Automorphy of Mod 2 Galois Representations Associated to Certain Genus 2 Curves Over Totally Real Fields

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AUTOMORPHY OF MOD 2 GALOIS REPRESENTATIONS ASSOCIATED TO CERTAIN GENUS 2 CURVES OVER TOTALLY REAL FIELDS

ALEXANDRU GHITZA AND TAKUYA YAMAUCHI

Abstract. Let C be a genus two hyperelliptic curve over a totally real ﬁeld F . We show that the

mod 2 Galois representation ρC,2 : Gal(F /F ) −→ GSp4(F2) attached to C is automorphic when the

image of ρC,2 is isomorphic to S5 and it is also a transitive subgroup under a ﬁxed isomorphism

GSp4(F2) ' S6. To be more precise, there exists a Hilbert–Siegel Hecke eigen cusp form h on

GSp4(AF ) of parallel weight two whose mod 2 Galois representation ρh,2 is isomorphic to ρC,2.

1. Introduction

Let K be a number ﬁeld in an algebraic closure Q of Q and p be a prime number. Fix an embedding Q ,→ C and an isomorphism Qp ' C. Let ι = ιp : Q −→ Qp be an embedding that is compatible with the maps Q ,→ C and Qp ' C ﬁxed above. In proving automorphy of a given geometric p-adic Galois representation

ρ : GK := Gal(Q/K) −→ GLn(Qp),

a standard way is to observe automorphy of its residual representation

ρ : GK −→ GLn(Fp)

after choosing a suitable integral lattice. However, proving automorphy of ρ is hard unless the image of ρ is small. If the image is solvable, one can use the Langlands base change argument to find an automorphic cuspidal representation as is done in many known cases. Beyond this, one can ask that p and the image of ρ be small enough that one can apply the known cases of automorphy of Artin representations (cf. [30], [35], [7], [24]). The natural problem is then to find geometric objects whose residual Galois representations have suitably small images for this purpose. In this paper, we study automorphy of mod 2 Galois representations associated to certain hyperelliptic curves of genus 2. Let us fix some notation to explain our results. Let F be a totally real field and fix an embedding F ⊂ Q. Let us consider the hyperelliptic curve over F of genus 2 defined by arXiv:2010.10021v1 [math.NT] 18 Oct 2020 2 6 5 4 3 2 (1.1) C : y = f(x) := x + a1x + a2x + a3x + a4x + a5x + a6, a1, . . . , a6 ∈ F.

By abusing the notation, we also denote by C a unique smooth completion of the above hyperelliptic curve. Let J = Jac(C) be the Jacobian variety of C and for each positive integer n, let J[n] be the

group scheme of n-torsion points. Let TJ,2 be the 2-adic Tate module over Z2 associated to J. Let

2010 Mathematics Subject Classiﬁcation. 11F, 11F33, 11F80. Key words and phrases. genus 2 curves, mod 2 Galois representations, automorphy. The second author is partially supported by by JSPS KAKENHI Grant Number (B) No.19H01778. 1 h∗, ∗i : TJ,2 × TJ,2 −→ Z2(1) be the Weil pairing, which is GF -equivariant, perfect, and alternating. It yields an integral 2-adic Galois representation

ρC,2 : GF −→ GSp(TJ,2, h∗, ∗i) ' GSp4(Z2) where the algebraic group GSp = GSp is the symplectic similitude group in GL4 associated to ! ! 4 J 02 s 0 1 , s = . Put T J,2 = TJ,2 ⊗Z2 F2 = J[2](Q). Thus it yields a mod 2 Galois −s 02 1 0 representation

(1.2) ρC,2 : GF −→ GSp(T J,2, h∗, ∗iF2 ) ' GSp4(F2).

On the other hand, we can embed the Galois group Gal(Ff /F ) of the splitting ﬁeld of f over F into

S6 by permutation of the roots of f(x); we denote this embedding by ιf : Gal(Ff /F ) ,→ S6. Here Sn stands for the n-th symmetric group. Since T J,2 is generated (cf. [27]) by divisors associated to the points (x, 0) in C(Q), ρC,2 factors through Gal(Ff /F ); we denote it by ρC,2 : Gal(Ff /F ) ,→ GSp4(F2) again. A subgroup H of S6 is transitive if it acts on the set {1, 2, 3, 4, 5, 6} transitively. Let us ﬁx an isomorphism GSp4(F2) ' S6 commuting with ρC,2 and ιf . In this situation we will prove the following:

Theorem 1.1. Keep the notation as above. Suppose that Im(ρC,2) ' S5 is a transitive subgroup of

S6 via the ﬁxed isomorphism GSp4(F2) ' S6. Further assume that for any complex conjugation c in GF , ρC,2(c) is of type (2, 2) as an element of S5. Then there exists a Hilbert–Siegel Hecke eigen cusp form h on GSp4(AF ) of parallel weight 2 such that ρh,2 ' ρC,2 as a representation to GL4(F2), where ρh,2 is the reduction modulo 2 of the 2-adic representation ρh,2 associated to h(see [23] for the construction of ρh,2).

