thus . k(1&;* nL )= &n; kn =0
if n divides k
otherwise.
The proposition follows. K Recall that G=Gal(KQP ). Let %: G Ä F* P be the Teichmuller character, i.e., %(_ A )#A(P). Let G be the set of morphisms /: G Ä F* =( %). P . Then G 2.3. Proposition. Let k be an integer, 1 k q d &2. Let _ # G. Then for all : # U K , we have . k(_(:))=% k(_) . k(:). Proof.
Recall that * P = f P (* L ). Therefore _(* P )= f P (_(* L )).
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But, there exists ` # F* P such that _(* L )=`* L . If we apply Lemma 1.1, we get _(L P )= f p(`* L ) = f P ([`] L (* L )) =[`] C (* P ). But _(* P )=[%(_)] C (* P ). Thus _(* L )=%(_) * L .
(2.4)
Now, let : # U K and write := g(* L ) for some g # Z P[[X ]]. We have _(:)= g(%(_) * L ). Write g _(X )= g(%(_) X ) # Z P[[X ]]. Then _(:)= g _(* L ). We have D(_(:))# #
g$_(* L ) g _(* L ) %(_) g$(%(_) * L ) _(:)
#%(_) _(D(:))
d
mod P qK &2 .
Write q d &2
D(:)# : . k(:) * k&1 L
d
mod P qK &2 ,
k=1
then q d &2
_(D(:))# : % k&1(_) . k(:) * k&1 L
d
mod P qK &2 .
k=1
The proposition follows. K 2.5. Proposition.
Let : # U K . Then
(. k(:)=0 for k=1, ..., q d &2) (: # (U K ) p U (q K Proof. If : # (U K ) p U K(q for k=1, ..., q d &2.
d &1)
d
d &1)
).
, then D(:)#0 mod P qK &2 . Thus, . k(:)=0
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Now, let : # U K such that . k(:)=0 for k=1, ..., q d &2. Note that q d &2 D(F* . Thus, we can suppose that : # U (1) P )#0 mod P K K . There, exist d (q d &1) such that ` i # F P , i=1, ..., q &2 and = # U K q d &2
:== ` (1&` i * iL ). i=1
/Ker . k . We have Note that U (k+1) K . 1(:)=&` 1 =0. Let j be an integer 2 j q d &2 and suppose that we have proved: for all l j&1, ` l =0 or p divides l. We have . j (:)=&j` j =0. Thus ` j =0 or p divides j. The proposition follows. K
3. THAKUR'S GAUSS SUMS In this section, we use Thakur's Gauss sums to obtain an explicit reciprocity law for the symbol defined in Section 1. The Thakur's Gauss sums are defined as k
G k =& : % &q (_) _(* P ) # PK , _#G
for k=0, ..., d&1. 3.1. Proposition. (ii) Proof.
(i)
j
q For all _ # G, _(* P )= d&1 j=0 % (_) G j
(G 0 } } } G d&1 ) q&1 =(&1) d P. It comes from [TH, Proposition I and Theorem II]. K
Let k be an integer, 1 k q d &1. We set d&1
' k = : G kj D k&1 . j j=0
3.2. Theorem. (ii)
(i)
' 1 =* P .
Let k be an integer, 1 k q d &1. Then ' k #f P (* kL )
d
mod P qK .
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267
Proof. The assertion (i) is a consequence of Proposition 3.1. Let's prove (ii). i Write f P (X )= i 0 B i X q with B 0 =1 and B i # Z P for all i. We have fP b Exp L =Exp C . But, by [AN, Proposition 2.2], we have d
Exp L #X
mod X q Q P[[X ]],
f P #Exp C
mod X q Q P[[X ]].
thus d
Therefore, we have Bi =
1 , Di
for i=0, ..., d&1, and we get d&1
1 qi * Di L
*P # : i=0
d
mod P qK .
Let j be an integer, 0 j d&1. Using formula (2.4), we have j
G j # & : % &q (_) _(* P ) _#G d&1
_#G d&1
#& : i=0 d&1
#& : i=0
#
qj L
* Dj
\:
1 i (_(* L )) q Di
+ 1 : % (_)(_(* )) + D \ 1 * \ : % (_)+ D j
# & : % &q (_)
i=0
&q j
qi
L
i
_#G
qi L
i
q i &q j
_#G d
mod P qK .
We get d&1
'k # : j=0
(* kL ) q Dj
# f P (* kL )
j
d
mod P qK . K
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3.3. Proposition.
Let k be an integer, 1 k q d &1. Let _ # G. Then d
_(' k )#[% k(_)] C (' k ) Proof.
mod P qK .
By Theorem 3.2 and formula (2.4), we have _(' k )# f P (_(* kL )) # f P (% k(_) * kL ) # f P ([% k(_)] L (* kL )) #[% k(_)] C ( f P (* kL )) d
#[% k(_)] C (' k )
mod P qK .
K
3.4. Proposition. Let k be an integer, 1 k q d &1. Let n be the g.c.d. of k and q d &1. Then [Q P (' k ) : Q P ]= Proof.
Let _ # G ( g
d &1)
q d &1 . n
n. We have d&1
_(' k )= : (_(G j )) k D k&1 j j=0 d&1
j
= : % q k(_) G kJ D k&1 j j=0
=' k . Therefore G (q
d &1)n
/Gal(KQ P (' k )).
Now let _ # G such that _(' k )=' k . By Proposition 3.3, we have ' k #% k(_) ' k
mod P k+1 . K
Therefore % k(_)=1. This implies _ # G ( g
d &1)
n. The proposition follows. K
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THE ORTHOGONAL OF CYCLOTOMIC UNITS
3.5. Lemma.
Let z # PK . Then there exist ` 1 , ..., ` q d &1 in F P such that q d &1
d
mod P qK .
z# : [` k ] C (' k ) k=1
Furthermore ` k is unique for k=1, ..., q d &1. Proof.
Let ` # F P . By Theorem 3.2, we have [`] C (' k )#`* kL
mod P k+1 . K
The uniqueness follows. Let z # PK . Since * L is a prime of K, we have O K =F P[[* L ]]. Therefore we can write q d &1
g P (z)# : ` k * kL
d
mod P qK ,
k=1
where ` k # F P for k=1, ..., q d &1. Thus z# f P ( g P (z)) q d &1
# fP
\:
` k * kL
k=1
+
q d &1
# : [` k ] C ( f P (* kL )) k=1 q d &1
# : [` k ] C (' k )
d
mod P qK . K
k=1 d
d
&1 [` k ] C (' k ) mod P qK , where 3.6. Theorem. Let z # P 2K . Write z# qk=2 d ` k # F P for k=2, ..., q &1. Let u # U K . Then q d &1
( z, u) C =
_:
k=2
Proof.
` k . q d &k(u)
&
(* P ). C
By Proposition 1.4 and Lemma 1.5, we get q d &1
( z, u) C = : [` k ] C (( ' k , u) C ). k=2
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Now, by Theorem 2.1 and Theorem 3.2, we have ( ' k , u) C =
1
_P Tr
KQP
d
(* kL D(u))
&
(* P ). C
d
&2 q &2 Write D(u)= qi=1 . i (u) * i&1 mod P K , and note that, by formula (1.2) L we have:
Tr KQP (* nL )#0(P 2 ) if n 1 and n{q d &1, d
Tr KQP (* qL &1 )=P. We get (' k , u) C =[. q d &k(u)] C (* P ). The theorem follows. K 3.7. Corollary.
Let u # U K . Then
(\z # P 2K , ( z, u) C =0) (u # (U K ) p U (q K Proof.
d &1)
).
By (i) of Proposition 1.4, Lemma 1.5, and Lemma 3.5, we have (\z # P 2K , ( z, u) C =0) (( ' k u) C =0 for k=2, ..., q d &1).
Now, by Theorem 3.6 (' k , u) C =0 . q d &k(u)=0. It remains to apply Proposition 2.5.
