An exceptional isomorphism between modular curves of level 13

Journal of Number Theory 145 (2014) 273–300

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Journal of Number Theory www.elsevier.com/locate/jnt

An exceptional isomorphism between modular curves of level 13 Burcu Baran Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, United States

a r t i c l e

i n f o

Article history: Received 14 November 2013 Accepted 15 May 2014 Available online 11 July 2014 Communicated by Kenneth A. Ribet Keywords: Elliptic curve Galois representations Modular curve Non-split Cartan group

a b s t r a c t In this note we use representation theory to compute an explicit equation over Q for the modular curve Xns (13) associated to the normalizer of a non-split Cartan subgroup of level 13. We also prove that the same equation defines the modular curve Xs (13) associated to the normalizer of a split Cartan subgroup of level 13. Hence, these two modular curves are isomorphic over Q. This isomorphism does not seem to have a “modular” explanation. For instance, Xs (13)(Q) contains a cusp but Xns (13)(Q) does not. © 2014 Elsevier Inc. All rights reserved.

1. Introduction For a positive integer n, let X(n) be the modular curve over Q with full level n structure. Let C+ ns (n) be the normalizer of a non-split Cartan subgroup Cns (n) of GL2 (Z/nZ). The corresponding modular curve Xns (n), defined as the quotient X(n)/C+ ns (n), is geometrically connected over Q. A detailed discussion of these curves is in [1]. The modular curve Xns (n) is useful for two interesting problems. For suitably chosen n, the determination of all integral points of the non-cuspidal locus of Xns (n) gives new solutions to the class number one problem which was solved by Baker–Heegner–Stark E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jnt.2014.05.017 0022-314X/© 2014 Elsevier Inc. All rights reserved.

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(see [10, Appendix]). Secondly, the rational points of Xns (n) are related to Serre’s uniformity problem over Q. This problem grew out of: Theorem. (See Serre [9].) If an elliptic curve E over Q does not have complex multiplication then there exists a constant CE > 0 such that for every prime p > CE , the mod-p Galois representation ρE,p : GQ → GL2 (Fp ) is surjective. Serre asked if CE can be chosen independently of E. He predicted an affirmative answer: Serre’s uniformity problem over Q . There is a constant C > 0 so that if E is an elliptic curve over Q without complex multiplication then ρE,p is surjective for all p > C. Past work by J.-P. Serre [9], B. Mazur [7], and Y. Bilu, P. Parent and M. Rebolledo [3] led to important advances but has not solved the problem. The remaining and most difficult part is to exclude the possibility that ρE,p has image contained in the normalizer C+ ns (p) of a non-split Cartan subgroup of GL2 (Fp ) for “large” p. More precisely: as the curve Xns (n) has no Q-rational cusps when n > 2, does there exist a constant C such that for every prime number p > C, the only Q-points of the modular curve Xns (p) over Q are CM points? There has been extensive work on finding explicit equations for Xns (n) over Q in low genus cases (≤ 2), and determining their Q-points (see [1] for a detailed discussion). There exists only one curve Xns (n) with genus 3 and it is the one with n = 13. In this paper we focus on this curve. We obtain an equation over Q for this curve as a plane quartic and explicitly compute its map to the j-line. Finding an equation for the modular curves Xns (n) is difficult, as they never have a Q-rational cusp for n > 2. The curve Xns (13) has genus > 2, so it is even more complicated. We use a new method resting on representation theory to overcome these difficulties. In the end this led us to a surprising isomorphism with another modular curve over Q. The canonical map of any genus 3 curve X over Q is either 2 to 1 onto a conic (the hyperelliptic case) or it is an isomorphism onto a plane quartic (the non-hyperelliptic case). Thus, a Q-basis of the vector space Ω 1 (X) of regular 1-forms on X either satisfies a nonzero quadratic relation or it does not satisfy a quadratic relation but it satisfies a nonzero quartic relation. In the case that it satisfies a quartic relation, it is unique up to scaling and that gives an equation for X in P2Q . Let Cs (n) be a split Cartan subgroup of GL2 (Z/nZ) and let C+ s (n) be its normalizer in GL2 (Z/nZ). We define the modular curve Xs (n) to be the quotient X(n)/C+ s (n). It is 2πiτ /13 ∗ ∗ geometrically connected over Q. Let q := e for τ ∈ H where H is the extended upper half-plane. Our main results are as follows: Main Results. (1) By using representation theory, we determine a Q-basis for Ω 1 (Xns (13)) in terms of the q-expansions at the point ∞ which is only Q(ζ13 )+ -rational on Xns (13). Here

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Table 1.1 Imaginary quadratic discriminants associated to the known rational points on Xs (13) and on Xns (13). (x, y, z) on (1.1)

(0, 0, 1)

(0, 1, 0)

on Xs (13) on Xns (13)

−27 −67

−12 −11

(0, 3, 2) −3 −163

(1, 0, −1) −16 −7

(1, 0, 0) −43 −8

(1, 0, 1) −4 −28

(1, 1, 0) cusp −19

Q(ζ13 )+ is the maximal real subfield of Q(ζ13 ). Using this, we deduce that the modular curve Xns (13) is defined by the equation     (−y − z)x3 + 2y 2 + zy x2 + −y 3 + zy 2 − 2z 2 y + z 3 x   + 2z 2 y 2 − 3z 3 y = 0

(1.1)

in P2Q . The discriminant of this equation is −136 . For |x| < 104 , |y| < 104 and |z| < 104 we search for the rational points and we obtain seven rational points. (2) In an easier way, we also determine a Q-basis for Ω 1 (Xs (13)) in terms of the q-expansions at the point ∞ which is Q-rational on Xs (13). This lets us show that the same equation also defines the genus-3 modular curve Xs (13) in P2Q . Hence, these two modular curves are isomorphic over Q. In [3] the “split Cartan” case of Serre’s uniformity problem is settled by showing that all such points of Xs (p) are CM points for p ≥ 11 with p = 13. Their method failed to determine the rational points of Xs (13). This surprising isomorphism shows that Xs (13) also behaves like a “non-split Cartan” curve. (3) We explicitly compute the j-line map Xns (13) → X(1) of degree 78 and the j-line map Xs (13) → X(1) of degree 91. We evaluate these at the known Q-rational points. There are 13 imaginary quadratic orders with class number one; in six of them 13 is split and in seven of them 13 is inert. We find that six of the known rational points on Xs (13) correspond to elliptic curves with CM by imaginary quadratic orders in which 13 is split and the other rational point corresponds to the unique Q-rational cusp. We also find that the seven known rational points of Xns(13) correspond to elliptic curves with CM by an imaginary quadratic order in which 13 is inert. Table 1.1 gives the known rational points on Eq. (1.1), the corresponding discriminants and cusp on Xs (13) and on Xns (13). This data supports Serre’s uniformity conjecture. In Section 2 we prove some properties of cuspidal representations of GL2 (Fp ) which are used in the rest of the paper. In Section 3 we associate a representation to any element of S2 (Γs (p)) that arises from a newform in S2 (Γ0+ (p2 )) under the isomorphism (3.4). In Section 4 we compute a Q-basis for Ω 1 (Xs (13)) in terms of q-expansions at ∞ ∈ Xs (13)(Q) and use it to compute a quartic relation which defines Xs (13) in P2Q . Since Xs (13) has a Q-rational cusp, the method that we use to determine a Q-basis for Ω 1 (Xs (13)) is standard and easy. In Section 5, for K := Q(ζ13 , ζ7 ), we consider the K[GL2 (F13 )]-span of each element in the basis of Ω 1 (Xs (13)K ) and show that it is an irreducible cuspidal representation of GL2 (F13 ). In Section 5 we obtain a concrete description of these

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irreducible cuspidal representations by determining which regular characters of F∗169 correspond to them. In Section 6, by using these cuspidal parameters we obtain a basis for the K-vector space Ω 1 (Xns (13)K ). In Section 7, we use more representation theory to describe how the group Gal(K/Q) acts on specific elements of Ω 1 (Xns (13)K ) in terms of the K-basis from Section 6, and thereby obtain a Q-basis for Ω 1 (Xns (13)). This leads us to a quartic relation which defines Xns (13) in P2Q . In Section 8, we compute j-line maps for our explicit models of Xs (13) and Xns (13) over Q. A less computational proof of the fact that Xs (13) and Xns (13) are isomorphic over Q is given in [2]. We know that the Jacobians of Xs (13) and Xns (13) are isogenous over Q (see [4,5]). In [2] we construct a Q-isomorphism between these Jacobians which respects their canonical principal polarizations. By Torelli’s theorem we conclude that Xs (13) and Xns (13) are isomorphic over Q. This is another way to obtain an equation for Xns (13). But our representation-theoretic method in this paper gives us something that the abstract Torelli’s theorem cannot: the computation of the j-line map Xns (13) → X(1). This is needed to determine if a point in Xns (13)(Q) is CM or not. 2. Cuspidal representations of GL2 (Fp ) In this section we review some properties of irreducible cuspidal representations of GL2 (Fp ). We will use these results in the following sections. Let p > 2 be a prime. The irreducible cuspidal representations of GL2 (Fp ) are all of dimension p − 1 and are parametrized, up to isomorphism, by unordered pairs {θ, θp } of characters θ : F∗p2 → C∗ such that θ = θp . Let πθ denote the representation associated to such a character. In [8, §13] (see especially formulas (3) on p. 35 and (14)–(16) on p. 40, but beware that in (16) there is a sign error) there is constructed an explicit model for πθ on the C-vector space   W = Maps F∗p , C .

