Theta Lifts of Bianchi Modular Forms and Applications to Paramodularity

We explain how the work of Johnson-Leung and Roberts on lifting Hilbert modular forms for real quadratic fields to Siegel modular forms can be adapted to imaginary quadratic fields. For this we use archimedean results from Harris, Soudry, Taylor and replace the global arguments of Roberts by the non-vanishing result of Takeda. As an application of our lifting result, we exhibit an abelian surface $B$ defined over $\mathbb{Q}$, which is not restriction of scalars of an elliptic curve and satisfies the Brumer-Kramer Paramodularity Conjecture.


Introduction
The following is a special case of [5,Conjecture 1.4], known as the Brumer-Kramer paramodularity Conjecture. (For definitions and terminology, we refer the reader to Sections 2 and 3.) quadratic fields that are not Q-curves provide a large supply of abelian surfaces which satisfy Conjecture 1.1.
In fact, the above strategy to produce evidence for Conjecture 1.1 was elaborated by Brumer and Kramer themselves. They also speculated that further evidence could be gathered by using abelian surfaces B with trivial endomorphism ring over Q such that End Q (B) Z (see the paragraph following the statement of [5,Conjecture 1.4]). In [9], Dembélé and Kumar provide such numerical evidence. They give explicit examples of paramodular abelian surfaces B defined over Q, which become of GL 2 -type over some real quadratic fields. Like with the surfaces B E above, the Johnson-Leung-Roberts' lift also plays a crucial rôle in their work.
The goal of this paper is twofold. First, we show that one can generalize the construction in [23] to imaginary quadratic fields. For this, one needs to replace the theta correspondence between GO (2,2) and GSp 4 by the one between GO(3, 1) and GSp 4 used by Harris, Taylor and Soudry in their work on Galois representations associated to cusp forms for GL 2 over imaginary quadratic fields (see [21,37]). Our analysis is then an adaptation of the one in [23,29] using archimedean results from [21] and the local-global non-vanishing result of Takeda [35]. Our lift can be seen as a type of Yoshida lift, although, strictly speaking, Yoshida [41] only considered the groups O(4) and O (2,2).
Secondly, we combine our lift with explicit computations of Bianchi modular forms to exhibit an abelian surface B defined over Q, which satisfies Conjecture 1.1 but is not a restriction of scalars of an elliptic curve. Because of the difficulties in constructing such examples (see Subsection 5.1) and the paucity of modularity results for GL 2 over imaginary quadratic fields, we had to limit ourselves to one example. However, it should be clear to the reader that our method, which borrows from [9], can be used in principle to generate more cases of the conjecture when further modularity results become available in this case.
As in the real quadratic case, if E is a modular over an imaginary quadratic field that is not a Q-curve, then the associated surface B E is paramodular by our lifting result. We note that there are several examples of such elliptic curves in the literature (see [10] and references therein).
The paper is organized as follows. In Sections 2 and 3, we recall the necessary background on Bianchi and Siegel modular forms. In Section 4, we prove the existence of our theta lift for any Bianchi modular form of even weight k that is non-base change. In Section 5, we give an example of an abelian surface B defined over Q which satisfies Conjecture 1.1. We prove the modularity of our surface (over an imaginary quadratic field) using the so-called Faltings-Serre method [34].

Bianchi modular forms
Let K be an imaginary quadratic field, and O K be its ring of integers. Let Z be the finite adèles of Z. The adèles of K are given by Let k 2 be an even integer and consider the GL 2 (C)-representation V k = Sym 2k−2 (C 2 ). Let χ : . We define the space S k (N, χ) of Bianchi cusp forms of level U 0 (N), central character χ and weight k to be the set of all functions (d) f , viewed as a function on GL 2 (C), is an eigenfunction of the complexification (in sl 2 (C) ⊗ R C) of the Casimir operator of sl 2 (C) with eigenvalue (k 2 − 2k)/8; (e) for all g ∈ GL 2 (A K ), one has We invite the reader to see [16,22] for details. In the notation of [16], we take for i, c : K → C, to ensure that our forms have trivial action by C × . The space S k (N, χ) is endowed with the so-called Petersson inner product.