K
4. CYCLOTOMIC UNITS Let H=Q(* P )/0 be the P th cyclotomic function field. Let H +Q be the maximal subextension of HQ such that 1T splits completely in H +. The field H + is called the maximal totally real subfield of H and H + =Q( q&1 ). We fix a subextension FQ of H +Q and we set l=[F : Q], l 2. Recall that FQ is totally ramified at P and we denote the unique prime ideal of O F above P by PF . We denote the PF -adic completion of F /K. Then Gal(FQ) & Gal(FQ P ) & GG l.
(4.1)
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271
Let B be a subgroup of U K . We set B = =[z # PK , \u # B, ( z, u) C =0]. By Proposition 1.4, B = is a Z P -module via the Carlitz module. Furthermore, by Lemma 1.5, we have d
P qK /B =. Note that, if z # PK and if a # PZ P then d
[a] C (z) # P qK . Therefore, via the Carlitz module, PK B = is a F P -vector space. In this section, our aim is to calculate dim FP PK Cyc = F . Recall that the group of cyclotomic units of F is the subgroup of E F (recall E F =O*F ) generated by F * q and N H + F (_(* P )* P ) for all _ # G. We denote the group of cyclotomic units of F by Cyc F . Set d
= P =&
* qP &1 , P
and = F =N H + F (= P ). Note that = F # Cyc F and N FQ(= F )=1. Let UF be the subgroup of E F generated by F * q and _(= F ) for all _ # Gal(FQ). We have UF /Cyc F /E F . 4.2. Lemma. Proof.
= U= F =Cyc F .
= Since UF /Cyc F , we have Cyc = F /U F . Note that
= P =& ` _#G
*P . _(* P )
Now, let { # G. We have {(= P )= & ` _#G
=& ` _#G
=
{(* P ) *P
{(* P ) _{(* P ) {(* P ) * P * P _{(* P )
\ +
q d &1
=P .
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Therefore (Cyc F ) q
d &1
K
It remains to apply Proposition 1.4. 4.3. Lemma.
/UF .
d
= P #(* L * P ) mod P qK &1 . i
Proof. Write [P] C (X )= di=0 (P) c, i X q , with (P) C, 0 =P and (P) C, d =1. By [GO, Proposition 3.3.10], for 1 i d&1, we have i
(T q &T )(P) C, i =(P) qC, i&1 &(P) C, i&1 . This implies that, for 0 i d&1, v P ((P) C, i )=1. Furthermore, for i=1, ..., d&1, we get (P) C, i P
#
1 qi
T &T
#&
\
(P) qC, i&1
1
P
&
(P) C, i&1
qi
T &T
P
(P) C, i&1 P
+
(P).
Therefore, for i=0, ..., d&1, we have (P) C, i (&1) i (P). # P Li We obtain d&1
=P # : i=0
(&1) i q i &1 *P Li
d
mod P qK &1 .
It remains to apply Lemma 1.3. K 4.4. Proposition.
Let k be an integer, 1 k q d &2. Then . k(= P )#
Proof. Recall that fP (* L ). We have
B(k) (P), 1k
f P (X )=Exp C b Log L(X ) # Z P[[X ]] and * P =
Log L(X ) B(k) (Log L(X )) k. = : f P (X ) 1k k 0
THE ORTHOGONAL OF CYCLOTOMIC UNITS
273
Therefore, we get d
q &2 X B(k) k X # : f P (X ) k=0 1 k
mod X q
d &1
Q P[[X ]].
But X f P (X ) # Z P[[X ]], thus this congruence holds in Z P[[X ]]. In particular, for k=0, ..., q d &2, we have B(k) # Z P . Furthermore d
* L q &2 B(k) k # : * * P k=0 1 k L
d
mod P qK &1 .
If we apply Lemma 4.3, we have D(= P )#D
*L
\* +
d
mod P qK &2 .
P
Now D
*L
\* + #D(* )&D(* ) L
P
P
#
1 1 & *L *P
#
1 *L 1& *L *P
\
q d &2
# : k=1
+
B(k) k&1 * 1k L
d
mod P qK &2 .
The proposition follows. K 4.5. Corollary.
Let k be an integer, 1 k q d &2. Then:
. k(= F )=0 if k0((q d &1)l), . k(= F )#(1l)(B(k)1 k )(P) if k#0((q d &1)l). Proof.
We have =N KF (= P )= = q&1 F
` _ # Gal(KF )
_(= P ).
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But, by (4.1), Gal(KF )=G l. Thus, we get &. k(= F )= : . k(_(= P )) _ # Gl
=
1 : . (_ l(= P )) l _#G k
=
1 : % k(_ l ) . k(= P ) l _#G
=
1 l
\ : % (_)+ . (= ). kl
k
P
_#G
But _ # G % kl(_)=0 if k0((q d &1)l), _ # G % kl(_)=&1 if k#0((q d &1)l). The Corollary follows. 4.6. Proposition.
K
* P # Cyc = F .
&1 . Then Proof. Let's prove that * P # Cyc = H + . Let _ # G and set {=_ {(* P )=[A] C (* P ) for some A # Z*P . We have
* , P
_(* P ) *P
=( * P , _(* P )) C &( * P , * P ) C C
=( * P , _(* P )) C =_(( {(* P ), * P ) C ) =_(( [A] C (* P ), * P ) C ) =_([A] C (( * P , * P ) C )) =0. The proposition follows because Cyc F /Cyc H+ .
K
4.7. Proposition. Let k be an integer, 2 k q d &1. Then ' k # Cyc = F if and only if (' k , = F ) C =0.
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THE ORTHOGONAL OF CYCLOTOMIC UNITS
= Recall that Cyc = F =U F . Let _ # G. By Proposition 3.3, we have
Proof.
( ' k , _(= F )) C =_(( _ &1(' k ), = F ) C ) =_(( [% k(_ &1 )] C (' k ), = F ) C ) =_([% k(_ &1 )] C (( ' k , = F ) C )). The proposition follows. K 4.8. Proposition.
Let k be an integer, 2 k q d &1. Then:
( ' k , = F ) C =0 if k1((q d &1)l), ( ' k , = F ) C =[(1l)(B(q d &k)1 q d &k )] C (* P ) if k#1((q d &1)l). Proof.
By Theorem 3.6, we have (' k , = F ) C =[. q d &k(= F )] C (* P ).
If remains to apply Corollary 4.5.
K
Let Rl(P)=[ j, 1 j l&1, B(((q d &1)l)(l& j))0(P)]. Let r l(P) be the cardinal of Rl(P). Then 0 r l(P) l&1. 4.9. Theorem. dim FP PK Cyc = F =r l(P). = Proof. Recall that, by Lemma 4.2, U = F =Cyc F . Set r=r l(P) and write Rl(P)=[i 1 , ..., i r ]. For j=1, ..., r, set
; j =' 1+ij ((q d &1)l) . d
Let L be a F P -subspace of PK L qK generated by ; 1 , ..., ; r . Then dim FP L=r. By Proposition 4.8, we know that if k1(q d &1l) or if k#1(q d &1l) and B(q d &k)#0(P) then ' k # Cyc = F . Thus d
d
q PK P qK =Cyc = F P K +L.
Let _ # G such that the restriction of _ to F generates Gal(FQ). Write d /=% (q &1)l. Let , : L Ä l&2 k=0 F P be the map defined by ,
\
r
+ \
r
: [` j ] C ( ; j ) = : ` j . q d &1&ij ((q d &1)l)(_ k(= F )) j=1
j=1
By Theorem 3.6 d
q L Cyc = F P K =Ker ,.