(2.1)

This explicit model is well-suited to doing computations (see (3.11)–(3.13) below). The coefficients in the formulas in this explicit model only involve ζp and values of θ so, if K is a field of characteristic zero containing ζp and the image of θ then πθ is realizable over K. Proposition 2.1. Let V be an irreducible cuspidal representation of GL2 (Fp ) associated to a character θ : F∗p2 → C∗ such that θ = θp . Then the following hold. (a) V contains a nonzero Cs (p)-invariant vector if and only if θ = 1 on F∗p . If this happens, then dim V Cs (p) = 1. ∗ (b) V contains a nonzero C+ s (p)-invariant vector if and only if θ = 1 on Fp and ∗ ∗ θ(v) = −1, where v is the unique element of order 2 in Fp2 /Fp . If this happens, +

then dim V Cs (p) = 1.

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Proof. We prove this proposition by using the explicit model for irreducible cuspidal representations given in [8, §13]. The proof of (a) is a direct computation with formula (3) in [8, p. 35]. It suffices to do this computation with one split Cartan subgroup, so we take the standard one (the diagonal matrices in GL2 (Fp )). In this case we see that the functions in the space W (see (2.1)) which are fixed by this subgroup are the constant functions. Hence, the dimension of the space of Cns (p)-invariant vectors is 1. For the proof of (b) we consider the normalizer of the diagonal split Cartan subgroup; this consists of the diagonal and anti-diagonal matrices in GL2 (Fp ). Let f be a nonzero element of W fixed by this subgroup. By (a), the function f is a constant function. But f is also fixed by an anti-diagonal matrix. By using formula (15) in [8, p. 35], this is equivalent to the condition that the equation −p = −

 v



θ−1 (v) + p TrF

/Fp p2

θ−1 (v)

(v)=0

holds, where v runs through coset representatives of F∗p2 /F∗p . Since θ is a nontrivial character, the first sum is zero. The second sum runs over the unique element v of order 2 in the quotient group F∗p2 /F∗p , so we have −p = pθ(v). Hence, θ(v) = −1. Under this condition, the functions in W which are fixed by the normalizer of the standard split Cartan subgroup are the constant functions. Therefore, the dimension of the space of C+ s (p)-fixed vectors is 1. 2 Notation 2.2. In the rest of the paper we fix p and denote the group GL2 (Fp ) by G. It is well-known (see [5, proof of Theorem 2]) that there exists an isomorphism of left Q[G]-modules     Q G/Cs (p)  Q G/Cns (p) ⊕ StG ⊕ StG     + Q G/C+ s (p)  Q G/Cns (p) ⊕ StG

(2.2)

where StG is the Steinberg representation of G. By using these isomorphisms and the above proposition we prove the following corollary. Corollary 2.3. Let V be an irreducible cuspidal representation of G associated to a character θ : F∗p2 → C∗ such that θ = θp . Then the following hold. (a) V contains a nonzero Cns (p)-invariant vector if and only if θ = 1 on F∗p . If this happens, then dim V Cns (p) = 1. (b) V contains a nonzero C+ ns (p)-invariant vector if and only if in addition θ(u) = −1, where u is the unique element of order 2 in F∗p2 /F∗p . If this happens, then +

dim V Cns (p) = 1.

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Proof. Suppose the representation space V is realizable over the characteristic zero field K. By using the isomorphisms (2.2) we have K-linear isomorphisms:     V Cs (p)  HomK[G] K G/Cs (p) , V        HomK[G] K G/Cns (p) , V ⊕ HomK[G] St2G , V ,     + V Cs (p)  HomK[G] K G/C+ s (p) , V      HomK[G] K G/C+ ns (p) , V ⊕ HomK[G] (StG , V ). Since V is an irreducible cuspidal representation, we have HomK[G] (St2G , V ) = 0, so dimK V Cs (p) = dimK V Cns (p)

+

+

dimK V Cs (p) = dimK V Cns (p) .

and

Hence, using Proposition 2.1 the corollary follows. 2 Proposition 2.4. Let (V, θ) be an irreducible cuspidal representation of G over C and let U ⊂ G be the standard upper-triangular unipotent subgroup. For each r ∈ F∗p , let ψr : U → μp be the nontrivial character of U defined by  ψr :

1

c

0

1

 → ζprc

with ζp = e2πi/p . Viewing V as a representation of U , each nontrivial ψr occurs in V with multiplicity 1. In particular, for each of the p −1 characters ψr there is a unique line Vψr ⊂ V on which U acts through ψr . Therefore there is an isomorphism of U -modules: V =



Vψr .

(2.3)

r∈F∗ p

Proof. Since dim V = p − 1 and there are p − 1 such characters, it suffices to show that each Vψr is nonzero. By using the formula for the character θ on U , we compute   1 θ(u)ψr u−1 p u∈U      1 (p − 1) + −ψr u−1 = p

θ, ψr U =

u=1

= 1. This shows that each Vψr is a line in V and hence the proposition follows. 2

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Proposition 2.5. Let V be an irreducible cuspidal representation of G over C. Let Z be the center of G and suppose that Z acts trivially on V . Consider the quotient group Cs (p)/Z 

 1 [a] := 0

0 a

   a ∈ F∗p .

Then the action of Cs (p)/Z permutes the U -eigenlines Vψ in (2.3) via [a]Vψr = Vψar . Proof. In order to prove the proposition, for v ∈ Vψr we need to check that ψar (c)([a].v) for all a ∈ F× p and c ∈ Fp . In GL2 (Fp ) we have 

1 0

c

1  1 ac 

  so 10 1c .([a].v) = [a].( follows. 2

0 1



1

0

0 a



 =

1

0

0 a



1 ac 0

1 c .([a].v) = 01

 ,

1

.v) = [a].(ψar (c)v) = ψar (c)([a].v). Hence the proposition

3. From newforms to cuspidal representations In this section we apply the representation theory that we discussed in the previous section to analyze the G-span of certain cusp forms. Notation 3.1. The preimage under SL2 (Z) → GL2 (Fp ) of the normalizer of a split (resp. non-split) Cartan subgroup of G is denoted by Γs (p) (resp. Γns (p)). Notation 3.2. Let X be a smooth projective curve over Q and F be any field extension of Q. We denote the fiber product X ×Spec Q Spec F by XF . The F -vector space Ω 1 (XF ) of regular 1-forms is naturally identified with Ω 1 (X) ⊗Q F . Consider the modular curve X(p)C with full level-p structure over C. Its analytification

is naturally isomorphic over C to the disjoint union ζ∈μ∗p H∗ /Γ (p). Here, μ∗p is the set of primitive p-th roots of unity in C and H∗ is the completed upper half-plane. By using this isomorphism, we obtain an isomorphism     Ω 1 X(p)C  S2 Γ (p)

(3.1)

ζ∈μ∗ p

between two C-vector spaces: regular 1-forms on X(p)C and the direct sum of copies of the space of weight-2 cusp forms on Γ (p) indexed by ζ ∈ μ∗p . With this isomorphism, we identify these two C-vector spaces. The isomorphism (3.1) is G-equivariant when the right side is identified with the G-representation space IndG SL2 (Fp ) S2 (Γ (p)). Remark 3.3. In X(p)C , we call the connected component in

ζ∈μ∗ p

H∗ /Γ (p) indexed by

ζp = e2πi/p the classical component. We denote this component by X(p)0C , on which the

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fiber over τ ∈ H/Γ (p) is given by (C/[τ, 1], (1/p, τ /p)). Here [τ, 1] := Zτ ⊕ Z is a lattice in C, so that C/[τ, 1] is an elliptic curve and the pair (1/p, τ /p) is a basis for the p-torsion points of this elliptic curve. We have natural quotient maps π1 : X(p)C → Xns (p)C and π2 : X(p)C → Xs (p)C . The restrictions of these quotient maps to X(p)0C are the classical quotient maps between the corresponding quotients of H∗ . Notation 3.4. We denote the identified C-vector spaces in (3.1) by S(p). The action of G on X(p)C induces an action on S(p). This space is the G-representation space inside of which we will do all of our work. Remark 3.5. The action on S(p) by G and by the Hecke operators T for prime = p commute. This comes from the fact that an -isogeny between two elliptic curves induces an isomorphism between p-torsion subgroups for p = . We have natural identifications of C-vector spaces     Ω 1 Xns (p)C = S2 Γns (p)

    and Ω 1 Xs (p)C = S2 Γs (p) .