Newforms and L-series
For each p N, write the double coset decomposition where p is a uniformizer at p N. We define the Hecke operator Similarly, one can define the operator U p for p | N.
We refer the reader to [39, Section 1.6] for the theory of newforms for Bianchi modular forms. When f ∈ S k (N, χ) is a newform, we have The L-series attached to f is then defined by The newform f determines a unique cuspidal irreducible automorphic representation π of GL 2 (A K ) of level N, which admits a restricted tensor product (see [13]) such that π ∞ has L-parameter (see Section 4) and the central character of π is χ. Conversely, any cuspidal automorphic representation π of GL 2 (A K ) whose infinity component has such an L-parameter corresponds to a newform f of weight k.

Connections with cohomology
It is standard to pass to the cohomological framework when working with Bianchi automorphic forms. In fact, this is the only approach that is currently suitable for the algorithmic methods that were used to gather the data in Subsection 5.1 (see [8,28,40]). So we now give a quick review of this framework.
Let W n = Sym n (C 2 ) ⊗ C Sym n (C 2 ) where the action of GL 2 (C) on the second factor is twisted with complex conjugation. Consider the adelic locally symmetric space where W n is the local system induced by W n , carry a natural Hecke action on them. Let Y 0 (N) BS denote the Borel-Serre compactification of Y 0 (N). This is a compact space with boundary ∂Y 0 (N) BS , made of finitely many 2-tori, that is, homotopy equivalent to Y 0 (N). We define the cuspidal cohomology H i cusp (Y 0 (N), W n ) as the kernel of the restriction map The Hecke action stabilizes the cuspidal cohomology.
By the strong approximation theorem, the determinant map induces a canonical bijection where Cl(K) is the class group of K. Let c i , 1 i h, be a complete set of representatives for the classes in Cl(K). For each i, let t i be a finite idèle which generates c i and set Γ 0 where H 3 GL 2 (C)/K ∞ is the hyperbolic 3-space. It follows that the cohomology of the adelic space decomposes as The cohomology spaces on the right-hand side lend themselves very well to explicit machine calculations. We refer the reader to [28,40] for more details on this, and on the Bianchi newforms related to the examples in this paper. We recall that the diagonal embedding This action is compatible with the Hecke action as it is via the diamond operators. Therefore, these cohomology spaces decompose accordingly.
For k 2 even, the Generalized Eichler-Shimura Isomorphism (see [20], in our setting [22, Proposition 3.1]) says that as Hecke modules, where χ runs over all Hecke characters of trivial infinity type that are unramified everywhere (that is, characters of Cl(K)).
Remark 2.1. Our Theorem 4.1 only applies to the newforms in S k (N, 1), that is, the newforms whose central character is trivial. We refer the reader to [25, Sections 2.4 and 2.5; 28; 40] for details on how the cohomology corresponding to this space can be computed.

Siegel modular forms
We recall that the symplectic group of genus 2 is the Q-algebraic group GSp 4 defined by setting for any Q-algebra R. The map ν : GSp 4 → G m is called the similitude factor, and its kernel is the symplectic group Sp 4 . Let GSp 4 (R) + be the subgroup of GSp 4 (R) which consists of the matrices γ such that ν(γ) > 0; also let be the Siegel upper half plane of degree 2. We recall that GSp 4 (R) + acts on H 2 by Let k 2 be an even integer, and set V (k,2) = Sym k−2 (C 2 ) ⊗ det 2 (C 2 ); this is a GL 2 (C)representation which we denote by ρ (k,2) . The group GSp 4 (R) + acts on the space of functions F : We fix a positive integer N , and consider the paramodular group A Siegel modular form of genus 2, weight (k, 2) and paramodular level N is a holomorphic function F : Let F be a Siegel modular form of genus 2, weight (k, 2) and paramodular level N . Then, we see that By the Koecher principle [17], F admits a Fourier expansion of the form where Q runs over all the 2 × 2 symmetric matrices in M 2 (Q) that are positive semi-definite. We say that F is a cusp form if, for all γ ∈ GSp 4 (Q), we have We denote by S (k,2) (Γ p (N )) the space of paramodular Siegel cusp forms of level N and weight (k, 2).