We give some remarks on our main theorem. The transitivity of the image implies f(x) is irreducible over F . Let L/F be the quadratic ﬁeld corresponding to the sign character of Im(ρC,2) ' S5.

By assumption, L/F has to be a totally real quadratic extension. Then it will turn out that ρC,2 is an induced representation to GF of a certain totally odd 2-dimensional mod 2 Galois representation τ : GL −→ GL2(F2) whose image is isomorphic to A5. Applying Sasaki’s results [29], the Jacquet– Langlands correspondence, and a suitable congruence method, one can ﬁnd a Hilbert cusp form g of parallel weight 2 on GL2(AL) whose corresponding mod 2 Galois representation ρg,2 is isomorphic to τ. Then the theta lift of the corresponding automorphic representation πg to GSp4(AF ) yields a cuspidal automorphic representation Π of GSp4(AF ) which is generated by a Hilbert–Siegel Hecke eigen cusp form of parallel weight 2 on GSp4(AF ). We note that J is potentially automorphic by Theorem 1.1.3 of [3] and the potential version of Theorem 1.1 is already known in a more general setting.

Remark 1.2. (1) In the main theorem, we do not specify the levels of the Hilbert–Siegel forms because of the lack of level-lowering results (see Remark 3.6). (2) In the course of the proof we do not use the results in [15] to associate a form with a Galois representation. Instead, we apply the unconditional result in [25] to construct a corresponding automorphic cuspidal representation. Hence Theorem 1.1 is true unconditionally. 2 This paper is organized as follows. In Section 2, we study some basic facts about mod 2 Galois representations to GSp4(F2). We devote Section 3 through Section 4 to the study of the automorphic forms in question and to various congruences between several automorphic forms. We prove automorphy of our representations in Section 5. Finally, we describe how to obtain explicit families of examples using two constructions: the ﬁrst from a classical result of Hermite, and the second from 5-division points on elliptic curves due to Goins [16]. Acknowledgment. This work started when the second author visited the ﬁrst author from December 2018 to January 2019. We thank everything in this work for making this opportunity and incredible hospitality.

2. Certain mod 2 Galois representations to GSp4(F2)

In this section, we study some elementary properties of mod 2 Galois representations to GSp4(F2) related to our work. We refer to [4, Section 5]. We denote by Sn the n-th symmetric group, by An the n-th alternating group, and by Cn the cyclic group of order n.

2.1. GSp4. Let us consider the smooth group scheme GSp4 = GSpJ over Z which is deﬁned as the ! ! 02 s 0 1 symplectic similitude group in GL4 associated to J := , s = . Explicitly, −s 02 1 0

n t ∃ o GSp4 = X ∈ GL4 | XJX = ν(X)J, ν(X) ∈ GL1 .

Put Sp4 = Ker(ν : GSp4 −→ GL1,X 7→ ν(X)).

6 2.2. An identiﬁcation between GSp4(F2) and S6. Let s : F2 −→ F2 be the linear functional 6 deﬁned by s(x1, . . . , x6) = x1 + ··· + x6 and put V = {x ∈ F2 | s(x) = 0} and W = V/U where 6 U = h(1, 1, 1, 1, 1, 1)i. Let us consider the bilinear form on F2 given by the formula 6 hx, yi = x1y1 + ··· x6y6, x, y ∈ F2.

It induces a non-degenerate, alternating pairing h∗, ∗iW on W , where being alternating means that 6 hx, xiW = 0 for each x ∈ W . The symmetric group S6 acts naturally on F2 and yields a group homomorphism

ϕ : S6 −→ GSp(W, h∗, ∗iW ).

It is easy to see that the action of S6 on W is faithful and ϕ can be seen to be an isomorphism by computing the cardinality of both sides. It is easy to ﬁnd a basis of W . For instance, we see that

(2.1) e1 = (1, 1, 0, 0, 0, 0), e2 = (0, 0, 0, 1, 1, 0), e3 = (0, 0, 0, 1, 0, 1), e4 = (1, 0, 1, 0, 0, 0) make up a basis of W .

2.3. An identiﬁcation between A5 and SL2(F4). It is well-known that A5 ' SL2(F4) as a group. × 2 × Notice that PGL2(F4) ' PSL2(F4) = SL2(F4) since (F4 ) = F4 . Then we have an embedding SL ( ) ' PGL ( ) −→∼ (Aut ( 1 ))( ) ,→ Aut ( 1( )) 2 F4 2 F4 alg PF4 F4 set P F4 where Aut ( 1 ) stands for the group scheme of algebraic automorphisms of the variety 1 while alg PF4 PF4 1 1 Autset(P (F4)) stands for the set-theoretic automorphisms of the set of F4-rational points on P . We 1 1 have Autset(P (F4)) ' S5 since |P (F4)| = 5. By counting elements, we can check that the image 3 × of the above embedding is, indeed, A5. Fix a generator a of F4 . Here are representatives for the conjugacy classes in SL2(F4):

• the identity matrix I2; ! 1 0 • the class of elements of order 2; 1 1 ! a 0 • the class of elements of order 3; 0 a−1 ! 0 1 • one class of elements of order 5; 1 a ! 0 1 • the other class of elements of order 5. 1 a2

2.4. S5-representations to GSp4(F2). We refer to [4, Section 5] (in particular, the table in Lemma

5.1.7 in loc.cit.). Let G be a group and ρ : G −→ GSp4(F2) be a representation.