The injective pullback maps   + Ω 1 Xns (p)C → S(p)Cns (p)

  and Ω 1 Xs (p)C → S(p)Cs (p)

induced from the quotients maps π1 and π2 (in Remark 3.3) are isomorphisms. Thus we have     + S2 Γns (p) = Ω 1 Xns C (p) = S(p)Cns (p) ,     + S2 Γs (p) = Ω 1 Xs C (p) = S(p)Cs (p) .

(3.2) (3.3)

In the rest of this section we focus on the C-vector space S2 (Γs (p)). First, we relate this vector space to a very well known space. In order to do this, for any positive integer N let  wN :=

0

−1

N

0

 .

The matrix wN acts on the space S2 (Γ0 (N )) of weight-2 cusp forms for Γ0 (N ) and it is an involution on that space. Now, fix an odd prime p. Let S2 (Γ0+ (p2 )) be the space of wp2 -invariants in the space S2 (Γ0 (p2 )). We have the C-linear isomorphism       ϕ : S2 Γ0+ p2 −− → S2 Γs (p) , g → g|2 wp .

(3.4)

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This isomorphism is equivariant with respect to the Hecke operators T for prime = p. ∞ Direct calculations give that for g = n=1 an e2πiτ n in S2 (Γ0+ (p2 )) we have

g|2 wp = (1/p)

∞ 

  an e2πiτ n/p ∈ S2 Γs (p) ,

(3.5)

n=1

where τ ∈ H. Proposition 3.6. For f ∈ S2 (Γs (p)) which is the image of a newform in S2 (Γ0+ (p2 )) under the isomorphism (3.4), the C[G]-span Wf of f inside the G-representation space S(p) is an irreducible representation. Proof. Let V be an irreducible subrepresentation of Wf . Choose a subrepresentation H of Wf such that V maps isomorphically onto the quotient Wf /H. The image of f in the quotient Wf /H is nonzero, as otherwise f would be in H and that would force V to be + zero. Thus, using (3.3), we see that V Cs (p) is nonzero. The G-action and the T action on S(p) commute for prime = p (see Remark 3.5). Therefore, for = p the Hecke operator T acts on V via a -scaling where a is the + T -eigenvalue of f . In particular, it also acts on the nonzero V Cs (p) = V ∩ S2 (Γs (p)) with eigenvalue a . The isomorphism ϕ (see (3.4)) is T -equivariant for = p and f = ϕ(g) where g is a newform in S2 (Γ0+ (p2 )) by assumption. Thus, the strong multiplicity one property implies that Cf is the space of all elements of S2 (Γs (p)) on which T acts via + + a -scaling for all = p. Therefore, as V Cs (p) is nonzero it follows that V Cs (p) = Cf . In other words, we have f ∈ V , which implies that Wf = V . This says that Wf is irreducible. 2 Fix f ∈ S2 (Γs (p)) whose inverse image under the isomorphism (3.4) is a newform in S2 (Γ0+ (p2 )). Then its C[G]-span Wf is an irreducible representation by the above proposition. There are four types of irreducible representations of G. In the rest of this section we assume that Wf is cuspidal in the sense of [6, p. 720]. Let K = Q(ζp , ζp+1 ) ⊂ C. From now on, we will work over the field K. One of the motivations for working over K is that the explicit model πθ for an irreducible cuspidal representation can be realized over that number field (see the discussion prior to Proposition 2.1). Consider the K[G]-submodule C+ (p)

Vf generated by f inside Wf . Since dim Vf s = 1, the G-representation space Vf is an absolutely irreducible cuspidal K-linear representation of G and it is a K-structure for Wf , in other words Vf ⊗K C = Wf . Consider the standard upper triangular unipotent subgroup U of G. For each r ∈ F∗p , let ψr : U → μp be the associated nontrivial character defined by  ψr :

1

c

0 1

 → ζprc

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with ζp = e2πi/p . Since the characters of U are valued in K ∗ , by using Proposition 2.4  we have the decomposition Vf = Vf,ψr into U -lines over K for the nontrivial r∈F∗ p characters ψr of U . There is a corresponding expression f=



fψr

(3.6)

r

where fψr is the ψr -isotypic component of f ; that is, fψr is the projection of f into Vf,ψr . There is a representation-theoretic relationship among the isotypic components fψr of f . To see this, define  [a] :=

1

0

 ∈ GL2 (Fp )

0 a

(3.7)

for any a ∈ F∗p . Since f ∈ S2 (Γs (p)), we have f = [a].f , which gives us the equality p−1 

fψr

p−1  = [a].fψr .

r=1

(3.8)

r=1

As the center Z acts trivially on Vf , by Proposition 2.5 it follows that [a].fψr = fψar

(3.9)

for all a, r ∈ F∗p . Some fψr is nonzero since f = 0, and it is important to observe that all fψr are nonzero. In particular, every fψr is a basis of the line Vf,ψr . In other words, from f (which is unique up to scaling), we get an entire collection (fψr )r of nonzero vectors in Vf which (i) constitute a K-basis of Vf , and (ii) are well-defined up to a common K ∗ -scaling factor. The non-vanishing of all fψr is due to the fact that they constitute a single Cs -orbit in the representation space, by (3.9). In the following proposition we describe q-expansions of fψr ’s concretely depending on the q-expansion of f for q = e2πiτ /p . Before doing this we make the following remark which we will use in the proof of Proposition 3.8.   ∞ Remark 3.7. For any u = 10 1c ∈ G with c ∈ Fp , the action of u on g = n=1 an × e2πiτ n/p ∈ S2 (Γs (p)) ⊂ S(p) is given on the restriction to the classical component X(p)0C by u.g = g(z +  c) =

∞  n=1

for ζp := e2πi/p and any  c ∈ Z lifting c ∈ Fp .

an ζpnc e2πinz/p

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Proposition 3.8. Let U be the standard upper triangular unipotent subgroup of G. For each r ∈ F∗p , let ψr : U → μp be the associated nontrivial character defined by  ψr :

1

c

0

1

 → ζprc

 with ζp = e2πi/p . Let f = an q n ∈ S2 (Γs (p)) with q = e2πiτ /p and let V be the K[G]-span of f in S(p), where K = Q(ζp , ζp+1 ). The ψr -isotypic component fψr of f in V (as in (3.6)) has q-expansion on the “classical” component of X(p)K at ∞ given by 

an q n .

n≡r mod p

  Proof. Define the isomorphism Fp  U via c → u(c) = 10 1c . In order to prove the proposition, we use the standard formula for projection to an isotypic component. For the U -representation space V , the projection of f into the ψr -isotypic factor is 1  −rc 1   −1  ψr u u.f = ζp u(c).f. p p u∈U

Remark 3.7 gives u(c).f = (3.10) yields the q-expansion fψr =

(3.10)

c∈Fp



an ζpnc q n on the classical component X(p)0K . Hence,

  1  1  −rc  ζp an ζpnc q n = ζp(n−r)c an q n p p n n c∈Fp

c∈Fp

by switching the order of summation. The inner sum vanishes when n ≡ r mod p and is 1 when n ≡ r mod p. Hence, the proposition follows. 2 Now we want to determine a concrete description of Vf as an irreducible cuspidal representation space. We have Vf  πθ for a regular character θ of F∗p2 that is trivial on F∗p , by Proposition 2.1(a). Hence we consider the explicit model for πθ on W = Maps(F∗p , C) in [8, §13] that we talked about in Section 2. The formulas (3) in [8, p. 35] and (14)–(16) in [8, p. 40] that describe the action of πθ (g) on W for each g ∈ GL2 (Fp ) in terms of the character θ come out as follows. Let ζp = e2πi/p ∈ C∗ . For any function h : F∗p → C in W and any  g=

α

β

γ

δ

 ∈ GL2 (Fp ),

if γ = 0 then we have   (g.h)(r) = ζp(β/δ)r h (α/δ)r

(3.11)

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(see [8, (3), p. 35]) and if γ = 0 then we have (g.h)(r) =



k(r, s; g)h(s)

(3.12)

s∈F∗ p

where 

(αr+δs)/γ

k(r, s; g) = −

ζp

p

ζp−(s/γ)(u+¯u) θ(u)

(3.13)

u¯ u=(r/s) det(g)

with u varying through F∗p2 and u ¯ := up (see [8, (15), (16); p. 40] but beware that in (16) there is a sign error). The following example is crucial.  0 1 Example 3.9. Consider w = −1 , and for a ∈ F∗p let δa : F∗p → C be the function which 0 carries a to 1 and vanishes elsewhere. Then (w.δa )(r) = k(r, a; w) = −

1  a(u+¯u) ζp θ(u). p u¯ u=r/a

In other words, 

w.δa = −

 ζpa(u+¯u) θ(u) δr . p

r∈F∗ u=r/a p u¯

Remark 3.10. Suppose the central character θ|Z of πθ is trivial (which is the case for Vf ). Thus, πθ is invariant under F∗p -scaling on GL2 (Fp ). In particular, for computations with πθ it suffices to consider (3.11) with δ = 1 (and γ = 0) and (3.13) with γ = 1. In these cases the formulas look a bit simpler:  

where for g =

α β 1 δ

α

β

0

1

α

β

1

δ

 .h : r → ζpβr h(αr),  .h : r →



k(r, s; g)h(s),

s∈F∗ p

we have

k(r, s; g) = −

ζpαr+δs p



ζp−s(u+¯u) θ(u).

u¯ u=(r/s) det(g)

Choose θ so that πθ  Vf . The following proposition which gives us a concrete description of Vf as a representation space, is the key to everything.