Newforms and L-series
We consider the double coset decompositions and following [23, p. 546] define the Hecke operators (As noted in [23], T (1, p, p, p 2 ) agrees with the classical T (p 2 ) for p N . We also note that our Hecke operators are scaled so as to match the definition in [1, p. 164].) We refer the reader to [23, p. 547] or [31] for the definition of the operators U p for p | N .
For the paramodular group Γ p (N ), the theory of newforms developed in [30,31] for scalar weights carries over to the vector-valued setting. The old subspace is generated by the images of the level-raising maps of [30]. One then defines the new subspace S new (k,2) (Γ p (N )) to be its orthogonal complement with respect to the Petersson inner product for vector-valued forms given in [1].
, we can associate an automorphic representation Π to F which admits a restricted tensor product The L-series attached to F is defined by where the local Euler factors L p (s, F ) are obtained as follows: (a) for val p (N ) = 0, we use the classical Euler factor in [1, p. 173]: For val p (N ) 1, these definitions are motivated by the results of [31] and the definitions in [23].

Theta lifts of Bianchi modular forms
We briefly recall the definition of L-parameters following [32,Section 4.1]. Let E be a number field, v be a place of E and E v be the completion of E at v. Let G be the group GL 2 or GSp 4 . Then, the local Langlands correspondence is known for G (see [6,15,24]). It yields a corresponding finite-to-one surjective map Theorem 4.1. Let K/Q be an imaginary quadratic field of discriminant D. Let O K be the ring of integers of K and N ⊆ O K be an ideal. Let π be a tempered cuspidal irreducible automorphic representation of GL 2 (A K ) of level N and trivial central character such that π ∞ has L-parameter (c) the following equality of L-parameters holds for all places v: Consequently, there exists a Siegel newform F of weight (k, 2) and paramodular level N = D 2 N K/Q (N) with Hecke eigenvalues, epsilon factor and (spinor) L-function determined explicitly by π (and described in the proof below).
Remark 4.2. The L-parameters for GSp 4 (Q v ) on the right-hand side of (4.1) are defined as follows.
(i) If v is split, then as in [23] (ii) If v is non-split and finite, then as in [23] (6), let g 0 ∈ W K w \W Qv be non-trivial. Then can also be described explicitly as follows: For odd weights k, one still has a lift to a cuspidal automorphic representation Π of GSp 4 , with central character given by the quadratic character ω K/Q corresponding to K such that conditions (a) and (b) in the theorem are satisfied. ( [22,Lemma 12] also gives a lift with trivial central character, but Π ∞ must then be generic of highest weight (k, 1).) We also note that one can, in fact, lift any non-Galois-invariant π with cyclotomic central character to an irreducible cuspidal representation of GSp 4 (A Q ) (as discussed in [21,35]). However, the local calculations of [23] and the paramodular newform theory of [31] apply only for trivial central character, which is the reason why we exclude odd weights in the theorem and impose the condition of a trivial central character.
Proof. The strategy of the proof is very similar to that of [23], but replaces the theta correspondence between GO(2, 2) and GSp 4 by that between GO(3, 1) and GSp 4 . There are four steps in the construction of the lift, which are outlined as follows.
(b) Choosing suitable extensions of the local components of σ promotes this to a representationσ of GO (3,1). (In this step, we follow [29] rather than [21] at the non-archimedean places.) (c) As in [21] (but using the non-vanishing result of [35]), we then use the theta correspondence between GO(3, 1) and GSp 4 to liftσ to the automorphic representation Π of GSp 4 described in the theorem.
(d) To produce the paramodular Siegel modular form, one takes the automorphic form for v ∞ are the paramodular newform vectors defined by Roberts and Schmidt [31]. By referring to the classical treatment in [1], we transfer the local nonarchimedean calculations of Hecke eigenvalues, epsilon and L-factors of Π in [23] to those of the corresponding vector-valued Siegel modular form on H 2 .
We now give precise details for each of these steps.
Before we come to Step (b), we review some details of the theta correspondence between GO(X, A Q ) and GSp 4 (A Q ): Following [21,29,35], we consider the extension of the Weil representation for Sp 4 ×O(X) to the group R = {(g, h) ∈ GSp 4 × GO(X) : ν(g)ν(h) = 1} and denote this representation by ω. (As explained in [35,Remark 4.3], there is a difference in the definition of R in [21,29]. But, since we are working with trivial central characters, this does not matter here.) Let G be a reductive group over Q, and v be a place of Q. We denote by Irr(G(Q v )) the collection of equivalence classes of irreducible smooth admissible representations of is the unique generic representation of GSp 4 (Q p ) with L-parameter satisfying (4.1).
By [35,Proposition 5.4], we obtain an irreducible cuspidal automorphic representation of GO(X, A Q ) by settingσ (c) We now consider the global theta lift Θ(Vσ), which is the space generated by the GSp 4 (A Q ) automorphic forms θ(f ; ϕ) for all f ∈ Vσ and ϕ ∈ S(X(A Q ) 2 ). (For details of this definition, we refer the reader to [29, Section 5; 35, p.11].) We know by the local non-vanishing and [35, Theorem 1.2] that Θ(Vσ) = 0. Since π ∼ = π c , we also know that Θ(Vσ) occurs in the space of cuspforms by Takeda [ where the last equality holds by [29, Proposition 1.10]). (d) The existence of the Siegel paramodular form is now proved exactly as in [23], but using the argument from [33, Section 3.1] for defining the vector-valued Siegel modular form F . The Hecke eigenvalues, epsilon and L-factors for the finite part of the automorphic representation Π are identical to those in the main theorem of [23]. (Note that [23] uses the notation π 0 and π instead of our π and Π.) To match the classical spinor L-factor of [1, p. 173] at unramified places, we see that the shift in the argument of the Euler factors of the local representations Π v given in [23, Proposition 4.2] to those of F is given by s → s − (k − 1)/2. By the calculations in [23], this means that, for all primes p, we have an equality of Euler factors where the L p (s, F ) were defined in Section 3 and L p (s + 1 2 , f) in Section 2 for the newform f corresponding to π. As in [23], the functional equation of π implies that the completed L-function satisfies the functional equation Remark 4.4. We end this section with two remarks.
(a) Arthur's multiplicity formula and [26, Theorem 2.2] imply that the multiplicity of Π in the space of cuspforms is 1. This has been proved in [29,Theorem 8.6] in the real quadratic case using the multiplicity preservation of the theta correspondence [29,Proposition 5.3].
(b) An alternative construction of the lift of automorphic representations has been proved by Chan [7] using trace formulas (under some local conditions on π). Recently, Mok [26] has also described how to obtain this lift from Arthur's endoscopic classification, based on work of Chan and Gan which relates Arthur's local correspondence with that of Gan-Takeda.