+
. k=0, ..., l&2
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Thus dim PP PK Cyc = F =dim FP LKer ,. For j=1, ..., r, let B j in F P such that B j #
1 B(q d &1&i j ((q d &1)l)) (P). l 1 q d &1&ij ((q d &1)l)
Then, by the results of Section 3 and Proposition 4.8, we have r
r
: ` j . q d &1&ij ((q d &1)l)(_ k(= F ))= : ` j B j / l&ij (_ k ). j=1
j=1
Let M be the matrix (/ l&ij (_ k ) B j ) 0 k l&2, 1 j r . Then dim FP LKer , is the rank of the matrix M. Recall that, for j=1, ..., r, B j {0. Therefore, the rank of M is the rank of the matrix (/ l&ij (_ k )) 0 k l&2, 1 j r . But recall that the determinant of (/ n(_ k )) 0 k l&2, 1 n l&1 is not equal to zero. Thus Ker ,=[0]. K 4.10. Corollary.
= dim FP Cyc = F E F l&1&r l(P).
Proof. Let k be an integer, 1 k q d &2. Suppose that there exists u # E F such that . k(u){0. Recall that, for _ # G, we have . k(_(u))=% k(_) . k(u). Therefore, we get \_ # Gal(KF ),
% k(_)=1.
Thus \_ # G,
% kl(_)=1.
THE ORTHOGONAL OF CYCLOTOMIC UNITS
277
This implies k#0((q d &1)l). Therefore, for k 2 and k1((q d &1)l), we have 'k # E = F. Now, let _ # G and u # U K . We have _(( * P , u) C )=[%(_)] C (( * P , u) C ), because ( * P , u) C # 4 P . But _(( * P , u) C )=( _(* P ), _(u)) C =[%(_)] C ((* P , _(u)) C ). Therefore ( * P , u) C =( * P , _(u)) C . Thus &( * P , u) C =( * P , N KQP (u)) C . This implies that * P # E = F . By Lemma 3.5, we get dim FP PK E = F l&1. It remains to apply Theorem 4.9.
K
It would be interesting to calculate the dimension of the F P -vector space = = = Cyc = F E F . In particular, if p does not divide h F , we have E F =Cyc F .
5. IDEAL CLASS NUMBERS In this last section, we will apply our previous results to obtain informations about ideal class numbers. In particular, we study the case of quadratic extensions of F q(T ) and we prove an analogue to the AnkenyArtinChowla theorem. Let FQ be a subextension of H +Q and set l=[F : Q], l 2. By [FEYI], we have h F =(E F : Cyc F ). Recall that the analogue of the KummerVandiver conjecture is not true for function fields (see [IRSM]). Let B be a subgroup of E F ; we set B [ p] =[: # B, there exists x # O F such that :#x p mod P lF ].
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5.1. Lemma.
(q B [ p] =B & (U K ) p U K
d &1)
.
Proof. It is clear that B [ p] /B & (U K ) p U (q K d &1) (U K ) p U (q . Then there exists $ # U K such that K
d &1)
d
u#$ p mod P qK &1 . But Tr KF (u)=
&1 q d &1 u= u. l l
Thus d
u# &lTr KF ($) p
mod P qK &1 .
Set $$=&lTr KF ($). Then $$ # U F and u#($$) p
d
mod P qK &1 .
Thus u#($$) p
mod P lF .
Choose x # O F such that x#$$ mod P lF . We have u#x p The lemma follows. 5.2. Proposition.
mod P lF .
K We have l&1 dim Fp UF U F[ p] r l(P).
Proof.
We have (UF ) p /U F[ p] and dim Fp UF (UF ) p =l&1.
Thus dim Fp UF U [Fp] l&1.
. Now, let u # B &
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THE ORTHOGONAL OF CYCLOTOMIC UNITS
Set r=r l(P) and let : UF Ä rj=1 F P be the map defined by (u)=(. q d &1&ij ((q d &1)l)(u)) 1 j r , where Rl(P)=[i 1 , ..., i r ]. Then, by Proposition 2.5, Corollary 4.5, and Lemma 5.1, Ker =U F[ p] . Let _ # G such that the restriction of _ to F generates Gal(FQ). Then, Im is the F p -vector space generated by e k =(_ k(= F )) for k=0, ..., l&2. We keep the notations of the proof of Theorem 4.9 and we have e k =(/ l&ij (_ k ) B j ) 1 j r . But, by the proof of Theorem 4.9, the dimension over F P of the F P -vector space generated by e k , k=0, ..., l&2, is equal to r. This implies that at least r vectors e k are linearly independent over F p . The proposition follows. K 5.3. Corollary. The p-rank of E F Cyc F is less than or equal to l&1&r l(P). In particular, if p divides h F then r l(P)
Proof. Recall that (Cyc F ) q &1 /UF , thus E F Cyc F and E F UF have the same p-rank. Note that the p-rank of E F UF is equal to dim Fp E F ((E F ) p UF ). But (E F ) p & UF /U [Fp] . It remains to apply Proposition 5.2.
K
Now, we assume that p is an odd prime number and that d=deg P is 2 (q d &1)2 . Let F=Q(- P), even. Fix + # F* P such that + =&1. Set - P=+* L + then F/H . Let # F be a generator of E F modulo F q* . Write # F = v 1 +v 2 - P, with v 1 , v 2 in Z=F q[T]. 5.4. Proposition.
Let k be an integer, 1 k q d &2. Then:
. k(# F )=0 if k{(q d &1)2, . (q d &1)2(# F )#(&12)(v 2 v 1 )(P). Proof.
Note that v 1 # Z* P , thus D(v 1 )#0
d
mod P qK &2 .
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We have
\
D(# F )#D 1+ #
v 1 (q d &1)2 +* v1 L
&1 v 2 (q d &3)2 * 2 v1 L
+ d
mod P qK &2 .
The proposition follows. K Now, we are able to prove an analogue of the AnkenyArtinChowla Theorem. 5.5. Theorem. Let p be an odd prime number and let F q be a finite field of characteristic p. Let P be a monic irreducible polynomial in F q[T] of degree d, d even. Let F=F q(T, - P) be the quadratic subfield of the P th cyclotomic function field. Let # F =v 1 +v 2 - P, v 1 , v 2 # F q[T ], be a fundamental unit of F q[T, - P]. Then (D 0 } } } D d&1 ) q&1 h 2F
v2 v1
\ +
2
\ \
# &4 B
q d &1 2
2
++ (P),
where h F is the ideal class number of F q[T, - P]. Proof.
Set /=% (q
d &1)2
. Set :=
_(* P ),
` _ # G, /(_)=1
and :$=
_(* P ).
` _ # G, /(_)=&1
Let { # G such that the restriction of { to F generates Gal(FQ). We have :$ {(* P ) . =N KF : *P
\ +
One can see that :$: generates (Cyc F ) q&1. Now let \ # Cyc F such that \ generates Cyc F modulo F * q . Then \ q&1 =`
:$ , :
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THE ORTHOGONAL OF CYCLOTOMIC UNITS
or \ q&1 =`
: , :$
for some ` # F * q . Recall that h F =(E F : Cyc F ). Therefore, there exists `$ # F * q such that # hFF =`$\, or # hFF =`$\ &1. We get h 2F . (q d &1)2(# F ) 2 =. (q d &1)2(\) 2 =. (q d &1)2
:$ 2 . :
\+
Recall that (see the proof of Proposition 4.4) d
* L q &2 B(k) k # : * * P k=0 1 k L
d
mod P qK &1 .
Thus, by formula (2.4), for _ # G, we have d
%(_) * L q &2 k B(k) k # : % (_) * _(* P ) 1k L k=0
d
q &1 mod P K .
Note that, for _ # G, we have D(_(* P ))#
%(_) _(* P )
d
mod P qK &2 .
Let _ # G. We have D
\
_(* P ) %(_) 1 # & *P _(* P ) * P
+
#
1 %(_) * L * L & * L _(* P ) *P
\
q d &2
# : (% k(_)&1) k=1
+
B(k) k&1 * 1k L
d
mod P qK &2 .
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Thus . (q d &1)2
\
_(* P ) B((q d &1)2) #(/(_)&1) (P). *P 1 (q d &1)2
+
We obtain . (q d &1)2
:$ B((q d &1)2) # (P). : 1 (q d &1)2
\+
But: 1 (q d &1)2 =(D 0 } } } D d&1 ) (q&1)2. It remains to apply Proposition 5.4.