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Proposition 3.11. Let Vf be as above. There is an isomorphism of representation spaces πθ  Vf as K[G]-modules via h →



h(r)fψr

r∈F∗ p

where fψr is the isotypic component of f . In particular, δr → fψr for all r = 0, where δr (s) is 1 for s = r and is 0 otherwise. Proof. Let T : πθ  Vf be an abstract K[G]-linear isomorphism. We will deduce that a K ∗ -multiple of T is given by the proposed formula in the proposition. In Vf , the vector fψr spans the ψr -isotypic line for each r ∈ F∗p . Likewise, by (3.11), we have u(c).δr = ψ(cr)δr = ψr (c)δr for all c ∈ Fp , so δr spans the ψr -isotypic line in πθ . Hence, for each r ∈ F∗p we have T (δr ) = a(r)fψr for some nonzero a(r). Then for  h = r∈F∗ h(r)δr , we have p

T (h) =



h(r)a(r)fψr .

r∈F∗ p

Using Remark 3.10, we see that ([r].h)(s) = h(sr) for any h ∈ πθ and any s ∈ F∗p (for the definition of [r], see (3.7)). Thus, taking h = δ1 gives [r].δ1 = δr for all r ∈ F∗p . Now we may replace T with (1/a(1))T , so T (δ1 ) = fψ1 and we aim to prove a(r) = 1 for all r ∈ F∗p . This follows from (3.9) and the equivariance of T :   fψr = [r].fψ1 = [r].T (δ1 ) = T [r].δ1 = T (δr ) = a(r)fψr .

2

4. Eigenforms for Γs (13) and an equation for Xs (13) In the rest of the paper we specialize to the case p = 13. Thus we use the following notation. Notation 4.1. We denote the non-split Cartan subgroup Cns (13), its normalizer C+ ns (13), + the diagonal split Cartan subgroup Cs (13), and its normalizer C+ s (13) by Cns , Cns , Cs , + and Cs respectively. Also, we denote the modular curves Xns (13) and Xs (13) by Xns and Xs respectively. In this section we compute an equation for the modular curve Xs over Q in P2 . We do this by computing a basis for S2 (Γs (13)). For our purposes the standard parameter e2πiτ will not be used, so we instead define the parameter q := e2πiτ /13 for τ in H, as this will be used very often. Consider the isomorphism ϕ in (3.4) for p = 13. Since S2 (Γ0 (13)) = 0, the 3-dimensional space S2 (Γ0+ (169)) has a basis consisting of newforms. By using Sage,

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we explicitly determine the q 13 -expansion at ∞ for the basis {g1 , g2 , g3 } of S2 (Γ0+ (169)) consisting of newforms. Thus, the elements fi := 13 · ϕ(gi )

for i = 1, 2, 3,

(4.1)

form a basis for S2 (Γs (13)). By (3.5) for p = 13, the n-th coefficient of the q-expansion of fi at ∞ is the n-th Hecke eigenvalue of gi . For f1 we have       f1 = q + α1 q 2 + −α12 − 2α1 q 3 + α12 − 2 q 4 + α12 + 2α1 − 2 q 5 + (−α1 − 1)q 6       + α12 − 3 q 7 + −2α12 − 3α1 + 1 q 8 + α12 + 3α1 − 1 q 9 + (−α1 + 1)q 10       + −α12 − 2α1 − 2 q 11 + α12 + 3α1 q 12 + −2α12 − 2α1 + 1 q 14     + α12 + α1 − 2 q 15 + O q 16 , where α1 is a root of the polynomial x3 + 2x2 − x − 1. The number field Q(ζ7 )+ is the splitting field of x3 +2x2 −x −1. The q-expansions of f2 and f3 at ∞ are the Q-conjugates of the q-expansion of f1 . Since ∞ is a Q-point of Xs , the above shows that all q-expansion coefficients of fi lie in Q(ζ7 )+ . Thus, the 1-forms fi lie in the three dimensional space Ω 1 (Xs Q(ζ7 )+ ) and are a basis over Q(ζ7 )+ . Moreover, {f1 , f2 , f3 } constitutes an orbit under the action of Gal(Q(ζ7 )+ /Q) on Ω 1 (Xs Q(ζ7 )+ ). Let α2 and α3 be the other two roots of the polynomial x3 + 2x2 − x − 1, corresponding to f2 and f3 respectively. Let f1 := TrQ(ζ7 )+ /Q (f1 ) = f1 + f2 + f3 , f2 := TrQ(ζ7 )+ /Q (α1 f1 ) = α1 f1 + α2 f2 + α3 f3 ,   f3 := TrQ(ζ7 )+ /Q α12 f1 = α12 f1 + α22 f2 + α32 f3 . The set {f1 , f2 , f3 } is a Q-basis for the Q-vector space Ω 1 (Xs ). Now we consider the canonical map for Xs . Since the curve Xs is a genus-3 curve, {f1 , f2 , f3 } either satisfies a nonzero quadratic relation over Q or it does not satisfy a nonzero quadratic relation but it satisfies a nonzero quartic relation over Q. By using Magma, we first check if the elements {f1 , f2 , f3 } satisfy a nonzero quadratic relation. By Riemann–Roch, it is enough to consider the q-expansions of the fi up to degree 9, as the degree of the canonical divisor of Xs is 4. We see that they do not satisfy a nonzero quadratic relation and hence Xs is not a hyperelliptic curve. Thus, Xs is defined by a quartic in P2Q . Next, again by using Magma we seek a quartic relation over Q satisfied by the elements {f1 , f2 , f3 }. Using the q-expansions up to degree 17, we find that the following equation defines Xs in P2Q : 6x4 − 133x3 y − 82x3 z + 343x2 y 2 + 434x2 yz + 135x2 z 2 − 119xy 3 − 322xy 2 z − 252xyz 2 − 59xz 3 − 51y 4 − 61y 3 z + y 2 z 2 + 22yz 3 + 6z 4 = 0.

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The linear change of coordinates with the matrix ⎛

2

⎜ ⎝ −6 11

1 4 −5

1

⎞

⎟ −3 ⎠ 2

yields the following simpler equation for Xs in P2Q :       (−y − z)x3 + 2y 2 + zy x2 + −y 3 + zy 2 − 2z 2 y + z 3 x + 2z 2 y 2 − 3z 3 y = 0.

(4.2)

5. Determining the character θ Consider the basis {f1 , f2 , f3 } of S2 (Γs (13)) that we obtained in the previous section. They are the images (up to scaling by 13) of newforms of S2 (Γ0+ (169)) under ϕ (see (4.1)). In this section we associate a G-representation space to each fi as we did in Section 3 and determine these spaces explicitly. We will do this by applying the representation-theoretic results in Sections 2 and 3 to the representation space that we will associate to f . From now on, assume that f is one of the basis elements {f1 , f2 , f3 } of S2 (Γs (13)). In Section 3 we did our work over the field K = Q(ζp , ζp+1 ). Since now we are focusing on p = 13, from now on we consider the field Q(ζ13 , ζ14 ) = Q(ζ13 , ζ7 ). Notation 5.1. From now on we denote the field Q(ζ13 , ζ7 ) that we will work over by K. We embed S2 (Γs (13)) into the representation space S(13) via (3.3) for p = 13. We denote the K[G]-span of f inside S(13) by Vf . Proposition 5.2. Let f be one of the basis elements {f1 , f2 , f3 } of S2 (Γs (13)). Consider the K[G]-span Vf of f inside S(13). The representation space Vf is an irreducible cuspidal representation space. Proof. In Proposition 3.6 we showed that Vf ⊗K C is an irreducible representation. This implies that Vf itself is also irreducible. It remains to show that Vf is cuspidal. Suppose it is not. If it is 1-dimensional, it is fixed by SL2 (F13 ). This is impossible since the genus of X(1) is zero. By the classification of irreducible GL2 (Fp )-representations, Vf is therefore contained in a representation that is induced from a 1-dimensional character of a Borel subgroup. It follows that a twist of Vf by a character of F∗13 has a non-zero Γ1 (13)-fixed point. This means that f is a twist of an eigenform in S2 (Γ1 (13)) by a character of F∗13 . (I thank Bas Edixhoven for this argument.) The space of cuspforms for Γ1 (13) has dimension 2. In William Stein’s modular forms database tables [11], we find that these eigenforms have coefficients in Q(ζ3 ). If we twist them with a character of F∗13 , then we get an eigenform with coefficients in Q(ζ12 ). But f has coefficients in Q(ζ7 ). Thus the representation space Vf is an irreducible cuspidal representation. 2