Lemma 2.1. Assume that Im(ρ) ' S5. Then ρ is absolutely irreducible and there are two types of representations whose images are characterized by

(1)( type S5(a)) the elements of order 3, 6 have trace 0, or

(2)( type S5(b)) the elements of order 3, 6 have trace 1. Furthermore, there exist an index 2 subgroup H of G with a generator ι of G/H and an absolutely ι irreducible two-dimensional representation τ : H −→ GL2(F4) which is not equivalent to τ such that G (1) if Im(ρ) is of type S5(a), then ρ ' IndH τ; (2) if Im(ρ) is of type S5(b), then ρ is isomorphic to the twisted tensor product with respect to G/H (see [34, Section 3]).

−1 Proof. Put H := ρ (A5). Clearly it is an index 2 subgroup of G. Since the order of Im(ρ) is divisible by 5, it follows from [34, Proposition 3.4] and [4, Lemma 5.1.7] that ρ|H is either reducible to two 2-dimensional representations or irreducible. First we consider the former case. Since A5 ' SL2(F4), there is a representation τ : H −→ SL2(F4) such that Im(ρ|H ) ' Im(τ). Under this isomorphism, by comparing the characteristic polynomials of all elements in S5 = S5(a) using results in §2.2 and §2.3, G we conclude that ρ ' IndH τ by the Brauer–Nesbitt theorem. The latter (irreducible) case follows from [34, Proposition 3.5].

Fix an isomorphism GSp4(F2) ' S6.

Remark 2.2. Keep the notation as above (recall that Im(ρ) ' S5). The type of the image Im(ρ) can also be characterized in the following way:

(1)( type S5(a)) it is a transitive subgroup of S6;

(2)( type S5(b)) it is not transitive, hence up to conjugacy, it ﬁxes 6 as a subgroup of S6, or equivalently,

(1)( type S5(a)) it contains an element of type (3, 3) in S6;

(2)( type S5(b)) it does not contain any elements of type (3, 3) in S6.

This article focuses on the case of type S5(a). Type S5(b) is studied in [34]. 4 Proposition 2.3. Let F be a totally real ﬁeld. Let ρ : GF −→ GSp4(F2) be an irreducible mod 2 Galois representation. Suppose that Im(ρ) is of type S5(a). Assume further that for each complex conjugation c of GF , ρ(c) is of type (2, 2). Then there exist a totally real quadratic extension L/F with Gal(L/F ) = hιi and an irreducible totally odd Galois representation τ : GL −→ GL2(F4) which is not equivalent to ιτ satisfying (1) ρ ' IndGF τ as a representation to GL ( ); FL 4 F4 (2) Im(τ) ' SL2(F4) ' A5; ! 0 1 (3) for each complex conjugation c of GF , τ(c) is conjugate to s := . 1 0

Proof. Let L/F be the quadratic extension corresponding to the sign character sgn : Im(ρ) ' S5 −→

{±1}. By assumption L/F is a totally real quadratic extension and we have ρ(GL) ' A5. The ﬁrst two claims follow from Lemma 2.1. Further, by assumption, for each complex conjugation c of GF , ! s 02 ρ(c) is conjugate to either J or . They are not conjugate in GSp4(F2) but conjugate each 02 s other in GL4(F2). The third claim follows from this.

3. Automorphy

3.1. Automorphic Galois representations. Let F be a totally real ﬁeld. For each place v of F , let Fv be the completion of F at v. In this section we recall basic properties of cuspidal automorphic representations of GSp4(AF ) such that each inﬁnite component is a limit of a discrete series representation. We basically follow the notation of Mok’s article [23] and add ingredients necessary for our purpose.

For any place v of F , we denote by WFv the Weil group of Fv. Let m1, m2, w be integers such that m1 > m2 ≥ 0 and m1+m2 ≡ w+1 (mod 2). Consider the L-parameter φ(w;m1,m2) : WR −→ GSp4(C) deﬁned by z m1+m2 z m1−m2 z − m1−m2 z − m1+m2 φ (z) = |z|−wdiag 2 , 2 , 2 , 2 (w;m1,m2) z z z z and ! ! 0 s 0 1 φ (j) = 2 , s = . (w;m1,m2) w (−1) s 02 1 0