K
5.6. Corollary. We keep the hypothesis of Theorem 5.5. Then p divides h F if and only if B((q d &1)2)#0(P). Proof.
This is a consequence of [YUYU]. K
Ichimura [IC] has proved that if p is odd, q 2 +1 square free and q{ p if p 7, then there exist infinitely many primes P # F q[T ] of even degree s such that B((q d &1)2)#0(P). 5.7. Proposition.
We keep the hypothesis of Theorem 5.5. Then = dim FP Cyc = F E F =1&r 2(P).
Proof. By [YUYU] and Proposition 5.4, ' 1+((q d &1)2) Â E = F . Note that = r 2(P)=0 if and only if ' 1+((q d &1)2) # Cyc = . But * # E . The proposition P F F follows. K
REFERENCES [AN] [FEYI] [GE] [GO] [IC]
B. Angles, On explicit reciprocity laws for the local CarlitzKummer symbols, Number Theory, in press. K. Feng and L. Yin, On maximal independent systems of cyclotomic units in cyclotomic function fields, Scientia Sinica 34 (1991), 252261. E.-U. Gekeler, On regularity of small primes in function fields, J. Number Theory 34 (1990), 114127. D. Goss, ``Basic Structures of Function Field Arithmetic,'' Springer-Verlag, New YorkBerlin, (1996). H. Ichimura, On the class numbers of the maximal real subfields of cyclotomic function fields, II, J. Number Theory 72 (1998), 140149.
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283
[IRSM] K. F. Ireland and R. D. Small, Class numbers of cyclotomic function fields, Math. Comput. 46 (1986), 337340. [LA] S. Lang, ``Cyclotomic Fields, I and II,'' Springer-Verlag, New YorkBerlin, 1990. [OK] S. Okada, Kummer's theory for function fields, J. Number Theory 38 (1991), 212215. [SC] F. Schultheis, Local and global residue symbols for algebraic function fields, J. Number Theory 52 (1995), 119124. [TH] D. S. Thakur, Gauss sums for F q[T], Invent. Math. 94 (1988), 105112. [VO] S. V. Vostokov, ArtinHasse exponentials and Bernoulli numbers, in ``Amer. Math. Soc. Transl.,'' Vol. 166, pp. 149156, Amer. Math. Soc., Providence, 1995. [YUYU] J. Yu and J.-K. Yu, A note on a geometric analogue of AnkenyArtinChowla's conjecture, in ``Contemp. Math.,'' Vol. 210, pp. 101105, Amer. Math. Soc., Providence, 1998.
On the Orthogonal of Cyclotomic Units in Positive Characteristic Bruno Angles Laboratoire de Mathematiques Emile Picard, C.N.R.S. U.M. 5580, U.F.R. M.I.G., Universite Toulouse III, 118 Route de Narbonne, 31062 Toulouse Cedex 4, France E-mail: anglespicard.ups-tlse.fr Communicated by D. Goss Received October 20, 1998
INTRODUCTION Let P be a monic irreducible polynomial in F q[T ] of degree d. Let FF q(T ) be a subfield of the maximal real subfield of the P th cyclotomic function field. In this paper, we study the orthogonal of the cyclotomic units of F for the local symbol at P attached to the Carlitz module (see [SC, AN]). Let l=[F : F q(T )] and let r l(P) be the number of BernoulliCarlitz numbers B(k) such that k#0((q d &1)l), 0
THE ORTHOGONAL OF CYCLOTOMIC UNITS
259
Let v P be the normalized P-adic valuation on Q P , then v P extends to a valuation on 0 which is also denoted by v P . Note that v P (P)=1. We view Q as contained in Q P . Let FQ P be a finite extension, F/0. We set: O F , the valuation ring of F; U F , the group of units of F ; PF , the maximal ideal of O F ; n for n 1, U (n) F =1+P F ;
e F is the ramification index of FQ P and f F is the inertia degree of FQ P ; let ? # O F , we say that ? is a prime of F if v P (?)=1e F . Let A be an unitary commutative ring; we denote the set of invertible elements of A by A*. Let F be a field and let LF be a finite extension. Then N LF is the norm map from L* to F* and Tr LF is the trace map from L to F. Let kQ be a finite extension. We set: O k , the integral closure of Z in k ; E k =O * k ; h k , the class number of the ideal class group of O k .
1. PRELIMINARIES In this section, we recall the definitions of the BernoulliCarlitz numbers and of the symbol attached to the P th cyclotomic function field. Let Z P Ä EndG a , b [ [b] C , be the local Carlitz module, i.e., it is a ring homomorphism such that [T ] C =TX+X q and for all b # Z P , [b] C = bX+higher terms (see [AN, Sect. 4], for more details). Let 4 P =[: # 0, [P] C (:)=0]. Then 4 P is a Z P -module via the Carlitz module and we have an isomorphism of Z P -modules, 4 P & Z P PZ P . Let * P be a fixed element in 4 P , * P {0. Set K=Q P (* P ). Recall that KQ P is an abelian extension, KQ P is totally ramified, * P is a prime of K and we have the isomorphism Gal(KQ P ) & (Z P PZ P )*,
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where _ maps to the class of A # Z P such that _(* P )=[A] C (* P ). We set G=Gal(KQ P ). For A # Z P "PZ P , we denote the element _ # G such that _(* P )=[A] C (* P ) by _ A . The limit lim n(1P n )[P n ] C exists in Q P[[X ]] [AN, Proposition 1.5] i and is denoted by Log C . Set L 0 =1 and for i 1, L i =(T q &T ) L i&1 . Then, by [AN, Proposition 4.1], we have (&1) i q X i. Li i 0
Log C (X )= :
Let Exp C be the unique element in Q P[[X ]] such that Log C b Exp C =Exp C b Log C =X. i
Set D 0 =1 and for i 1, D i =(T q &T ) D qi&1 . We have 1 qi X . Di i 0
Exp C (X )= :
Now, we recall the definition of the BernoulliCarlitz numbers B(k). Let k 0 be an integer and write k=a 0 +a 1 q+ } } } +a N q N where 0 a i ai q&1, for i=0, ..., N. Set 1 k => N i=0 D i . The BernoulliCarlitz numbers are defined by X B(k) k X . =: Exp C (X ) k 0 1 k We have B(0)=1 and if k0(q&1), B(k)=0. Furthermore, for k 1, we have 1k B(k+1&q i ). D 1 i k+1&q i i 1
B(k)=& :
We refer the reader to [Ge, GO] for more informations on Bernoulli Carlitz numbers. d Let L(X )=PX+X q be the basic Lubin-Tate polynomial, then L gives rise to a ring homomorphism (see [AN, Sect. 1]): Z P Ä EndG a , b [ [b] L . d Note that [P] L =PX+W q and for all ` # F P , [`] L =`X. We set Log L = n n lim n(1P )[P ] L , such a limit exists in Q P[[X ]]. Let Exp L be the unique element in Q P[[X ]] such that Exp L b Log L =Log L b Exp L =X.
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THE ORTHOGONAL OF CYCLOTOMIC UNITS
1.1. Lemma. (i) (ii)
Set f P =Exp C b Log L and g P =Exp L b Log C .
f P and g P are elements of Z P[[X ]]. For all b in Z P , we have f P b [b] L =[b] C b f P , g P b [b] C =[b] L b g P .
Proof.
This comes from [AN, Proposition 1.1 and Proposition 1.6].
K
We set * L = g P (* P ). Then * L {0 and [P] L (* L )=0.
(1.2)
Set 4 L =[`* L , ` # F P ]. Note that K=Q P (* L ). 1.3. Lemma. Proof. have
i
d
i q q * L # d&1 i=0 ((&1) L i ) * P (P K ). i
Write g P (X )= i 0 A i X q , with A 0 =1 and A i # Z P for all i. We Log L b g P =Log C .