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Now by Proposition 5.2 we know that Vf is an irreducible cuspidal representation. As an application of Proposition 3.11, we shall determine a character θ : F∗169 → K ∗ that satisfies πθ  Vf . Notation 5.3. To simplify the notation, define fr := fψr . The explicit description of fr is given in Proposition 3.8 and we know the q-expansion  0 1 ∈ an q n of f from Section 4. By Example 3.9 and Proposition 3.11, for w = −1 0 SL2 (F13 ) we have



w.f1 = −

 1  u+¯ u ζ13 θ(u) fr 13 u¯u=r ∗

(5.1)

r∈F13

where u varies through F∗169 and u ¯ = u13 . Here, ζ13 is as before e2πi/13 . This equality is true on the entire (disconnected) curve X(13)K . The operators [a] for a ∈ F∗13 (see (3.7) for the definition of [a]) permute the connected components of X(13)K transitively, so the identity (5.1) on the entirety of X(13)K is equivalent to the restriction to the “classical” component of the same identity after applying [a] for every a ∈ F∗13 . It is easy to verify that we have  [a]w =

a

0

0

a

 w[1/a].

Thus, since the center of GL2 (F13 ) acts trivially on Vf , we have [a].w.f1 = w.[1/a].f1 =  w.f1/a . Applying [a] to the right side of (5.1) yields   1    1  (u+¯u)/a u+¯ u  ζ θ(u) far = ζ θ(u) fr 13 u¯u=r 13 13 u¯u=ar 13 ∗ ∗

r∈F13

r∈F13

by using the change of variables u  au and r  ar. By renaming 1/a as a, this shows that the single identity (5.1) on the disconnected space X(13)K is equivalent to the collection of identities    1  a(u+¯u)  w.fa = − ζ13 θ(u) fr 13 ∗ r∈F13

u¯ u=r/a

on the connected “classical” component for all a ∈ F∗13 . Since det w = 1, on the “classical” component we have w.fa = fa |2 w = z −2 fa (−1/z). We consider such identities on the imaginary axis of H, so if we define Fr (y) := fr (iy) =

 n≡r mod 13

an e−2πny/13

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for y > 0 then (w.fa )(iy) = −y −2 fa (i/y) = −y −2 Fa (1/y). Hence the identity (5.1) on X(13)K is equivalent to the collection of identities −y −2 Fa (1/y) = −

  1  a(u+¯u) ζ13 θ(u) Fr (y) 13 r u¯ u=r/a

=−

1  a(u+¯u) ζ13 θ(u)Fau¯u (y) 13 ∗

(5.2)

u∈F169

for all a ∈ F∗13 . By using this identity, we shall do a numerical computation with f =  an q n to find out which θ satisfies πθ  Vf . The representation space Vf contains the C+ s (13)-invariant vector f (see (3.3)). Thus, ∗ by Proposition 2.1(b) we have θ = 1 on F13 and θ(u) = −1 for the unique element u of order 2 in F∗169 /F∗13 . In other words, the character θ = θ13 corresponding to the cuspidal representation Vf has exact order 14. Thus, there are three unordered pairs {θ, θ−1 } that can satisfy the isomorphism πθ  Vf . √ Let u be the generator 2 + 2 for F∗169 . The image of u under these three θ’s is exactly one of −ζ7±1 or −ζ7±2 or −ζ7±3 . For each f ∈ {f1 , f2 , f3 } we numerically test the identities in (5.2) for a = 1 using these three θ’s and some y > 0. We use Magma to do this. In these computations we took α1 = (ζ72 + ζ7−2 )−1 , α2 = (ζ7 + ζ7−1 )−1 , α3 = (ζ73 + ζ7−3 )−1 (see the q-expansions of fi in Section 4). We find that the pair {θ1 , θ1−1 } such that θ1 (u) = −ζ73 corresponds to Vf1 , the pair {θ2 , θ2−1 } such that θ2 (u) = −ζ72 corresponds to Vf2 , and the pair {θ3 , θ3−1 } such that θ3 (u) = −ζ7 corresponds to Vf3 . Hence, we have Vf 1  πθ 1 ,

Vf2  πθ2 ,

Vf3  πθ3

(5.3)

where πθi is the irreducible cuspidal representation corresponding to the pair of characters {θi , θi−1 }. 6. Determination of the q-expansion of a K-basis of Ω 1 (Xns K ) In this section we will determine the q-expansion at ∞ ∈ Xns (Q(ζ13 )+ ) for a basis of the K-vector space Ω 1 (Xns K ). This is the first step for us to determine a basis for the Q-vector space Ω 1 (Xns ) in order to write an equation for Xns over Q. Akin to (3.2) over C, the projection map π1 : X(13)K → Xns K induces an isomor+ phism Ω 1 (Xns K )  Ω 1 (X(13)K )Cns . We identify these spaces via this isomorphism. The K[G]-representation space Ω 1 (X(13)K ) decomposes into a direct sum of subrepresentation spaces   Ω 1 X(13)K = Vfe11 ⊕ Vfe22 ⊕ Vfe22 ⊕ W

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for some positive integers e1 , e2 , e3 and some subrepresentation W containing no copies of the Vfi . Thus, we have  C+  C+ e1  C+ e2  C+  e2 + Ω 1 (Xns K ) = Ω 1 X(13)K ns = Vf1 ns ⊕ Vf2 ns ⊕ Vf2 ns ⊕ W Cns . C+

The curve Xns K has genus 3 and by Corollary 2.3(b) each Vfi ns is one-dimensional. Thus, we have C+

C+

C+

Ω 1 (Xns K ) = Vf1 ns ⊕ Vf2 ns ⊕ Vf3 ns . For i = 1, 2, 3 we will compute the q-expansion at ∞ of a nonzero element hi of the C+ C+ C /Z 1-dimensional space Vfi ns . By Corollary 2.3 we have Vfi ns = VfCi ns = Vfi ns where Z is the center of GL2 (F13 ). We define a projection map C /Z

Π : Vfi → Vfi ns  v → t.v t∈Cns /Z

In the previous section we calculated the character θi corresponding to the irreducible cuspidal representation Vfi (see (5.3)). In Section 3 we also gave an explicit K[G]-linear isomorphism between Vfi and πθi (Proposition 3.11), where πθi is an explicit model over K for the cuspidal irreducible representation corresponding to θi (see (3.11) and (3.12)). Hence, for any v ∈ Vfi and any t ∈ Cns /Z we can compute the action t.v explicitly. For any v, the projection Π(v) lies in Ω 1 (Xns K ). In Proposition 6.1 we will compute Π(fi,r ) for every r, where fi,r is the ψr -isotypic component of fi . Since Π is a nonzero K-linear map and the fi,r ’s span Vfi over K, we know in advance that Π(fi,r ) is nonzero C+

for some r. That must then be a basis of the line Vfi ns . Now we give an explicit description of the classes in Cns /Z. We have the non-square element 2 ∈ F∗13 , and  m :=

0 2

 ∈ Mat2 (F13 )

1 0

satisfies m2 = 2I. Hence,

 F13 ⊕ F13 · m =

a 2b b

a

   a, b ∈ F13

is a copy of F169 as an F13 -subalgebra of Mat2 (F13 ). Thus, its subset of nonzero elements

 Cns :=

a 2b b

a

    = F∗169 ⊂ GL2 (F13 )  (a, b) ∈ F213 − (0, 0)

(6.1)

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291

is an explicit non-split Cartan subgroup. The non-identity classes in Cns /Z are uniquely represented by the matrices  ga :=

a

2

1

a



for a ∈ F13 . Hence, we have Π(v) = v +



ga .v

a∈F13

for any v ∈ Vfi . Proposition 6.1. Let f be one of {f1 , f2 , f3 }. Consider the basis {fr }r∈F∗13 of Vf , where fr is the ψr -isotypic component of f . Let θ be the character corresponding to the irreducible C+ cuspidal representation Vf . Then each Π(fr ) ∈ Vf ns is given by     Π fr = fr − s∈F13

a∈F13





a(r+s)

u¯ u=(s/r)(a2 −2)

ζ13 13

·

−r(u+¯ u) ζ13 θ(u)

fs .