By the Local Langlands Correspondence the archimedean L-packet Π(φ(w;m ,m )) corresponding to n o 1 2 φ consists of two elements πH , πW whose central characters both satisfy (w;m1,m2) (w;m1,m2) (w;m1,m2) −w × z 7→ z for z ∈ R>0. These are essentially tempered unitary representations of GSp4(R) and tempered exactly when w = 0. When m ≥ 1 (resp. m = 0) the representation πH is called 2 2 (w;m1,m2) a (resp. limit of) discrete series representation of minimal K-type k = (k1, k2) := (m1 + 1, m2 + 2) k1−k2 k2 which corresponds to an algebraic representation Vk := Sym St2 ⊗ det St2 of KC = GL2(C). Here K is the maximal compact subgroup of Sp4(R). If m2 ≥ 1 (resp. m2 = 0), the representation πH is called a (resp. limit of) discrete series representation of minimal K-type V . (w;m1,m2) (m1+1,−m2) 0 Fix an integer w. Let π = ⊗vπv be an automorphic cuspidal representation of GSp4(AF ) such that for each infinite place v, πv has L-parameter φ(w;m1,v,m2,v) with the parity condition m1,v + m2,v ≡ w + 1 (mod 2). Let Ram(π) be the set of all finite places at which πv is ramified. Thanks to [23] and [15] we can attach Galois representations to π: 5 Theorem 3.1 (cf. Theorem 3.1 and Remark 3.3, Theorem 1.1 of [23] with Theorem C of [36]). ∼ Assume that π is neither CAP nor endoscopic. For each prime p and ιp : Qp −→ C there exists a continuous, semisimple Galois representation ρπ,ιp : GF −→ GSp4(Qp) such that

(1) ν ◦ ρπ,ιp (c∞) = −1 for any complex conjugation c∞ in GF .

(2) ρπ,ιp is unramified at all finite places that do not belong to Ram(π) ∪ {v|p}; (3) for each finite place v not lying over p, the local-global compatibility holds:

F −ss GT − 3 WD ρ | ' rec π ⊗ |ν| 2 π,ιp GFv v v

GT with respect to ιp, where recv stands for the local Langlands correspondence constructed by Gan–Takeda [13];

(4) for each v|p and each embedding σ : Fv ,→ Qp, there is a unique embedding vσ : F,→ C such that ι ◦ σ| = v . Then the representation ρ | is Hodge–Tate of weights p F σ π,ιp GFv HT ρ | = {δ , δ + m , δ + m , δ + m + m } σ π,ιp GFv vσ vσ 2,vσ vσ 1,vσ vσ 2,vσ 1,vσ 1 where δ = (w +3−m −m ). If we assume either π is discrete series for all inﬁnite vσ 2 1,vσ 2,vσ v places v or genuine, then ρ | is crystalline and the local-global compatibility also holds π,ιp GFv up to semi-simpliﬁcation.

Deﬁnition 3.2. (1) Let ρ : GF −→ GSp4(Qp) be an irreducible p-adic Galois representation.

We say ρ is automorphic if there exists a cuspidal automorphic representation π of GSp4(AF )

with πv a (limit of) discrete series representation for any v|∞, and such that ρ ' ρπ,ιp as a

representation to GL4(Qp).

(2) Let ρ : GF −→ GSp4(Fp) be an irreducible mod p Galois representation. We say ρ is auto-

morphic if there exists a cuspidal automorphic representation π of GSp4(AF ) with πv a (limit

of) discrete series representation for any v|∞, and such that ρ ' ρπ,ιp as a representation to GL4(Fp).

Remark 3.3. Let h be a holomorphic Hilbert–Siegel Hecke eigen cusp form on GSp4(AF ) of parallel weight 2. Let πh be the corresponding cuspidal automorphic representation of GSp4(AF ). Then for each inﬁnite place v, the local Langlands parameter at v is given by φ(w;1,0) and πh,∞ is a limit of holomorphic discrete series. Conversely, if π is neither CAP nor endoscopic and its local Langlands parameter at v|∞ is given by φ(w;1,0) for some integer w, one can associate such a form h by using [15]. Note that the results in [15] are conditional on the trace formula (see the second paragraph on p.472 of [15]). However, in the course of the proof of Theorem 1.1, we apply the unconditional result in [25] to construct a corresponding automorphic cuspidal representation. Hence Theorem 1.1 is true unconditionally.

