Computing Automorphic Forms on Shimura Curves over Fields with Arbitrary Class Number

We extend methods of Greenberg and the author to compute in the cohomology of a Shimura curve defined over a totally real field with arbitrary class number. Via the Jacquet-Langlands correspondence, we thereby compute systems of Hecke eigenvalues associated to Hilbert modular forms of arbitrary level over a totally real field of odd degree. We conclude with two examples which illustrate the effectiveness of our algorithms. The development and implementation of algorithms to compute with automorphic forms has emerged as a major topic in explicit arithmetic geometry. The first such computations were carried out for elliptic modular forms, and now very large and useful databases of such forms exist [2, 13, 14]. Recently, effective algorithms to compute with Hilbert modular forms over a totally real field F have been advanced. The first such method is due to Dembélé [4, 5], who worked initially under the assumption that F has even degree n = [F : Q] and strict class number 1. Exploiting the Jacquet-Langlands correspondence, systems of Hecke eigenvalues can be identified inside spaces of automorphic forms on B, where B is the quaternion algebra over F ramified precisely at the infinite places of F—whence the assumption that n is even. Dembélé then provides a computationally efficient theory of Brandt matrices associated to B. This method was later extended (in a nontrivial way) to fields F of arbitrary class number by Dembélé and Donnelly [6]. When the degree n is odd, a different algorithm has been proposed by Greenberg and the author [8], again under the assumption that F has strict class number 1. This method instead locates systems of Hecke eigenvalues in the (degree one) cohomology of a Shimura curve, now associated to the quaternion algebra B ramified at all but one real place and no finite place. This method uses in a critical way the computation of a fundamental domain and a reduction theory for the associated quaternionic unit group [16]; see Section 1 for an overview. In this article, we extend this method to the case where F has arbitrary (strict) class number. Our main result is as follows; we refer the reader to Sections 1 and 2 for precise definitions and notation. Theorem 1. There exists an (explicit) algorithm which, given a totally real field F of degree n = [F : Q], a quaternion algebra B over F ramified at all but one real place, an ideal N of F coprime to the discriminant D of B, and a weight k ∈ (2Z>0), computes the system of eigenvalues for the Hecke operators Tp with p DN and the Atkin-Lehner involutions Wpe with p e ‖ DN acting on the space of quaternionic modular forms S k (N) of weight k and level N for B. In other words, there exists an explicit finite procedure which takes as input the field F , its ring of integers ZF , a quaternion algebra B over F , an ideal N ⊂ ZF , and the vector k encoded in bits (each in the usual way), and outputs a finite set of number fields Ef ⊂ Q and sequences (af (p))p encoding the Hecke eigenvalues for each cusp form constituent f in S k (N), with af (p) ∈ Ef . From the Jacquet-Langlands correspondence, applying the above theorem to the special case where D = (1) (and hence n = [F : Q] is odd), we have the following corollary. Corollary 2. There exists an algorithm which, given a totally real field F of odd degree n = [F : Q], an ideal N of F , and a weight k ∈ (2Z>0), computes the system of eigenvalues for the Hecke operators Tp and Atkin-Lehner involutions Wpe acting on the space of Hilbert modular cusp forms Sk(N) of weight k and level N. This corollary is not stated in its strongest form: in fact, our methods overlap with the methods of Dembélé and his coauthors whenever there is a prime p which exactly divides the level; see Remark 5 for more detail. Combining these methods, Donnelly and the author [7] are systematically enumerating tables of Hilbert modular forms, and the details of these computations (including the dependence on the weight, level, and class number, as well as a comparison of the runtime complexity of the steps involved) will be reported there [7], after further careful optimization. A third technique to compute with automorphic forms, including Hilbert modular forms, has been advanced by Gunnells and Yasaki [9]. They