(6.2)

Proof. Using Proposition 3.11 and the formula in (3.12), we claim that ga .fr =



k(s, r; ga )fs .

s∈F∗ 13

Indeed, since the identification of Vf with πθ in Proposition 3.11 carries fr to δr , it is equivalent to prove the identity ga .δr =



k(s, r; ga )δs

s∈F∗ 13

as functions from F∗13 to C∗ . Evaluating both sides at an arbitrary c ∈ F∗13 converts this into verifying if the equality (ga .δr )(c) = k(c, r; ga ) holds for all r, c ∈ F∗13 or not. By (3.12), (ga .δr )(c) =



    k c, c ; ga δr c = k(c, r; ga ),

c ∈F∗ 13

as desired. We conclude that      Π fr = fr + ga .fr = fr + k(s, r; ga )fs , a∈F13

a∈F13 s∈F∗ 13

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where (by the final displayed expression in Remark 3.10 and the definition of ga ) 

a(s+r)

k(s, r; ga ) = −

ζ13 13

−r(u+¯ u)

ζ13

θ(u).

u¯ u=(s/r)(a2 −2)

Rearranging the order of summation, we get the desired formula for Π(fr ).

2

By using Magma, we use Proposition 6.1 to compute q-expansions at ∞ for Π(fi,1 ) ∈ VfCi ns , where fi,1 is the ψ1 -isotypic component of fi . We see that they are all nonzero. Notation 6.2. We denote Π(f1,1 ), Π(f2,1 ) and Π(f3,1 ) by h1 , h2 , and h3 respectively. C+

Since each hi is a K-basis for the line Vfi ns , the elements h1 , h2 and h3 form a K-basis for Ω 1 (Xns K ). 7. Determination of the q-expansion of a Q -basis of Ω 1 (Xns ) In the previous section we computed a K-basis {h1 , h2 , h3 } for the K-vector space Ω 1 (Xns K ). With this basis, we can compute a quartic planar model for Xns over K. But our main interest is to find a quartic planar model for Xns over Q. This amounts to computing a Q-basis of Ω 1 (Xns ) inside the K-vector space Ω 1 (Xns K ). In this section we do this in terms of the q-expansions at the point ∞ that is a Q(ζ13 )+ -rational point of Xns . We define a surjective Q-linear trace map by T : Ω 1 (Xns K ) → Ω 1 (Xns ),  ω → γ.ω. γ∈Gal(K/Q)

In order to compute T on a Q-basis of Ω 1 (Xns K ) at the level of q-expansions at ∞, we need to know the action of Gal(K/Q) on Ω 1 (Xns K ) at the level of q-expansions at ∞. Since ∞ is not a Q-rational point of Xns , this is not given by acting on the coefficients of q-expansions. We will first determine this action. Once this is done, then using T we will obtain three Q-linearly independent q-expansions in Ω 1 (Xns ). This will be a Q-basis for Ω 1 (Xns ), from which we can determine a quartic planar model for Xns over Q. The Gal(K/Q)-action on Ω 1 (Xns K ) is semi-linear over the K-linear structure. Thus, computing this action amounts to computing it on the K-basis elements h1 , h2 and h3 at the level of q-expansions at ∞. Since ∞ ∈ Xns (Q(ζ13 )+ ), the action of the subgroup Gal(K/Q(ζ13 )) is given by acting on the coefficients on q-expansions of hi . Hence the decomposition     Gal(K/Q) = Gal K/Q(ζ13 ) × Gal K/Q(ζ7 )

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reduces our problem to giving a q-expansion description of the action of Gal(K/Q(ζ7 )) = Gal(Q(ζ13 )/Q) on the hi ’s. The representation-theoretic construction of hi will be the crucial tool in determination of how Gal(K/Q(ζ7 )) acts on it. Let U be the upper triangular unipotent subgroup of G. Recall that for r ∈ F∗13 the   jr characters ψr : U → μ13 are given by 10 1j → ζ13 , and fi,r denotes the ψr -isotypic component of fi ∈ Ω 1 (X(13)K ) (see (3.6)). By Proposition 3.8, the 1-form fi,r has q-expansion 

ai,n q n

n≡r mod 13

 + n at ∞ ∈ X(13)(Q(ζ13 )) where fi = n ai,n q . A formula for hi = Π(fi,1 ) is given by (6.2). Consider the Gal(K/Q)-equivariant embedding Ω 1 (Xns K ) → Ω 1 (X(13)K ). We will compute γ.hi for each γ ∈ Gal(K/Q(ζ7 )) by using the G-action on Ω 1 (X(13)K ). By (6.2), this is reduced to the computation of γ.fi,r for every r ∈ F∗13 and γ ∈ Gal(K/Q(ζ7 )). We do this in the following proposition. Proposition 7.1. For γ ∈ Gal(K/Q(ζ7 )) = Gal(Q(ζ13 )/Q), let nγ ∈ (Z/13Z)∗ be such that the action of γ on μ13 is given by the nγ -power map. Then we have γ.fi,r = fi,nγ r . Proof. Consider the K[G]-span Vfi of fi in Ω 1 (X(13)K ). The GL2 (F13 )-action and Gal(K/Q)-action on Ω 1 (X(13)K ) commute and the action of Gal(K/Q(ζ7 )) leaves fi invariant since fi ∈ Ω 1 (Xs Q(ζ7 )+ ). Thus, the action of Gal(K/Q(ζ7 )) on Ω 1 (X(13)K ) preserves Vfi . Consider the decomposition

Vf i =

12

Kfi,r

r=1

of Vfi into its basis of ψr -isotypic lines for r ∈ F∗13 . For v ∈ Kfi,r we have for every j ∈ F∗13 . We have 

1

j



0 1

 1 .(γ.v) = γ. 0

j 1

1 j  01

jr .v = ζ13 v

   jr   jr  n jr .v = γ. ζ13 v = γ.ζ13 (γ.v) = ζ13γ (γ.v).

Hence, it follows that γ.Kfi,r = Kfi,nγ r . The identity γ.fi = fi for γ ∈ Gal(K/Q(ζ7 )) therefore implies that γ.fi,r is the ψrnγ -isotypic component of fi . Thus, we have γ.fi,r = fi,nγ r as desired. 2

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Now we know via (6.2) how the group Gal(K/Q) acts on each hi . By using Magma, we thereby obtain a Q-basis {b1 , b2 , b3 } for Ω 1 (Xns ) where bi = T (hi ). The first three q-expansion coefficients of bi are as below:  11 10 9 8 7 6 5 4 3 13b1 = −36ζ13 − 76ζ13 + 40ζ13 − 88ζ13 + 36ζ13 + 36ζ13 − 88ζ13 + 40ζ13 − 76ζ13   2 11 10 9 8 7 6 5 − 36ζ13 + 144 q + −284ζ13 + 36ζ13 − 8ζ13 − 180ζ13 − 28ζ13 − 28ζ13 − 180ζ13   4 3 2 11 10 9 8 7 − 8ζ13 + 36ζ13 − 284ζ13 − 268 q 2 + 104ζ13 + 120ζ13 + 124ζ13 − 100ζ13 + 32ζ13    6 5 4 3 2 + 32ζ13 − 100ζ13 + 124ζ13 + 120ζ13 + 104ζ13 − 144 q 3 + O q 4 ,  11 10 9 8 7 6 5 4 3 13b2 = −24ζ13 + 66ζ13 + 36ζ13 + 16ζ13 − 46ζ13 − 46ζ13 + 16ζ13 + 36ζ13 + 66ζ13   2 11 10 9 8 7 6 5 − 24ζ13 − 44 q + 100ζ13 + 24ζ13 + 32ζ13 + 20ζ13 + 14ζ13 + 14ζ13 + 20ζ13   4 3 2 11 10 9 8 7 + 32ζ13 + 24ζ13 + 100ζ13 + 36 q 2 + −80ζ13 − 144ζ13 − 62ζ13 + 36ζ13 − 58ζ13    6 5 4 3 2 − 58ζ13 + 36ζ13 − 62ζ13 − 144ζ13 − 80ζ13 + 44 q 3 + O q 4 ,  11 10 9 8 7 6 5 4 3 2 13b3 = 58ζ13 + 54ζ13 + 32ζ13 + 64ζ13 + 68ζ13 + 68ζ13 + 64ζ13 + 32ζ13 + 54ζ13 + 58ζ13   11 10 9 8 7 6 5 4 + 20 q + 50ζ13 − 58ζ13 − 26ζ13 + 38ζ13 − 42ζ13 − 42ζ13 + 38ζ13 − 26ζ13   3 2 11 10 9 8 7 6 − 58ζ13 + 50ζ13 + 102 q 2 + −12ζ13 + 40ζ13 + 4ζ13 − 52ζ13 − 22ζ13 − 22ζ13    5 4 3 2 − 52ζ13 + 4ζ13 + 40ζ13 − 12ζ13 − 20 q 3 + O q 4 . By using Magma, we find that bi ’s do not satisfy a nonzero quadratic relation (so Xns is not hyperelliptic). Thus we compute the quartic equation 503x4 − 1451x3 y + 6253x3 z − 11 212x2 y 2 − 6584x2 yz + 20 638x2 z 2 − 14 760xy 3 − 65 676xy 2 z − 17 036xyz 2 + 27 500xz 3 − 5240y 4 − 50 776y 3 z − 75 344y 2 z 2 + 5672yz 3 + 10 440z 4 = 0 which defines Xns in P2Q . Making a change of variables to the coordinates of Xns with the matrix ⎛

20

⎜ M := ⎝ −3 −5

−18 10

⎞

9

⎟ 2 ⎠

1

1

(7.1)

we find the following simpler equation for Xns ,       (−y − z)x3 + 2y 2 + zy x2 + −y 3 + zy 2 − 2z 2 y + z 3 x + 2z 2 y 2 − 3z 3 y = 0. This is the same as Eq. (4.2) that defines Xs in P2Q !