3.2. Hilbert modular forms and the Jacquet–Langlands correspondence. We refer to [33] and [20, Section 3] for the theory of (p-adic) algebraic modular forms corresponding to (paritious) Hilbert modular forms via the Jacquet–Langlands correspondence. In this section, p is any rational prime but we always remind the readers that we will later take p = 2. Let M be a totally real field of even degree m. For each finite place v of M, let Mv be the completion of M at v, Ov its ring of integers, $v a uniformizer of Mv, and Fv the residue field of Mv. Let D be the quaternion algebra with center M which is ramified exactly at all the infinite 6 places of M and OD be the ring of integral quaternions of D. For each finite place v of M, we fix × an isomorphism ιv : Dv := D ⊗M Mv ' M2(Mv). We view D as an algebraic group over M so that × × for any M-algebra A, D (A) outputs (D ⊗M A) and similarly as an algebraic group scheme over × × OM such that D (R) = (OD ⊗OM R) for any OM -algebra R. Let K be a finite extension of Qp contained in Qp with residue field k and ring of integers O = OK , and assume that K contains the images of all embeddings M,→ Qp. For each finite place v of M lying over p, let τv be a smooth representation of GL2(Ov) on a finite free O-module Wτ . We also view it as a representation of Dv via ιv. Put τ := ⊗ ⊗ v v|p σv∈Hom(Mv,Qp) σv σv τv which is a representation of GL2(Op) acting on Wτ := ⊗ ⊗ Wτ . Suppose v|p σv∈Hom(Mv,Qp) v × ∞ × × × ψ : M \(AM ) −→ O is a continuous character so that for each v|p, ZD× (Ov) 'Ov acts on Wτv −1 × by ψ |O× where ZD× ' GL1 is the center of D as a group scheme over OM . Note that we put the Mv × Q discrete topology on O and therefore, such a character is necessarily of finite order. Let U = v Uv × ∞ ∞ × be a compact open subgroup of D (AM ) ' GL2(AM ) such that Uv ⊂ D (OMv ) for all finite places v of M. Put U := Q U and U (p) = Q U . For any local O-algebra A put W := W ⊗ A. p v|p v v-p v τ,A τ O × Let Σ be a finite set of finite places of M. For each v ∈ Σ, let χv : Uv −→ A be a quasi-character. × Define χΣ : U −→ A whose local component is χv if v ∈ Σ, the trivial representation otherwise.

Deﬁnition 3.4 (p-adic algebraic quaternionic forms). Let Sτ,ψ(U, A) denote the space of functions × × ∞ f : D \D (AM ) −→ Wτ,A such that −1 (p) (p) × ∞ • f(gu) = τ(up) f(g) for u = (u , up) ∈ U = U × Up and g ∈ D (AM ); ∞ × ∞ • f(zg) = ψ(z)f(g) for z ∈ ZD× (AM ) and g ∈ D (AM ). × × ∞ Similarly, let Sτ,ψ,χΣ (U, A) denote the space of functions f : D \D (AM ) −→ Wτ,A such that −1 −1 (p) (p) × ∞ • f(gu) = χΣ (u)τ(up) f(g) for u = (u , up) ∈ U = U × Up and g ∈ D (AM ); ∞ × ∞ • f(zg) = ψ(z)f(g) for z ∈ ZD× (AM ) and g ∈ D (AM ). We call a function belonging to these spaces a p-adic algebraic quaternionic form.

× Let S be a finite set of finite places of M containing all places v - p such that Uv 6= D (Ov). We define the (formal) Hecke algebra

S (3.1) TA := A[Tv,Sv]v6∈S∪{v|p} × −1 × × −1 × where Tv = [D (Ov)ιv (diag(πv, 1))D (Ov)],Sv = [D (Ov)ιv (diag(πv, πv))D (Ov)] are the usual

Hecke operators. It is easy to see that both of Sτ,ψ(U, A) and Sτ,ψ,χΣ (U, A) have a natural action of S TA (cf. [10, Deﬁnition 2.2]). (p) ∞ Let U = U × Up be as above. As explained in [33, Section 1], if we write GL2(AM ) = ` × ∞ i D tiUZD× (AM ), then ∞ −1 × × M (UZD× (AM )∩ti D ti)/M Sτ,ψ(U, A) ' Wτ . i ∞ −1 × × The group (UZD× (AM ) ∩ ti D ti)/M is trivial for all t when U is suﬃciently small (see [20, p.623]). Henceforth, we keep this condition until the end of the section. It follows from this that the functor Wτ 7→ Sτ,ψ(U, A) is exact (cf. [19, Lemma 3.1.4]).

Fix an isomorphism ι : Qp ' C. Let S (U ,A) := lim S U (p) × U ,A τ,ψ p −→ τ,ψ p U (p) 7 where U (p) tends to be small. Hom(M, ) Qp Hom(M,Qp) We consider (k, w) = ((kσ)σ, (wσ)σ) ∈ Z>1 × Z with the property that kσ + 2ωσ is independent of σ. Set δ := kσ + 2wσ − 1 for some (any) σ. A special case we will encounter later

Hom(M,Qp) is (k, w) = (k · 1, w · 1) with k ∈ Z>1, w ∈ Z, and 1 = (1,..., 1) ∈ Z , where δ = k + 2w − 1. × × × Let ψC : M \AM −→ C be the character deﬁned by

δ−1 ∞ 1−δ (3.2) ψC(z) = ι N(zp) ψ(z ) N(z∞)

(p) × for z = (zp, z , z∞) ∈ AM where the symbol N stands for the norm. For each σ ∈ Hom(M, Qp), there exists a unique pair of v|p and an embedding σv : Mv −→ Qp such that σv|M = σ. There we can rewrite (σ) = (σv) . σ∈Hom(M,Qp) v|p, σv∈Hom(Mv,Qp) Since wσ ∈ Z for each σ, we can deﬁne the algebraic representation τ(k,w),A of GL2(Op) = Q v|p GL2(Ov) by