But by [AN, Proposition 2.2], we have d
Log L #X
mod X q Q P[[X ]].
g P #Log C
mod X q Q P[[X ]].
Thus d
Therefore, for 0 i d&1, we get Ai =
(&1) i . Li
K
Let z # PK and let t 1 , t 2 in 0 such that [P] C (t 1 )=z, [P] L (t 2 )=z. Set E 1 =K(t 1 ) and E 2 =K(t 2 ). Then E 1 and E 2 are abelian extensions of K. Let ( } , E 1 K ) and ( } , E 2 K ) be the local Artin maps associated to these abelian extensions. Let : # K*; we set ( z, :) C =(:, E 1 K )(t 1 )&t 1 # 4 P ,
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and ( z, :) L =(:, E 2 K )(t 2 )&t 2 # 4 L . In this paper, we are interested by the symbol ( } , } ) C which was introduced by Schultheis in [SC]. Let's give the basic properties of this symbol. 1.4. Proposition.
(i)
Let z 1 , z 2 in PK , a 1 , a 2 in Z P and : # K*,
( [a 1 ] C (z 1 )+[a 2 ] C (z 2 ), :) C =[a 1 ] C (( z 1 , :) C )+[a 2 ] C (( z 2 , :) C ). (ii)
Let z in PK and : 1 , : 2 in K*, ( z, : 1 : 2 ) C =( z, : 1 ) C +( z, : 2 ) C .
ab (iii) Let z # PK , : # k* and _ # Gal(Q ab P Q P ), where Q P /0 is the maximal abelian extension of Q P . Then
_(( z, :) C )=( _(z), _(:)) C . (iv)
Let z # PK , z{0. We have ( z, z) C =0.
(v)
Let z # PK and : # K*. Then ( z, :) C = f P (( g P (z), :) L ).
Proof.
See [AN, Sect. 1]. K
1.5. Lemma. d z # P qK . Proof.
Let z # PK . Then (z, :) C =0 for all : in U K if and only if
Let t # 0 such that [P] C (t)=z. Write E=K(t). We have ( z, :) C =0 : # N EK (E*).
Therefore (\: # U K , ( z, :) C =0) (U K /N EK (E*)). But U K /N EK (E*) if and only if EK is unramified. By [AN, Proposition 2.1], d EK is unramified if and only if z # P qK . K
THE ORTHOGONAL OF CYCLOTOMIC UNITS
263
2. THE CARLITZKUMMER MORPHISMS In this section, we give the definition of the CarlitzKummer morphisms and we prove the basic properties of these morphisms. The CarlitzKummer morphisms play a central ro^le in the next sections. Recall that K=Q P (* P )=Q P (* L ). Let : # K*. Write :=* rL : 1 , where r # Z and : 1 # U K . Since KQ P is totally ramified, there exists g 1(X ) # Z P[[X ]] such that : 1 = g 1(* L ). Write g(X )=X rg 1(X ). Let g$(X ) be the derivative of g(X ), we set D(:)=
g$(* L ) g(* L )
d
mod P qK &2 .
Note that D(:) is independent of the choice of g 1(X ). By [LA, Chap. 9, q d &2 Sect. 2], D: K* Ä P &1 is a morphism of groups and D(U K )/ K P K q d &2 O K P K . 2.1. Theorem. Let z # P 2K and : # U K . Then (z, :) C = Proof.
_
1 Tr KQP (g P (z) D(:)) P
&
(* P ). C
By [AN, Theorem 2.9], if z # P 2K , we have (z, :) L =
_
1 Tr KQP (Log L(z) D(:)) P
&
(* L ). L
Let z # P 2K . By [AN, Proposition 2.2], we have Log L( g P (z))#g P (z)
d
mod P dK .
d
d
q &2 Note that Tr KQP (P qK )/P 2Z P . Furthermore, if : # U K , D(:) # O K P K . Therefore, if : # U K , we get
( g P (z), :) L =
_
1 Tr KQP (g P (z) D(:)) P
&
(* L ). L
Note that f P (* L )=* P . It remains to apply Lemma 1.1 and Proposition 1.4. K
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Now we define the k th CarlitzKummer morphism . k , this morphism is a little bit different from the one defined in Okada's work [OK]. Let k be an integer, 1 k q d &2. Let : # U K ; write q d &3
d
D(:)# : a j * Lj
mod P qK &2 ,
j=0
where a j # Z P for j=0, ..., q d &3 (we can do so because O K =Z P[* L ]). Then . k(:) # F P is defined by . k(:)#a k&1(P). 2.2. Proposition. Let k be an integer, 1 k q d &2. Then . k : U K Ä F P is well-defined and is a morphism of groups. Furthermore, . k is surjective. d
Proof. Note that PO K =P qK &1 . Thus . k is well-defined. Since D is a morphism of groups, . k is a morphism of groups too. Let n 1 and let ; # F P . We have d
D(1&;* nL )# &n : ; l* ln&1 L
mod P qK &2 .
l 1
Thus . k(1&;* nL )= &n; kn =0
if n divides k
otherwise.
The proposition follows. K Recall that G=Gal(KQP ). Let %: G Ä F* P be the Teichmuller character, i.e., %(_ A )#A(P). Let G be the set of morphisms /: G Ä F* =( %). P . Then G 2.3. Proposition. Let k be an integer, 1 k q d &2. Let _ # G. Then for all : # U K , we have . k(_(:))=% k(_) . k(:). Proof.
Recall that * P = f P (* L ). Therefore _(* P )= f P (_(* L )).
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THE ORTHOGONAL OF CYCLOTOMIC UNITS
But, there exists ` # F* P such that _(* L )=`* L . If we apply Lemma 1.1, we get _(L P )= f p(`* L ) = f P ([`] L (* L )) =[`] C (* P ). But _(* P )=[%(_)] C (* P ). Thus _(* L )=%(_) * L .
(2.4)
Now, let : # U K and write := g(* L ) for some g # Z P[[X ]]. We have _(:)= g(%(_) * L ). Write g _(X )= g(%(_) X ) # Z P[[X ]]. Then _(:)= g _(* L ). We have D(_(:))# #
g$_(* L ) g _(* L ) %(_) g$(%(_) * L ) _(:)
#%(_) _(D(:))
d
mod P qK &2 .
Write q d &2
D(:)# : . k(:) * k&1 L
d
mod P qK &2 ,
k=1
then q d &2
_(D(:))# : % k&1(_) . k(:) * k&1 L
d
mod P qK &2 .
k=1
The proposition follows. K 2.5. Proposition.
Let : # U K . Then
(. k(:)=0 for k=1, ..., q d &2) (: # (U K ) p U (q K Proof. If : # (U K ) p U K(q for k=1, ..., q d &2.
d &1)
d
d &1)
).
, then D(:)#0 mod P qK &2 . Thus, . k(:)=0
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Now, let : # U K such that . k(:)=0 for k=1, ..., q d &2. Note that q d &2 D(F* . Thus, we can suppose that : # U (1) P )#0 mod P K K . There, exist d (q d &1) such that ` i # F P , i=1, ..., q &2 and = # U K q d &2
:== ` (1&` i * iL ). i=1
/Ker . k . We have Note that U (k+1) K . 1(:)=&` 1 =0. Let j be an integer 2 j q d &2 and suppose that we have proved: for all l j&1, ` l =0 or p divides l. We have . j (:)=&j` j =0. Thus ` j =0 or p divides j. The proposition follows. K
3. THAKUR'S GAUSS SUMS In this section, we use Thakur's Gauss sums to obtain an explicit reciprocity law for the symbol defined in Section 1. The Thakur's Gauss sums are defined as k
G k =& : % &q (_) _(* P ) # PK , _#G
for k=0, ..., d&1. 3.1. Proposition. (ii) Proof.
(i)
j
q For all _ # G, _(* P )= d&1 j=0 % (_) G j
(G 0 } } } G d&1 ) q&1 =(&1) d P. It comes from [TH, Proposition I and Theorem II]. K
Let k be an integer, 1 k q d &1. We set d&1
' k = : G kj D k&1 . j j=0
3.2. Theorem. (ii)
(i)
' 1 =* P .