(7.2)

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8. Computing j-maps In this section we briefly describe how we compute j-maps for our explicit models for Xns and Xs over Q. The explicit equation that we use to compute j-maps for the Q-isomorphic curves Xns and Xs is (7.2) (or equivalently (4.2)). In order to compute the j-map for Xns , consider the following commutative diagram. α

H∗ /Γns (13)

Xns j -map

γ β

H∗ /SL2 (Z)

P1

In terms of the notation of the previous section, the map α is the isomorphism given by α(z) = [d1 (z) : d2 (z) : d3 (z)] where ⎛

b1 (z)

⎞

⎛

d1 (z)

⎞

⎟ ⎜ ⎟ ⎜ M −1 ⎝ b2 (z) ⎠ = ⎝ d2 (z) ⎠ . b3 (z)

d3 (z)

We computed q-expansions of bi ’s in the previous section and the matrix M is given in (7.1). Recall that q = e2πiz/13 . The map β is also an isomorphism. For z ∈ H∗ /SL2 (Z), the image β(z) is given by the usual j-function at z, for which the first few coefficients of its q 13 -expansion are   1 + 744 + 196 884q 13 + 21 493 760q 26 + 864 299 970q 39 + O q 52 . q 13 The map γ is the natural projection map. Our aim is to compute the j-map. Since the index [SL2 (Z) : Γns (13)] is 78, the degree of the j-map is 78. Let {τi } be coset representatives for Γns (13) in SL2 (Z). Consider ρ := e2πi/3 and ∞ ∈ H∗ /SL2 (Z). We have β(ρ) = 0 and β(∞) = ∞. Thus, the zeros of the j-map on Xns are α(τi ρ) and the poles of the j-map on Xns are α(τi ∞). By using PARI and Magma, we do a numerical calculation and compute all α(τi ρ) and α(τi ∞). In order to do this, we use 5000 coefficients of the q-expansions of the di ’s. Next, by using Magma we compute the divisor D1 corresponding to the zeros of the j-map. The divisor D1 is of the form D1 = 3(P1 + P2 + P3 + P4 + P5 + P6 + Q) where each Pi is a closed point of degree 4 and Q is a closed point of degree 2. Similarly, we compute the divisor D2 corresponding to the poles of the j-map. It has the form D2 = 13R

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where R is a closed point of degree 6. Then again by using Magma, we compute a basis for the 1-dimensional Riemann–Roch space L(D2 − D1 ) which contains the j-map as an element. Any basis that we compute gives us the j-map up to a constant. Since we know the q-expansions of di and the q 13 -expansion of their image under j-map, we can easily obtain that constant and hence the j-map. This explicit j-map is given in Appendix A. A similar calculation for the modular curve Xs yields the explicit j-map for this curve. This time the j-map has degree 91. The explicit j-map for Xs is given in Appendix A. These explicit j-maps for Xns and Xs recover the expected j-invariants of CM elliptic curves at the known rational points (see Table 1.1). Acknowledgments I thank Brian Conrad, David Kohel, and René Schoof for several useful remarks, observations and suggestions. I also thank Stanford University for postdoctoral support and the Michigan Society of Fellows at the University of Michigan for the generous funding and stimulating working environment they provided for me. Appendix A. Explicit j-line maps The j-line map Xns → X(1) is given below. Let {x, y, z} be the homogeneous coordinates in P2Q where Xns is embedded as in (7.2). Using X = x/z and Y = y/z, the j-map is: j=

f (X, Y ) (Y + 1)24 ((−Y − 2)X 2 + (Y 2 + 5Y )X + (−Y 2 + Y ))13

where f (X, Y ) is given by 

884 736Y 55 + 49 489 920Y 54 + 1 012 211 712Y 53 + 11 898 450 496Y 52 + 95 148 841 568Y 51 + 564 433 986 856Y 50 + 2 616 961 812 219Y 49 + 9 822 086 693 987Y 48 + 30 642 386 133 997Y 47 + 81 359 505 328 755Y 46 + 188 611 532 400 778Y 45 + 392 863 091 096 565Y 44 + 750 115 492 349 610Y 43 + 1 285 021 297 306 328Y 42 + 1 751 719 154 883 360Y 41 + 1 150 764 689 309 236Y 40 − 2 039 149 007 306 334Y 39 − 5 717 108 163 092 124Y 38 + 7 099 555 968 682 235Y 37 + 77 776 536 294 315 788Y 36 + 240 934 894 546 612 597Y 35 + 395 125 111 510 297 175Y 34 + 89 165 142 545 864 768Y 33 − 1 550 321 042 833 254 477Y 32 − 5 305 245 647 359 351 740Y 31 − 10 446 908 162 336 304 678Y 30 − 12 871 375 409 731 491 993Y 29 − 4 395 668 686 472 870 357Y 28 + 24 906 193 210 193 909 056Y 27

B. Baran / Journal of Number Theory 145 (2014) 273–300

+ 81 237 769 117 029 166 717Y 26 + 161 268 611 204 135 399 411Y 25 + 249 955 252 927 207 983 657Y 24 + 324 483 552 163 739 360 618Y 23 + 363 683 177 940 575 393 974Y 22 + 358 040 773 388 507 611 446Y 21 + 313 163 052 354 472 751 348Y 20 + 244 627 586 482 873 074 382Y 19 + 170 321 443 968 359 565 942Y 18 + 105 533 439 538 375 007 957Y 17 + 59 274 776 728 757 700 369Y 16 + 31 575 303 637 834 908 430Y 15 + 16 108 641 614 130 173 307Y 14 + 6 982 396 840 373 322 235Y 13 + 1 946 153 985 121 206 304Y 12 + 125 999 715 600 544 888Y 11 − 17 233 649 727 426 000Y 10 + 97 243 737 410 324 752Y 9 + 60 793 442 160 963 072Y 8 + 4 429 798 119 424 768Y 7 − 5 098 133 689 396 992Y 6 − 508 472 036 045 568Y 5 + 376 022 069 526 528Y 4 + 11 471 461 152 768Y 3 − 15 545 480 048 640Y 2   + 1 852 894 310 400Y − 67 903 488 000 X 2 + −884 736Y 56 − 41 306 112Y 55 − 745 311 744Y 54 − 7 854 617 152Y 53 − 56 752 908 832Y 52 − 305 691 731 184Y 51 − 1 291 334 290 291Y 50 − 4 426 288 172 137Y 49 − 12 647 728 750 928Y 48 − 31 075 240 957 232Y 47 − 69 032 547 442 536Y 46 − 148 896 176 655 469Y 45 − 322 250 360 417 387Y 44 − 632 799 187 856 807Y 43 − 813 902 597 999 889Y 42 + 446 780 681 983 306Y 41 + 5 558 180 767 681 622Y 40 + 12 861 196 240 238 133Y 39 + 82 777 469 121 694Y 38 − 95 303 288 589 756 575Y 37 − 346 077 988 812 264 244Y 36 − 660 107 106 734 655 623Y 35 − 436 018 183 788 282 799Y 34 + 1 681 210 077 303 367 073Y 33 + 7 273 388 110 177 545 277Y 32 + 16 144 187 701 672 166 507Y 31 + 23 113 156 489 251 424 786Y 30 + 15 841 883 577 593 033 385Y 29 − 22 985 886 157 655 295 827Y 28 − 107 891 300 530 866 582 629Y 27 − 240 236 673 343 003 218 988Y 26 − 401 070 259 854 975 037 146Y 25 − 553 221 859 660 242 994 706Y 24 − 655 402 208 407 230 091 461Y 23 − 681 436 550 533 356 290 765Y 22 − 630 169 426 685 781 559 948Y 21 − 521 000 422 369 504 316 094Y 20 − 384 796 397 343 162 994 888Y 19 − 255 121 295 576 640 474 717Y 18 − 155 328 774 650 536 353 143Y 17 − 88 765 501 587 983 950 847Y 16 − 45 750 129 339 254 668 296Y 15 − 18 659 682 247 552 136 243Y 14 − 5 068 590 639 242 514 778Y 13 − 1 175 381 774 283 273 326Y 12 − 905 896 315 537 604 416Y 11 − 627 696 074 121 250 176Y 10