instead use the theory of Voronŏı reduction and sharbly complexes; their work is independent of either of the above approaches. This article is organized as follows. In Section 1, we give an overview of the basic algorithm of Greenberg and the author which works over fields F with strict class number 1. In Section 2, using an adelic language we address the complications which arise over fields of arbitrary class number, and in Section 3 we make this theory concrete and provide the explicit algorithms announced in Theorem 1. Finally, in Section 4, we consider two examples, one in detail; our computations are performed in the computer system Magma [1]. The author would like to thank Steve Donnelly and Matthew Greenberg for helpful discussions as well as the referees for their comments. The author was supported by NSF Grant No. DMS-0901971. 1 An overview of the algorithm for strict class number 1 In this section, we introduce the basic algorithm of Greenberg and the author [8] with a view to extending its scope to base fields of arbitrary class number; for further reading, see the references contained therein. Let F be a totally real field of degree n = [F : Q] with ring of integers ZF . Let F + be the group of totally positive elements of F and let Z × F,+ = Z × F ∩ F + . Let B be a quaternion algebra over F of discriminant D. Suppose that B is split at a unique real place v1, corresponding to an embedding ι∞ : B ↪→ B⊗R ∼= M2(R), and ramified at the other real places v2, . . . , vn. Let O(1) ⊂ B be a maximal order and let O(1)×+ = {γ ∈ O(1) : v1(nrd(γ)) > 0} = {γ ∈ O(1) : nrd(γ) ∈ Z×F,+} denote the group of units of O(1) with totally positive reduced norm. Let Γ (1) = ι∞(O(1)×+/Z×F ) ⊂ PGL2(R), so that Γ (1) acts on the upper half-plane H = {z ∈ C : Im(z) > 0} by linear fractional transformations. Let N ⊂ ZF be an ideal coprime to D, let O = O0(N) be an Eichler order of level N, and let Γ = Γ0(N) = ι∞(O0(N)×+/Z×F ). Let k = (k1, . . . , kn) ∈ (2Z>0) be a weight vector; for example, the case k = (2, . . . , 2) of parallel weight 2 is of significant interest. Let S k (N) denote the finite-dimensional C-vector space of quaternionic modular forms of weight k and level N for B. Roughly speaking, a form f ∈ S k (N) is an analytic function f : H →Wk(C) which is invariant under the weight k action by the group γ ∈ Γ , where Wk(C) is an explicit right B -module [8, (2.4)] and Wk(C) = C when k is parallel weight 2. The space S k (N) comes equipped with the action of Hecke operators Tp for primes p DN and Atkin-Lehner involutions Wpe for prime powers p ‖ DN. The Jacquet-Langlands correspondence [8, Theorem 2.9] (see Hida [10, Proposition 2.12]) gives an isomorphism of Hecke modules S k (N) ∼ −→ Sk(DN), where Sk(DN) D-new denotes the space of Hilbert modular cusp forms of weight k and level DN which are new at all primes dividing D. Therefore, as Hecke modules one can compute equivalently with Hilbert cusp forms or with quaternionic modular forms. We compute with the Hecke module S k (N) by identifying it as a subspace in the degree one cohomology of Γ (1), as follows. Let Vk(C) be the subspace of the algebra C[x1, y1, . . . , xn, yn] consisting of those polynomials q which are homogeneous in (xi, yi) of degree wi = ki − 2. Then Vk(C) has a right action of the group B given by q(x1, y1, . . . , xn, yn) = ( n ∏ i=1 (det γi) −wi/2 ) q((x1 y1)γ1, . . . , (xn yn)γn) (1) for γ ∈ B, where denotes the standard involution (conjugation) on B and γi = vi(γ) ∈ M2(C). By the theorem of Eichler and Shimura [8, Theorem 3.8], we have an isomorphism of Hecke modules S k (N) ∼ −→ H ( Γ, Vk(C) )+ where the group cohomology H denotes the (finite-dimensional) C-vector space of crossed homomorphisms f : Γ → Vk(C) modulo coboundaries and + denotes the +1-eigenspace for complex conjugation. By Shapiro’s lemma [8, §6], we then have a further identification S k (N) ∼ −→ H ( Γ, Vk(C) )+ ∼= H(Γ (1), V (C)), (2) where V (C) = Coind Γ (1) Γ Vk(C). In the isomorphism (2), the Hecke operators act as follows. Let p be a prime of ZF with p DN and let Fp denote the residue class field of p. Since F has strict class number 1, by strong approximation [15, Theorème III.4.3] there exists π ∈ O such that nrdπ is a totally positive generator for p. It follows that there are elements γa ∈ O + , indexed by a ∈ P(Fp), such that