Let k be an integer, 1 k q d &1. Then ' k #f P (* kL )
d
mod P qK .
THE ORTHOGONAL OF CYCLOTOMIC UNITS
267
Proof. The assertion (i) is a consequence of Proposition 3.1. Let's prove (ii). i Write f P (X )= i 0 B i X q with B 0 =1 and B i # Z P for all i. We have fP b Exp L =Exp C . But, by [AN, Proposition 2.2], we have d
Exp L #X
mod X q Q P[[X ]],
f P #Exp C
mod X q Q P[[X ]].
thus d
Therefore, we have Bi =
1 , Di
for i=0, ..., d&1, and we get d&1
1 qi * Di L
*P # : i=0
d
mod P qK .
Let j be an integer, 0 j d&1. Using formula (2.4), we have j
G j # & : % &q (_) _(* P ) _#G d&1
_#G d&1
#& : i=0 d&1
#& : i=0
#
qj L
* Dj
\:
1 i (_(* L )) q Di
+ 1 : % (_)(_(* )) + D \ 1 * \ : % (_)+ D j
# & : % &q (_)
i=0
&q j
qi
L
i
_#G
qi L
i
q i &q j
_#G d
mod P qK .
We get d&1
'k # : j=0
(* kL ) q Dj
# f P (* kL )
j
d
mod P qK . K
268
BRUNO ANGLES
3.3. Proposition.
Let k be an integer, 1 k q d &1. Let _ # G. Then d
_(' k )#[% k(_)] C (' k ) Proof.
mod P qK .
By Theorem 3.2 and formula (2.4), we have _(' k )# f P (_(* kL )) # f P (% k(_) * kL ) # f P ([% k(_)] L (* kL )) #[% k(_)] C ( f P (* kL )) d
#[% k(_)] C (' k )
mod P qK .
K
3.4. Proposition. Let k be an integer, 1 k q d &1. Let n be the g.c.d. of k and q d &1. Then [Q P (' k ) : Q P ]= Proof.
Let _ # G ( g
d &1)
q d &1 . n
n. We have d&1
_(' k )= : (_(G j )) k D k&1 j j=0 d&1
j
= : % q k(_) G kJ D k&1 j j=0
=' k . Therefore G (q
d &1)n
/Gal(KQ P (' k )).
Now let _ # G such that _(' k )=' k . By Proposition 3.3, we have ' k #% k(_) ' k
mod P k+1 . K
Therefore % k(_)=1. This implies _ # G ( g
d &1)
n. The proposition follows. K
269
THE ORTHOGONAL OF CYCLOTOMIC UNITS
3.5. Lemma.
Let z # PK . Then there exist ` 1 , ..., ` q d &1 in F P such that q d &1
d
mod P qK .
z# : [` k ] C (' k ) k=1
Furthermore ` k is unique for k=1, ..., q d &1. Proof.
Let ` # F P . By Theorem 3.2, we have [`] C (' k )#`* kL
mod P k+1 . K
The uniqueness follows. Let z # PK . Since * L is a prime of K, we have O K =F P[[* L ]]. Therefore we can write q d &1
g P (z)# : ` k * kL
d
mod P qK ,
k=1
where ` k # F P for k=1, ..., q d &1. Thus z# f P ( g P (z)) q d &1
# fP
\:
` k * kL
k=1
+
q d &1
# : [` k ] C ( f P (* kL )) k=1 q d &1
# : [` k ] C (' k )
d
mod P qK . K
k=1 d
d
&1 [` k ] C (' k ) mod P qK , where 3.6. Theorem. Let z # P 2K . Write z# qk=2 d ` k # F P for k=2, ..., q &1. Let u # U K . Then q d &1
( z, u) C =
_:
k=2
Proof.
` k . q d &k(u)
&
(* P ). C
By Proposition 1.4 and Lemma 1.5, we get q d &1
( z, u) C = : [` k ] C (( ' k , u) C ). k=2
270
BRUNO ANGLES
Now, by Theorem 2.1 and Theorem 3.2, we have ( ' k , u) C =
1
_P Tr
KQP
d
(* kL D(u))
&
(* P ). C
d
&2 q &2 Write D(u)= qi=1 . i (u) * i&1 mod P K , and note that, by formula (1.2) L we have:
Tr KQP (* nL )#0(P 2 ) if n 1 and n{q d &1, d
Tr KQP (* qL &1 )=P. We get (' k , u) C =[. q d &k(u)] C (* P ). The theorem follows. K 3.7. Corollary.
Let u # U K . Then
(\z # P 2K , ( z, u) C =0) (u # (U K ) p U (q K Proof.
d &1)
).
By (i) of Proposition 1.4, Lemma 1.5, and Lemma 3.5, we have (\z # P 2K , ( z, u) C =0) (( ' k u) C =0 for k=2, ..., q d &1).
Now, by Theorem 3.6 (' k , u) C =0 . q d &k(u)=0. It remains to apply Proposition 2.5.
K
4. CYCLOTOMIC UNITS Let H=Q(* P )/0 be the P th cyclotomic function field. Let H +Q be the maximal subextension of HQ such that 1T splits completely in H +. The field H + is called the maximal totally real subfield of H and H + =Q( q&1 ). We fix a subextension FQ of H +Q and we set l=[F : Q], l 2. Recall that FQ is totally ramified at P and we denote the unique prime ideal of O F above P by PF . We denote the PF -adic completion of F /K. Then Gal(FQ) & Gal(FQ P ) & GG l.
(4.1)
THE ORTHOGONAL OF CYCLOTOMIC UNITS
271
Let B be a subgroup of U K . We set B = =[z # PK , \u # B, ( z, u) C =0]. By Proposition 1.4, B = is a Z P -module via the Carlitz module. Furthermore, by Lemma 1.5, we have d
P qK /B =. Note that, if z # PK and if a # PZ P then d
[a] C (z) # P qK . Therefore, via the Carlitz module, PK B = is a F P -vector space. In this section, our aim is to calculate dim FP PK Cyc = F . Recall that the group of cyclotomic units of F is the subgroup of E F (recall E F =O*F ) generated by F * q and N H + F (_(* P )* P ) for all _ # G. We denote the group of cyclotomic units of F by Cyc F . Set d
= P =&
* qP &1 , P
and = F =N H + F (= P ). Note that = F # Cyc F and N FQ(= F )=1. Let UF be the subgroup of E F generated by F * q and _(= F ) for all _ # Gal(FQ). We have UF /Cyc F /E F . 4.2. Lemma. Proof.
= U= F =Cyc F .
= Since UF /Cyc F , we have Cyc = F /U F . Note that
= P =& ` _#G
*P . _(* P )
Now, let { # G. We have {(= P )= & ` _#G
=& ` _#G
=
{(* P ) *P
{(* P ) _{(* P ) {(* P ) * P * P _{(* P )
\ +
q d &1
=P .
272
BRUNO ANGLES
Therefore (Cyc F ) q
d &1
K
It remains to apply Proposition 1.4. 4.3. Lemma.
/UF .
d
= P #(* L * P ) mod P qK &1 . i
Proof. Write [P] C (X )= di=0 (P) c, i X q , with (P) C, 0 =P and (P) C, d =1. By [GO, Proposition 3.3.10], for 1 i d&1, we have i
(T q &T )(P) C, i =(P) qC, i&1 &(P) C, i&1 . This implies that, for 0 i d&1, v P ((P) C, i )=1. Furthermore, for i=1, ..., d&1, we get (P) C, i P
#
1 qi
T &T
#&
\
(P) qC, i&1
1
P
&
(P) C, i&1
qi
T &T
P
(P) C, i&1 P
+
(P).
Therefore, for i=0, ..., d&1, we have (P) C, i (&1) i (P). # P Li We obtain d&1
=P # : i=0
(&1) i q i &1 *P Li
d
mod P qK &1 .