297

298

B. Baran / Journal of Number Theory 145 (2014) 273–300

− 142 339 763 530 459 808Y 9 + 40 822 149 606 469 600Y 8 + 17 042 318 686 921 728Y 7 − 4 711 676 288 999 168Y 6 − 1 772 235 425 484 288Y 5 + 359 764 019 596 800Y 4 + 59 190 591 062 016Y 3 − 21 787 540 807 680Y 2 + 2 089 099 468 800Y   − 67 931 136 000 X + 1 769 472Y 55 + 81 727 488Y 54 + 1 448 432 640Y 53 + 14 896 074 368Y 52 + 104 157 512 704Y 51 + 536 681 652 096Y 50 + 2 133 314 691 734Y 49 + 6 717 439 350 959Y 48 + 17 014 862 466 548Y 47 + 35 253 668 331 292Y 46 + 62 993 289 306 098Y 45 + 112 164 904 174 479Y 44 + 225 104 747 950 936Y 43 + 392 976 024 258 106Y 42 + 41 136 059 444 059Y 41 − 3 005 798 979 372 973Y 40 − 11 839 374 362 691 297Y 39 − 21 190 199 356 139 685Y 38 + 6 071 591 688 951 686Y 37 + 156 325 146 877 196 166Y 36 + 498 229 160 275 748 461Y 35 + 823 122 591 905 419 376Y 34 + 242 278 135 736 846 249Y 33 − 2 876 324 971 563 594 929Y 32 − 9 820 974 562 808 688 564Y 31 − 19 019 630 528 554 710 562Y 30 − 23 294 749 281 381 561 713Y 29 − 9 486 631 575 477 114 590Y 28 + 36 945 136 929 714 190 121Y 27 + 122 510 790 902 234 508 091Y 26 + 237 163 604 354 617 256 618Y 25 + 354 702 102 863 369 258 879Y 24 + 443 147 794 960 495 122 187Y 23 + 477 688 281 307 028 925 796Y 22 + 450 443 978 678 593 045 613Y 21 + 374 770 111 259 894 186 150Y 20 + 278 602 970 647 572 826 148Y 19 + 187 430 241 355 450 245 572Y 18 + 113 084 138 555 091 016 698Y 17 + 58 445 464 663 933 323 835Y 16 + 24 688 966 627 452 318 352Y 15 + 9 359 950 076 743 879 942Y 14 + 4 515 705 377 792 888 754Y 13 + 2 518 265 993 855 063 344Y 12 + 865 069 239 507 223 888Y 11 − 38 745 965 042 080Y 10 − 90 275 758 453 065 632Y 9 + 96 762 513 188 096 ∗ Y 8 + 14 322 780 344 679 936Y 7 + 1 383 116 431 053 312Y 6 − 986 975 640 101 376Y 5 − 23 199 406 829 568Y 4  + 43 522 275 594 240Y 3 − 5 384 442 470 400Y 2 + 203 793 408 000Y . The j-line map Xs → X(1) is given below. Let {x, y, z} be the coordinates in P2Q where Xs is embedded as in (4.2). Using X = x/z and Y = y/z, the j-map is: j= where g(X, Y ) is given by

g(X, Y ) (X 2 Y + X 2 − X − 1)13

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299

−Y 44 − 478Y 43 − 32 789Y 42 − 762 363Y 41 − 7 547 176Y 40 − 29 112 957Y 39 + 27 142 557Y 38 + 516 914 701Y 37 + 812 780 834Y 36 − 3 217 396 462Y 35 − 9 372 779 891Y 34 + 10 412 400 391Y 33 + 51 281 877 696Y 32 − 21 282 455 533Y 31 − 192 026 183 004Y 30 + 36 250 779 915Y 29 + 569 052 119 000Y 28 − 70 791 418 862Y 27 − 1 392 885 057 099Y 26 + 180 780 916 888Y 25 + 2 828 775 392 943Y 24 − 517 934 371 345Y 23 − 4 803 267 109 749Y 22 + 1 263 736 631 868Y 21 + 6 827 480 292 072Y 20 − 2 364 214 517 403Y 19 − 8 006 349 554 252Y 18 + 3 419 622 432 067Y 17 + 7 623 483 793 639Y 16 − 3 825 699 399 717Y 15 − 5 822 086 727 508Y 14 + 3 183 399 803 606Y 13 + 3 469 776 050 052Y 12 − 1 874 600 063 602Y 11 − 1 524 399 772 882Y 10 + 753 079 641 994Y 9 + 455 793 324 718Y 8 − 201 254 434 444Y 7 − 83 894 215 212Y 6 + 34 538 107 094Y 5 + 8 027 212 027Y 4 − 3 468 719 981Y 3 − 204 764 859Y 2 + 159 120 623Y   − 12 145 116 X 2 + Y 45 + 446Y 44 + 27 904Y 43 + 571 184Y 42 + 4 618 918Y 41 + 10 200 852Y 40 − 54 213 498Y 39 − 267 763 439Y 38 + 165 832 967Y 37 + 2 368 932 904Y 36 + 395 119 701Y 35 − 12 956 472 125Y 34 − 4 150 166 662Y 33 + 53 633 145 847Y 32 + 12 575 521 646Y 31 − 180 183 851 743Y 30 − 10 692 143 892Y 29 + 494 822 838 826Y 28 − 75 462 285 877Y 27 − 1 111 012 707 457Y 26 + 427 597 488 033Y 25 + 2 035 065 625 744Y 24 − 1 283 704 039 956Y 23 − 2 989 648 304 276Y 22 + 2 716 215 586 599Y 21 + 3 407 000 739 960Y 20 − 4 339 306 349 468Y 19 − 2 827 171 518 873Y 18 + 5 300 271 455 330Y 17 + 1 407 694 779 775Y 16 − 4 900 196 317 351Y 15 + 18 526 414 704Y 14 + 3 345 715 118 980Y 13 − 643 829 576 586Y 12 − 1 623 126 310 814Y 11 + 509 263 027 702Y 10 + 527 959 900 322Y 9 − 203 278 262 234Y 8 − 104 991 650 269Y 7 + 45 780 617 674Y 6 + 10 651 697 173Y 5 − 5 549 670 458Y 4 − 270 006 722Y 3  + 269 295 673Y 2 − 26 210 553Y − 144 612 X − 2Y 44 − 891Y 43 − 55 361Y 42 − 1 113 897Y 41 − 8 622 455Y 40 − 14 660 030Y 39 + 130 330 785Y 38 + 526 492 528Y 37 − 657 396 776Y 36 − 5 427 279 992Y 35 + 569 333 662Y 34 + 32 403 186 745Y 33 + 8 217 537 249Y 32 − 137 416 119 935Y 31 − 44 405 672 291Y 30 + 459 234 305 539Y 29 + 116 561 534 270Y 28 − 1 261 675 920 964Y 27 − 154 320 890 766Y 26 + 2 870 046 210 200Y 25 − 102 772 109 156Y 24 − 5 399 242 483 910Y 23 + 1 112 801 469 826Y 22 + 8 364 611 871 299Y 21 − 3 167 545 239 322Y 20

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− 10 526 258 387 861Y 19 + 5 800 415 621 050Y 18 + 10 505 818 581 326Y 17 − 7 693 864 828 631Y 16 − 8 045 599 613 602Y 15 + 7 523 007 777 896Y 14 + 4 507 918 282 558Y 13 − 5 289 650 276 066Y 12 − 1 710 818 708 766Y 11 + 2 566 391 186 088Y 10 + 370 366 462 290Y 9 − 818 009 316 691Y 8 − 14 872 094 943Y 7 + 160 086 948 432Y 6 − 12 957 280 360Y 5 − 16 931 385 812Y 4 + 2 913 054 190Y 3 + 664 153 911Y 2 − 193 921 364Y + 12 288 000. References [1] B. Baran, Normalizers of non-split Cartan subgroups, modular curves and the class number one problem, J. Number Theory 130 (12) (2010) 2753–2772. [2] B. Baran, An exceptional isomorphism between modular curves of level 13 via Torelli’s Theorem, 2012, preprint. [3] Y. Bilu, P. Parent, M. Rebolledo, Rational points on X0+ (pr ), Ann. Inst. Fourier (2014), in press. [4] I. Chen, The Jacobians of non-split Cartan modular curves, Proc. Lond. Math. Soc. (3) 77 (1998) 1–38, Part 1. [5] B. de Smit, S. Edixhoven, Sur un résultat d’Imin Chen, Math. Res. Lett. 7 (2–3) (2000) 147–153. [6] S. Lang, Algebra, 3rd ed., Grad. Texts in Math., vol. 211, Springer-Verlag, New York, 2002. [7] B. Mazur, Rational isogenies of prime degree, Invent. Math. 44 (1978) 129–162. [8] I. Piatetski-Shapiro, Complex Representations of GL(2, K) for Finite Fields K, Contemp. Math., vol. 16, AMS, New York, 1983. [9] J.-P. Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972) 259–331. [10] J.-P. Serre, Lectures on the Mordell–Weil Theorem, 3rd ed., Aspects Math., vol. E15, Vieweg, Braunschweig, 1997. [11] W. Stein, The modular forms database, http://modular.math.washington.edu/Tables/tables.html.