Application to paramodularity
In this section, we use our lifting result to prove the following theorem.
Theorem 5.1. Let C /Q be the curve defined by and B the Jacobian of C . Then, B is a paramodular abelian surface of conductor 223 2 in the sense that it satisfies Conjecture 1.1.
We obtain Theorem 5.1 as a consequence of the following statement.
Let A = Jac(C) be the Jacobian of C. Then we have the following. Remark 5.3. Theorem 5.2(b) implies that A is modular in the sense that its L-series is given by the product of those of the forms f and f τ . So, this is an instance of the Eichler-Shimura Conjecture [38,Conjecture 3] in dimension 2. We think that this is the first non-trivial such example over an imaginary quadratic field. (There is a significant amount of numerical data going back to [8,19] which supports this conjecture in the case of elliptic curves.) The rest of this section is dedicated to proving Theorem 5.2. But first, we show how to derive Theorem 5.1 from it.
Proof. The equality f σ = f τ implies that the form f cannot be a base change. Moreover, as the abelian surface A is modular by f , the cuspidal irreducible automorphic representation associated to f is tempered. So by Theorem 4.1, it admits a lift g of weight 2 to GSp 4 /Q with paramodular level 223 2 . By construction, the identity L(A, s) = L(f, s)L(f τ , s) implies that L(g, s) = L(B, s). So B is paramodular.