It remains to apply Lemma 1.3. K 4.4. Proposition.
Let k be an integer, 1 k q d &2. Then . k(= P )#
Proof. Recall that fP (* L ). We have
B(k) (P), 1k
f P (X )=Exp C b Log L(X ) # Z P[[X ]] and * P =
Log L(X ) B(k) (Log L(X )) k. = : f P (X ) 1k k 0
THE ORTHOGONAL OF CYCLOTOMIC UNITS
273
Therefore, we get d
q &2 X B(k) k X # : f P (X ) k=0 1 k
mod X q
d &1
Q P[[X ]].
But X f P (X ) # Z P[[X ]], thus this congruence holds in Z P[[X ]]. In particular, for k=0, ..., q d &2, we have B(k) # Z P . Furthermore d
* L q &2 B(k) k # : * * P k=0 1 k L
d
mod P qK &1 .
If we apply Lemma 4.3, we have D(= P )#D
*L
\* +
d
mod P qK &2 .
P
Now D
*L
\* + #D(* )&D(* ) L
P
P
#
1 1 & *L *P
#
1 *L 1& *L *P
\
q d &2
# : k=1
+
B(k) k&1 * 1k L
d
mod P qK &2 .
The proposition follows. K 4.5. Corollary.
Let k be an integer, 1 k q d &2. Then:
. k(= F )=0 if k0((q d &1)l), . k(= F )#(1l)(B(k)1 k )(P) if k#0((q d &1)l). Proof.
We have =N KF (= P )= = q&1 F
` _ # Gal(KF )
_(= P ).
274
BRUNO ANGLES
But, by (4.1), Gal(KF )=G l. Thus, we get &. k(= F )= : . k(_(= P )) _ # Gl
=
1 : . (_ l(= P )) l _#G k
=
1 : % k(_ l ) . k(= P ) l _#G
=
1 l
\ : % (_)+ . (= ). kl
k
P
_#G
But _ # G % kl(_)=0 if k0((q d &1)l), _ # G % kl(_)=&1 if k#0((q d &1)l). The Corollary follows. 4.6. Proposition.
K
* P # Cyc = F .
&1 . Then Proof. Let's prove that * P # Cyc = H + . Let _ # G and set {=_ {(* P )=[A] C (* P ) for some A # Z*P . We have
* , P
_(* P ) *P
=( * P , _(* P )) C &( * P , * P ) C C
=( * P , _(* P )) C =_(( {(* P ), * P ) C ) =_(( [A] C (* P ), * P ) C ) =_([A] C (( * P , * P ) C )) =0. The proposition follows because Cyc F /Cyc H+ .
K
4.7. Proposition. Let k be an integer, 2 k q d &1. Then ' k # Cyc = F if and only if (' k , = F ) C =0.
275
THE ORTHOGONAL OF CYCLOTOMIC UNITS
= Recall that Cyc = F =U F . Let _ # G. By Proposition 3.3, we have
Proof.
( ' k , _(= F )) C =_(( _ &1(' k ), = F ) C ) =_(( [% k(_ &1 )] C (' k ), = F ) C ) =_([% k(_ &1 )] C (( ' k , = F ) C )). The proposition follows. K 4.8. Proposition.
Let k be an integer, 2 k q d &1. Then:
( ' k , = F ) C =0 if k1((q d &1)l), ( ' k , = F ) C =[(1l)(B(q d &k)1 q d &k )] C (* P ) if k#1((q d &1)l). Proof.
By Theorem 3.6, we have (' k , = F ) C =[. q d &k(= F )] C (* P ).
If remains to apply Corollary 4.5.
K
Let Rl(P)=[ j, 1 j l&1, B(((q d &1)l)(l& j))0(P)]. Let r l(P) be the cardinal of Rl(P). Then 0 r l(P) l&1. 4.9. Theorem. dim FP PK Cyc = F =r l(P). = Proof. Recall that, by Lemma 4.2, U = F =Cyc F . Set r=r l(P) and write Rl(P)=[i 1 , ..., i r ]. For j=1, ..., r, set
; j =' 1+ij ((q d &1)l) . d
Let L be a F P -subspace of PK L qK generated by ; 1 , ..., ; r . Then dim FP L=r. By Proposition 4.8, we know that if k1(q d &1l) or if k#1(q d &1l) and B(q d &k)#0(P) then ' k # Cyc = F . Thus d
d
q PK P qK =Cyc = F P K +L.
Let _ # G such that the restriction of _ to F generates Gal(FQ). Write d /=% (q &1)l. Let , : L Ä l&2 k=0 F P be the map defined by ,
\
r
+ \
r
: [` j ] C ( ; j ) = : ` j . q d &1&ij ((q d &1)l)(_ k(= F )) j=1
j=1
By Theorem 3.6 d
q L Cyc = F P K =Ker ,.
+
. k=0, ..., l&2
276
BRUNO ANGLES
Thus dim PP PK Cyc = F =dim FP LKer ,. For j=1, ..., r, let B j in F P such that B j #
1 B(q d &1&i j ((q d &1)l)) (P). l 1 q d &1&ij ((q d &1)l)
Then, by the results of Section 3 and Proposition 4.8, we have r
r
: ` j . q d &1&ij ((q d &1)l)(_ k(= F ))= : ` j B j / l&ij (_ k ). j=1
j=1
Let M be the matrix (/ l&ij (_ k ) B j ) 0 k l&2, 1 j r . Then dim FP LKer , is the rank of the matrix M. Recall that, for j=1, ..., r, B j {0. Therefore, the rank of M is the rank of the matrix (/ l&ij (_ k )) 0 k l&2, 1 j r . But recall that the determinant of (/ n(_ k )) 0 k l&2, 1 n l&1 is not equal to zero. Thus Ker ,=[0]. K 4.10. Corollary.
= dim FP Cyc = F E F l&1&r l(P).
Proof. Let k be an integer, 1 k q d &2. Suppose that there exists u # E F such that . k(u){0. Recall that, for _ # G, we have . k(_(u))=% k(_) . k(u). Therefore, we get \_ # Gal(KF ),
% k(_)=1.
Thus \_ # G,
% kl(_)=1.
THE ORTHOGONAL OF CYCLOTOMIC UNITS
277
This implies k#0((q d &1)l). Therefore, for k 2 and k1((q d &1)l), we have 'k # E = F. Now, let _ # G and u # U K . We have _(( * P , u) C )=[%(_)] C (( * P , u) C ), because ( * P , u) C # 4 P . But _(( * P , u) C )=( _(* P ), _(u)) C =[%(_)] C ((* P , _(u)) C ). Therefore ( * P , u) C =( * P , _(u)) C . Thus &( * P , u) C =( * P , N KQP (u)) C . This implies that * P # E = F . By Lemma 3.5, we get dim FP PK E = F l&1. It remains to apply Theorem 4.9.
K
It would be interesting to calculate the dimension of the F P -vector space = = = Cyc = F E F . In particular, if p does not divide h F , we have E F =Cyc F .
5. IDEAL CLASS NUMBERS In this last section, we will apply our previous results to obtain informations about ideal class numbers. In particular, we study the case of quadratic extensions of F q(T ) and we prove an analogue to the AnkenyArtinChowla theorem. Let FQ be a subextension of H +Q and set l=[F : Q], l 2. By [FEYI], we have h F =(E F : Cyc F ). Recall that the analogue of the KummerVandiver conjecture is not true for function fields (see [IRSM]). Let B be a subgroup of E F ; we set B [ p] =[: # B, there exists x # O F such that :#x p mod P lF ].
278
BRUNO ANGLES
5.1. Lemma.
(q B [ p] =B & (U K ) p U K
d &1)
.
Proof. It is clear that B [ p] /B & (U K ) p U (q K d &1) (U K ) p U (q . Then there exists $ # U K such that K
d &1)
d
u#$ p mod P qK &1 . But Tr KF (u)=
&1 q d &1 u= u. l l
Thus d
u# &lTr KF ($) p
mod P qK &1 .