O O kσv −2 wσv (3.3) τ(k,w),A = Sym St2(A) ⊗ det A

v|p σv∈Hom(M,Qp) where St2 is the standard representation of dimension two. We often drop the subscript A from

τ(k,w),A which should not cause any confusion. Notice that τ(k,w),O ⊗O,ι C is the algebraic represen- δ−1 tation of GL2(C) of highest weight (k, w) so that the center acts by z 7→ z , δ = kσ + 2wσ − 1 for × z ∈ C . We write

(3.4) Sk,w,ψ(Up,A) := Sτ(k,w),ψ(Up,A) for simplicity. × p,∞ By [33, Lemma 1.3-2], we have an isomorphism of D (A )-modules

triv M ∞,p Up (3.5) Sk,w,ψ(Up, O)/Sk,w,ψ(Up, O) ⊗O,ι C ' π ⊗ πp π where π runs over the regular algebraic cuspidal automorphic representations of GL2(AM ) such that triv π has central character ψC and Sk,w,ψ(Up, O) is deﬁned to be zero unless (k, w) = (2 · 1, w · 1), in which case we deﬁne it to be the subspace of Sτ(k,w),ψ(Up, O) consisting of functions that factor × through the reduced norm of D (AF ).

For τ(k,w),C as above we can also consider the space of Hilbert cusp forms on GL2(AM ) of level U and of weight (k, w) and their geometric counterparts (see [11, Section 1.5]). Thanks to many contributors (see [17] and the references there), to each (adelic) Hilbert Hecke eigen cusp form f of weight (k, w) and level U, one can attach an irreducible p-adic Galois representation ρf,ιp : GM −→

GL2(Qp) for any rational prime p and ﬁxed isomorphism Qp ' C. The construction also works for k ≥ 1 (in the lexicographic order). In particular, in the case when k = 1 (parallel weight one), its image is ﬁnite and it gives rise to an Artin representation ρf : GM −→ GL2(C) (see [26]). Taking a suitable integral lattice, we consider the reduction ρf,p : GM −→ GL2(Fp) of ρf,p or ρf modulo the maximal ideal of Zp. m Set 1m = (1,..., 1) ∈ Z for a positive integer m.

Proposition 3.5. Let p = 2. Let M be a totally real ﬁeld of even degree m. Let f be a Hilbert

Hecke eigen cusp form on GL2(AM ) of parallel weight one such that ρf,2 is irreducible. There exists a Hilbert Hecke eigen cusp form g on GL2(AM ) such that 8 × (1) ρf,2 ' ρg,2 ⊗ ψ for some continuous character ψ : GM −→ F2 ; (2) the character corresponding to the central character of g under (3.2) is trivial; m (3) g is of weight (k, w) with k = 2 · 1m (parallel weight 2) for some w ∈ Z .

Proof. Let U be a sufficiently small open compact subgroup of GL2(AM ) that fixes f. Since f has parallel weight one, we have (k, w) = (1, w · 1) for some w ∈ Z. Let K be a finite extension of S Q2 in Q2 with ring of integers O such that K contains all Hecke eigenvalues of f for TO. Let F be the residue field of K. As in [11], we can view f as a geometric Hilbert modular form over O via a classical Hilbert modular form associated to f and consider its base change f to F. After multiplying by a high enough power of the Hasse invariant, we have another form g1 of weight (k, w) with k = k · 1, k 0 such that

(1) h = g1;

(2) g1 is liftable to a geometric Hilbert Hecke eigenform g1 over O by enlarging K if necessary;

(3) ρg1,2 ' ρf,2.

We view g1 as a classical Hilbert Hecke eigen cusp form and by the Jacquet–Langlands correspondence, we have a 2-adic algebraic quaternionic form h1 in Sτ(k,w),ψ(U, O) corresponding to g1; here × ∞ × ψ : M \AM −→ O is the finite character corresponding to the central character of g1 under (3.2). For each finite place v of M lying over 2, let U1,v be the subgroup Uv consisting of all elements ! 1 ∗ Q (2) congruent to . Put U1,2 := U1,v. By definition, h1 ∈ Sτ ,ψ(U × U1,2, F). Since U1,2 0 1 v|2 (k,w) acts on τ(k,w),F unipotently, it can be written as a successive extension of the trivial representation of U1,2. This successive extension commutes with the Hecke action and one can find a Hecke eigenform (2) × 2 × h2 ∈ Sτ(2·1,w),ψ(U × U1,2, F). Further, notice that (F ) = F and by twisting, we may assume that the reduction ψ of ψ is trivial. Since U is taken to be sufficiently small, there exists a lift (2) triv h ∈ S triv (U × U , O) of h where ψ stands for the trivial character. In fact, the Hecke 2 τ(2·1,w),ψ 1,2 2 eigen system of h2 is liftable to O by the Deligne–Serre Lemma [9, Lemma 6.11] and by multiplicity one, we can recover a Hecke eigenform h2 from the Hecke eigensystem. Further h2 is non-trivial since τ is absolutely irreducible. Therefore, by [33, Lemma 1.3-2] the form g is taken to be the image of h2 under the Jacquet–Langlands correspondence.