The abelian surface
As we mentioned in Remark 5.3, the surface A satisfies the Eichler-Shimura Conjecture. It was in fact located via explicit computations of Bianchi modular forms. More specifically, we used the extensive data provided in [28].
To simplify notation, let S 2 (1) := S 2 (O K , 1) be the space of Bianchi cusp forms of weight 2, level N = (1) and central character χ = 1. Let S bc 2 (1) be the subspace of S 2 (1) which consists of twists of those Bianchi cusp forms which arise from classical elliptic newforms via base change, and of those Bianchi cusp forms which are CM (see [12]), and S bc 2 (1) ⊥ its orthogonal complement with respect to the Petersson inner product. We will call the newforms in S bc Of all the 186 imaginary quadratic fields K in [28] (including all those 153 for which |D| < 500), there are only 6 for which S bc 2 (1) S 2 (1). (Note that our S 2 (1) corresponds to S 0 (1) + of [28] by the arguments in [25, Sections 2.4 and 2.5].) In each case, S bc 2 (1) ⊥ is an irreducible Hecke module of dimension 2, except for |D| = 643 when there are two newforms, with rational Hecke eigenvalues, that are Gal(K/Q)-conjugate. Table 1 provides a summary of these data.
Remark 5.4. The index entries in Table 1 are based on finite sets of Hecke eigenvalues. Since there is no analogue of Sturm's bound for Bianchi modular forms, the last row entries are not proved to be correct. However, we strongly expect them to reflect the truth.
Let f be any of the genuine Bianchi newforms listed in Table 1. Let σ and τ denote the non-trivial elements of Gal(K/Q) and Gal(L/Q), respectively. Let f τ be Gal(L/Q)-conjugate of f , which is determined by the relation a p (f τ ) = τ (a p (f )) for all primes p, and f σ be the Gal(K/Q)-conjugate of f which is determined by Except for the discriminant |D| = 643, dimension considerations show that f τ = f σ , so we have a σ(p) (f ) = τ (a p (f )) for all primes p.
A refinement of the Eichler-Shimura Conjecture implies that there exists an abelian surface There are only two pairs (|D|, Disc(L)), where provably the Hecke eigenvalues of the newform f generate the ring of integers of L, namely (223, 8) and (1003, 28). For the first pair, the (conjectured) abelian surface attached to the form is principally polarizable (see [18,Corollary 2.12 and Proposition 3.11]) in contrast to the second pair, for which this does not seem to be the case. So we will only focus on the first pair, for which we found the associated abelian surface. In that case, Table 2 for the Hecke eigenvalues of the form, which we computed using Yasaki's algorithm [40] implemented in Magma [4]). Our current approach does not allow us to find the remaining surfaces. (We elaborate on this in Remark 5.5.) For the discriminant |D| = 223, since A is principally polarized and has real multiplication by , it corresponds to a K-rational point on the Humbert surface Y − (8) of discriminant 8. In their recent paper [11], Elkies and Kumar give an explicit rational model for Y − (8) as a double-cover of the weighted projective space P 2 r,s . We look for A using this model. In fact, the same heuristic as in [9,Proposition 6 and Remark 5] show that A must be the base change of a surface B defined over Q. Indeed, our newform satisfies the identity f τ = f σ . So by Theorem 4.1, it admits a lift to a classical Siegel newform g of genus 2, weight 2 and level D 2 , with integer coefficients. Moreover, g is not a Gritsenko lift. So, assuming Conjecture 1.1, g corresponds to an abelian surface B over Q such that End Q (B) = Z and L(B, s) = L(g, s).
So A = Jac(C) has everywhere good reduction.
Remark 5.5. For the discriminants |D| = 415, 1003, the class number of L is 2. The same heuristics as in [9,Remark 8] suggest that the surface A is likely not principally polarized. Further, for |D| = 415, 455, 571, A does not have real multiplication by the maximal order in L. So finding these surfaces will require working with more general Humbert surfaces for which no explicit models are yet available.