Set $$=&lTr KF ($). Then $$ # U F and u#($$) p
d
mod P qK &1 .
Thus u#($$) p
mod P lF .
Choose x # O F such that x#$$ mod P lF . We have u#x p The lemma follows. 5.2. Proposition.
mod P lF .
K We have l&1 dim Fp UF U F[ p] r l(P).
Proof.
We have (UF ) p /U F[ p] and dim Fp UF (UF ) p =l&1.
Thus dim Fp UF U [Fp] l&1.
. Now, let u # B &
279
THE ORTHOGONAL OF CYCLOTOMIC UNITS
Set r=r l(P) and let : UF Ä rj=1 F P be the map defined by (u)=(. q d &1&ij ((q d &1)l)(u)) 1 j r , where Rl(P)=[i 1 , ..., i r ]. Then, by Proposition 2.5, Corollary 4.5, and Lemma 5.1, Ker =U F[ p] . Let _ # G such that the restriction of _ to F generates Gal(FQ). Then, Im is the F p -vector space generated by e k =(_ k(= F )) for k=0, ..., l&2. We keep the notations of the proof of Theorem 4.9 and we have e k =(/ l&ij (_ k ) B j ) 1 j r . But, by the proof of Theorem 4.9, the dimension over F P of the F P -vector space generated by e k , k=0, ..., l&2, is equal to r. This implies that at least r vectors e k are linearly independent over F p . The proposition follows. K 5.3. Corollary. The p-rank of E F Cyc F is less than or equal to l&1&r l(P). In particular, if p divides h F then r l(P)
Proof. Recall that (Cyc F ) q &1 /UF , thus E F Cyc F and E F UF have the same p-rank. Note that the p-rank of E F UF is equal to dim Fp E F ((E F ) p UF ). But (E F ) p & UF /U [Fp] . It remains to apply Proposition 5.2.
K
Now, we assume that p is an odd prime number and that d=deg P is 2 (q d &1)2 . Let F=Q(- P), even. Fix + # F* P such that + =&1. Set - P=+* L + then F/H . Let # F be a generator of E F modulo F q* . Write # F = v 1 +v 2 - P, with v 1 , v 2 in Z=F q[T]. 5.4. Proposition.
Let k be an integer, 1 k q d &2. Then:
. k(# F )=0 if k{(q d &1)2, . (q d &1)2(# F )#(&12)(v 2 v 1 )(P). Proof.
Note that v 1 # Z* P , thus D(v 1 )#0
d
mod P qK &2 .
280
BRUNO ANGLES
We have
\
D(# F )#D 1+ #
v 1 (q d &1)2 +* v1 L
&1 v 2 (q d &3)2 * 2 v1 L
+ d
mod P qK &2 .
The proposition follows. K Now, we are able to prove an analogue of the AnkenyArtinChowla Theorem. 5.5. Theorem. Let p be an odd prime number and let F q be a finite field of characteristic p. Let P be a monic irreducible polynomial in F q[T] of degree d, d even. Let F=F q(T, - P) be the quadratic subfield of the P th cyclotomic function field. Let # F =v 1 +v 2 - P, v 1 , v 2 # F q[T ], be a fundamental unit of F q[T, - P]. Then (D 0 } } } D d&1 ) q&1 h 2F
v2 v1
\ +
2
\ \
# &4 B
q d &1 2
2
++ (P),
where h F is the ideal class number of F q[T, - P]. Proof.
Set /=% (q
d &1)2
. Set :=
_(* P ),
` _ # G, /(_)=1
and :$=
_(* P ).
` _ # G, /(_)=&1
Let { # G such that the restriction of { to F generates Gal(FQ). We have :$ {(* P ) . =N KF : *P
\ +
One can see that :$: generates (Cyc F ) q&1. Now let \ # Cyc F such that \ generates Cyc F modulo F * q . Then \ q&1 =`
:$ , :
281
THE ORTHOGONAL OF CYCLOTOMIC UNITS
or \ q&1 =`
: , :$
for some ` # F * q . Recall that h F =(E F : Cyc F ). Therefore, there exists `$ # F * q such that # hFF =`$\, or # hFF =`$\ &1. We get h 2F . (q d &1)2(# F ) 2 =. (q d &1)2(\) 2 =. (q d &1)2
:$ 2 . :
\+
Recall that (see the proof of Proposition 4.4) d
* L q &2 B(k) k # : * * P k=0 1 k L
d
mod P qK &1 .
Thus, by formula (2.4), for _ # G, we have d
%(_) * L q &2 k B(k) k # : % (_) * _(* P ) 1k L k=0
d
q &1 mod P K .
Note that, for _ # G, we have D(_(* P ))#
%(_) _(* P )
d
mod P qK &2 .
Let _ # G. We have D
\
_(* P ) %(_) 1 # & *P _(* P ) * P
+
#
1 %(_) * L * L & * L _(* P ) *P
\
q d &2
# : (% k(_)&1) k=1
+
B(k) k&1 * 1k L
d
mod P qK &2 .
282
BRUNO ANGLES
Thus . (q d &1)2
\
_(* P ) B((q d &1)2) #(/(_)&1) (P). *P 1 (q d &1)2
+
We obtain . (q d &1)2
:$ B((q d &1)2) # (P). : 1 (q d &1)2
\+
But: 1 (q d &1)2 =(D 0 } } } D d&1 ) (q&1)2. It remains to apply Proposition 5.4.
K
5.6. Corollary. We keep the hypothesis of Theorem 5.5. Then p divides h F if and only if B((q d &1)2)#0(P). Proof.
This is a consequence of [YUYU]. K
Ichimura [IC] has proved that if p is odd, q 2 +1 square free and q{ p if p 7, then there exist infinitely many primes P # F q[T ] of even degree s such that B((q d &1)2)#0(P). 5.7. Proposition.
We keep the hypothesis of Theorem 5.5. Then = dim FP Cyc = F E F =1&r 2(P).
Proof. By [YUYU] and Proposition 5.4, ' 1+((q d &1)2) Â E = F . Note that = r 2(P)=0 if and only if ' 1+((q d &1)2) # Cyc = . But * # E . The proposition P F F follows. K
REFERENCES [AN] [FEYI] [GE] [GO] [IC]
B. Angles, On explicit reciprocity laws for the local CarlitzKummer symbols, Number Theory, in press. K. Feng and L. Yin, On maximal independent systems of cyclotomic units in cyclotomic function fields, Scientia Sinica 34 (1991), 252261. E.-U. Gekeler, On regularity of small primes in function fields, J. Number Theory 34 (1990), 114127. D. Goss, ``Basic Structures of Function Field Arithmetic,'' Springer-Verlag, New YorkBerlin, (1996). H. Ichimura, On the class numbers of the maximal real subfields of cyclotomic function fields, II, J. Number Theory 72 (1998), 140149.
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283
[IRSM] K. F. Ireland and R. D. Small, Class numbers of cyclotomic function fields, Math. Comput. 46 (1986), 337340. [LA] S. Lang, ``Cyclotomic Fields, I and II,'' Springer-Verlag, New YorkBerlin, 1990. [OK] S. Okada, Kummer's theory for function fields, J. Number Theory 38 (1991), 212215. [SC] F. Schultheis, Local and global residue symbols for algebraic function fields, J. Number Theory 52 (1995), 119124. [TH] D. S. Thakur, Gauss sums for F q[T], Invent. Math. 94 (1988), 105112. [VO] S. V. Vostokov, ArtinHasse exponentials and Bernoulli numbers, in ``Amer. Math. Soc. Transl.,'' Vol. 166, pp. 149156, Amer. Math. Soc., Providence, 1995. [YUYU] J. Yu and J.-K. Yu, A note on a geometric analogue of AnkenyArtinChowla's conjecture, in ``Contemp. Math.,'' Vol. 210, pp. 101105, Amer. Math. Soc., Providence, 1998.