3.3. Mod 2 automorphy. In this section we prove Theorem 1.1.

Proof. By assumption and Lemma 2.1, Im(ρC,2) is of type S5(a). Applying Proposition 2.3 there exist a totally real quadratic extension L/F and an absolutely irreducible totally odd two-dimensional Galois representation τ : G −→ GL ( ) such that ρ ' IndGF τ. Notice that τ is not equivalent L 2 F4 GL to ιτ for any lift ι of the generator of Gal(L/F ). By [29, Theorem 2], τ is modular. Further, applying Proposition 3.5, there exists a Hilbert cusp form f of GL2(AF ) of parallel weight 2 with trivial central character such that ρf,2 ' τ. Let π be the cuspidal automorphic representation of GL2(AL) corresponding to f. Applying [25, Theorem 8.6 on p.251], we have a cuspidal automorphic representation Π of GSp4(AF ) such that for each inﬁnite place v of L, the local Langlands parameter is given by 1 1 0 z 2 z − 2 φ (z) = |z|−wv diag , πv z z 9 and Πv is holomorphic discrete series. It follows that the local Langlands parameter of Πv is φ(w;1,0) for some w ∈ Z.

Remark 3.6. In the course of the proof, we apply Sasaki’s modularity lifting theorem which requires raising the level. Therefore, we can not specify the levels of the Hilbert modular forms appearing there. This implies that we are also unable to specify the levels of our Hilbert–Siegel modular forms. However, there is some hope and a substantial expectation for further development of the level- lowering method. Indeed, [5, Conjectures 4.7 and 4.9] indicate that we can take the Hilbert cusp forms in question to have the correct level corresponding to the Artin conductor of ρ.

4. Examples

We are looking for monic irreducible polynomials f ∈ Q[x] of degree 6 such that

(1) there is an isomorphism σ from the Galois group of f to the symmetric group S5;

(2) for any complex conjugation c in the Galois group of f, the element σ(c) ∈ S5 has type (2, 2). We shall give two families of such polynomials. A general approach to produce them is to compute the sextic resolvent polynomial f of a given sextic polynomial over F so that f factors over F into a polynomial of degree 1 and an irreducible quintic [8, Algorithm 6.3.10 of Section 6.3, pp.316–324]. It is just done by symbolic computation but it would be tedious. Further, writing down an explicit form would take considerable space. To get around this, we apply classical results of Hermite and Klein.

4.1. Using generic polynomials. Let F = Q. Hermite proved that the polynomial P ∈ Q(s, t)[x] given by P (x) = x5 + sx3 + tx + t is generic for S5, which means that every S5-extension K/Q is the splitting ﬁeld of some specialization of P , and that the splitting ﬁeld of P has Galois group S5 over Q(s, t). A proof appears in [18, Proposition 2.3.8]. The discussion in Section 2.4 of the same book contains the following observation:

If h ∈ Q[x] is a quintic polynomial with Galois group S5, then the Weber sextic resolvent of h is an irreducible sextic with the same splitting ﬁeld, corresponding to the transitive embedding of S5 into S6. We can apply this to a specialization h of P with s, t ∈ Q, and get the Weber sextic resolvent 3 2 2 10 f(x) = (x + b4x + b2x + b0) − 2 disc(h)x where disc(h) is the discriminant of h and the coeﬃcients are given by

2 2 4 2 6 3 b0 = −176s t + 28s t + 4000st − s + 320t 4 2 2 b2 = 3s − 8s t + 240t 2 b4 = −3s − 20t

Taking different rational values of s and t, we can obtain lots of examples of f with property (1) above. Property (2) is easy to check once s and t are given. To give an idea of what to expect, note that the derivative of h is biquadratic: h0(x) = 5x4 + sx2 + t and elementary calculations show that: 10 • in the first quadrant of the (s, t)-plane, for all values of s and t that give a polynomial satisfying (1), condition (2) is also satisfied (because the derivative h0 has no zeros) • the lower half (s, t)-plane is cut into two regions by the curve with equation

(4.1) 108 s5 + 16 s4t − 900 s3t − 128 s2t2 + 2000 st2 + 256 t3 + 3125 t2 = 0;

in the region on the left of the curve (which takes up most, but not all, of the third quadrant), the corresponding polynomial does not satisfy condition (2); in the region on the right of the curve, the polynomial satisfies condition (2); this is because, for t < 0, the derivative h0 has precisely two real zeros ±ξ, and Equation (4.1) governs whether h(ξ) and h(−ξ) have the same sign. This leaves the second quadrant, which is more delicate; the region to the left of the parabola 9s2 −20t = 0 gives h0 without any real zeros, so condition (2) is satisfied; the rest gives h0 with 4 real zeros, and we did not find a simple expression that separates the cases where (2) is satisfied from those where it is not. A numerical experiment1 involving 100 000 randomly chosen rational numbers in the rectangle (−60, 0) × (0, 1000) gave rise to the picture in Figure 1.