Proof of modularity
The above discussion already proves Theorem 5.2(a). We will now show that A is modular, hence completing the proof of the theorem. In fact, we already have strong evidence that this is the case. Indeed, let p be a prime, and L p (A, s) (respectively, L p (f, s) and L p (f τ , s)) be the Euler factor of A (respectively, f and f τ ) at p. As a built-in of our search method, we already know that for each prime p listed in Table 2, we have Let λ ⊂ O L be a prime ideal. We recall that the λ-adic Tate module of A is given by where O L,λ is the completion of O L at λ, and This is naturally endowed with an action of Gal(Q/K) giving rise to the λ-adic representation In the rest of this section, we will drop the reference to the prime in our notation as we are only interested in the prime λ = λ 2 (above 2) for which O L,λ2 = Z 2 [ √ 2]. So our aim is to show that the λ 2 -adic Tate module is isomorphic to the representation associated to f by work of Taylor et al. [3,21,26,37]. But, in order to do so, we must first determine the coefficient field of ρ f .
Proof. By construction, the image of ρ f lies in an extension of Q 2 of degree at most 4. The prime 2 is split in K, and the eigenvalues of the Frobenii at the primes above it are distinct and do not add up to zero (see Table 2), so by [37, Corollary 1] we can take the coefficient field L f to be the one given by the polynomial x 4 + 2x 3 + 3x 2 + 4x + 4. But, there are two split primes above 2 in L f , and the completion of L f at either of them is isomorphic to From Lemma 5.6, we now have We denote their reductions modulo 2 byρ A andρ f , respectively. We will show that ρ A ρ f by making use of the following version of the so-called Faltings-Serre criterion [10,34].
Theorem 5.7 (Faltings-Serre). Let be two continuous representations, whose reductions modulo 2 areρ 1 andρ 2 . Suppose that There is an explicit description of the set T, which we recall here for the sake of completeness (see [10] for more details on this). Assume that the image lies in GL 2 (F 2 ), which will be the case in our example.
Let M be the fixed field of Im(ρ 1 ), the residual image of ρ 1 . (We have the same fixed field for Im(ρ 2 ) sinceρ 1 andρ 2 are isomorphic.) Let M 0 2 (F 2 ) be the set of all trace zero 2 × 2 matrices with coefficients in F 2 . We note that this is a 2-group of order 8, and we consider the set of all extensions M of M that are unramified outside S, such that M is Galois over K and Gal( M/K) M 0 2 (F 2 ) Im(ρ 1 ). Each such M is a compositum of quadratic extensions of M , so there is a canonical isomorphism ϕ M : Gal( M/K) M 0 2 (F 2 ) Im(ρ 1 ). For algorithmic purpose, this set is determined explicitly using class field theory (see [10,Lemma 5.6]). For each M , we then find a prime ideal p M ⊂ O K such that ϕ M (Frob p M ) = (A, B) with Tr(AB) = 0. The set {p M } thus obtained has the desired properties.
Proof of Theorem 5.2(b). We will use Theorem 5.7 to show that ρ A ρ f ; hence that A is modular.
We note that ρ A and ρ f are unramified away from λ 2 (for ρ f this uses the local-global compatibility results in [26]) so they satisfy (ii) with S = {p | 2}. Condition (i) is satisfied by ρ f since f has trivial Nebentypus. It is satisfied by ρ A by basic properties of Tate modules. So we only need to check (iii), find the set T and show that Tr(ρ A (Frob p )) = Tr(ρ f (Frob p )) for all p ∈ T.
Since O L /λ 2 = F 2 , Im(ρ f ) and Im(ρ A ) are both contained in GL 2 (F 2 ) S 3 , where S 3 is the symmetric group on three elements.
First, recall that Tr(ρ f (Frob p )) = a p (f ) mod λ 2 , and that since ρ f is unramified away from 2,ρ f (Frob p ) is either trivial or has order 3 for every odd prime p. In particular, for p = (3), Tr(ρ f (Frob p )) = −3 = 1 ∈ F 2 implies thatρ f (Frob p ) has order 3. Next, for p | 2, the representation ρ f is ordinary at p, so the restriction ofρ f to the decomposition group D p is of the formρ If Im(ρ f | Dp ) was trivial for both primes p | 2, thenρ f would be unramified at 2 (and hence everywhere). In that case, the fixed field of ker(ρ f ) would be an unramified cubic extension of K, which is impossible since the class number of K is 7. So Im(ρ f ) contains an element of order 2, hence Im(ρ f ) = GL 2 (F 2 ). Similarly, one shows that Im(ρ A ) = GL 2 (F 2 ).
Next, let M f and M A be the Galois extensions of K cut out by ρ f and ρ A , respectively. Then M A and M f are S 3 -extensions of K ramified at λ 2 only. To check (iii), we will show that there is a unique such extension. This will force the isomorphism M f M A .
Let N A and N f be the respective normal closures of M A and M f over Q. Since f σ = f mod λ 2 ,ρ f is a base change, so Gal(N f /Q) Z/2Z × S 3 . Similarly, we have Gal(N A /Q) Z/2Z × S 3 . So each extension comes from an S 3 -extension of Q which is unramified outside {2, 223}. Now we will show that there is a unique such S 3 -extension N/Q.