Figure 1. Second quadrant in the (s, t)-plane: blue (resp. red) points represent polynomials h satisfying (resp. not satisfying) condition (2).

4.2. Using 5-torsion on elliptic curves. Next we consider the quintic polynomials arising from 5-division points of elliptic curves. This has the advantage that an explicit form of the resolvent is well-known: it goes back to work of Klein. Almost everything that follows is described in [16]. Let F be a totally real ﬁeld. Let

2 3 2 E : y + a1xy + a3y = x + a2x + a4x + a6 be an elliptic curve over F whose mod 5 representation ρE,5 : GF −→ GL2(F5) is full, hence it is surjective. The projective image of Im(ρE,5) is PGL2(F5) and it is isomorphic to S5. The corresponding S5-extension of F is described by a quintic polynomial

5 2 2 θE,5(x) = x − 40∆x − 5c4∆x − c4∆

1These calculations were performed using the mathematical software SageMath [28]. 11 (see [16, Equation (1.24)]) where the invariants ∆, c4 are deﬁned in [31, §III.1, p.42]. The resolvent of θE,5 is computed as in [16, Equation (2.37)] in conjunction with [21, Proposition 3.2.1] and it is explicitly given by

6 5 4 3 2 2 2 fE,6(x) := x + b2x + 10b4x + 40b6x + 80b8x + 16(b2b8 − b4b6)x + (−b2b4b6 + b2b8 − 5b6) where b2, b4, b6, b8 are also deﬁned in [31, §III.1, p.42]. The Galois group of the splitting ﬁeld FfE,6 of fE,6(x) over F in Q is a priori embedded in S6 by permuting the roots but it is isomorphic to

S5 since FfE,6 = FθE,5 by construction. Let us consider the hyperelliptic curve C = CE deﬁned 2 by y = fE,6(x). The above argument shows there is a natural embedding Im(ρC,2) ,→ S6. It is well-known that ρE,5 is totally odd and it shows that ρC,2(c) is of type (2, 2) for each complex conjugation of GF . Further, if fE,6(x) is irreducible, then Im(ρC,2) is transitive in S6 and hence is of type S5(a). Summing up, we have

Theorem 4.1. Assume that ρ is full and f is irreducible. Then ρ is automorphic in the E,5 E,6 CE ,2 sense of Theorem 1.1.

2 3 2 We give an explicit example. Let us consider the elliptic curve E : y + y = x + x over Q with the conductor 43. By [22, Elliptic curve with label 43.a1], the representation ρE,5 : GQ −→ GL2(F5) 5 2 is full. On the other hand, we have θE,5 = x + 1720x + 3440x + 11008 and it is easy to check (e.g. using SageMath [28] or Magma [2]) that

6 5 3 2 (4.2) fE,6(x) = x + 4x + 40x + 80x + 64x + 11 is irreducible over with Galois group isomorphic to S . By Theorem 4.1, ρ is automorphic. Q 5 CE ,2

4.3. Databases of genus 2 curves. The methods used in §4.1 and §4.2 produce inﬁnite families of examples satisfying the conditions of Theorem 1.1, but they tend to give hyperelliptic curves with large conductors. For instance, the conductor of the curve (4.2) is2

4 786 321 400 000 = 26 · 55 · 7 · 434 and the curve corresponding to the choice of parameters s = 0, t = −1 in §4.1 has equation

y2 = x6 + 40x5 + 880x4 + 8960x3 + 44800x2 − 3091456x + 102400 and conductor 82 311 610 000 = 24 · 54 · 192 · 1512 There are however databases of genus 2 curves with conductor that is either small or has restricted prime factors. Currently the most comprehensive are based on [1], which produced: • 66 158 curves with conductor less than 1 000 000, to be found at [22, Curves of genus 2 over Q]; from this list, we veriﬁed that the curves labelled 50000.b.800000.1 180625.a.903125.1 382347.a.382347.1 681472.a.681472.1 64800.c.648000.1 378125.a.378125.1 506763.a.506763.1 703125.b.703125.1

satisfy the conditions of Theorem 1.1;

2We used the Magma package Genus2Conductor [12] by Tim Dokchitser and Christopher Doris for the computation of the conductors listed in this section. 12 • 487 493 curves with 5-smooth discriminant, to be found at [32]; we veriﬁed that 4 885 of these curves satisfy the conditions of Theorem 1.1.

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Alexandru Ghitza, School of Mathematics and Statistics, University of Melbourne, Parkville, VIC 3010, AUSTRALIA Email address: [email protected]

Takuya Yamauchi, Mathematical Inst. Tohoku Univ., 6-3,Aoba, Aramaki, Aoba-Ku, Sendai 980-8578, JAPAN Email address: [email protected]