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**Computing genus-2 Hilbert-Siegel modular forms over \(\mathbb Q(\sqrt{5})\) via the Jacquet-Langlands correspondence.**
*(English)*
Zbl 1246.11102

From the introduction: Let \(F\) be a real quadratic field of narrow class number one and let \(B\) be the unique (up to isomorphism) quaternion algebra over \(F\) which is ramified at both archimedean places of \(F\) and unramified everywhere else. Let \(\text{GU}_2(B)\) be the unitary similitude group of \(B^{\oplus 2}\). This is the set of \(\mathbb Q\)-rational points of an algebraic group \(G^B\) defined over \(\mathbb Q\). The group \(G^B\) is an inner form of \(G := \text{Res}_{F/\mathbb Q}(\text{GSp}_4)\) such that \(G^B(R)\) is compact modulo its centre.

In this paper the authors develop an algorithm which computes automorphic forms on \(G^B\) in the following sense: given an ideal \(N\) in \(\mathcal O_F\) and an integer \(k\) greater than 3, the algorithm returns the Hecke eigensystems of all automorphic forms \(f\) of level \(N\) and parallel weight \(k\). More precisely, given a prime \(\mathfrak p\) in \(\mathcal O_F\), the algorithm returns the Hecke eigenvalues of \(f\) at \(\mathfrak p\), and hence the Euler factor \(L_{\mathfrak p}(f, s)\), for each eigenform \(f\) of level \(N\) and parallel weight \(k\).

The algorithm is a generalization of the one developed by the second author in [Exp. Math. 14, No. 4, 457–466 (2005; Zbl 1152.11328)] to the genus 2 case. Although the algorithm is only described in the case of a real quadratic field in this paper, it should be clear that it can be adapted to any totally real number field of narrow class number one.

The Jacquet-Langlands correspondence predicts the existence of a transfer map \(JL: \Pi(G^B) \to\Pi(G)\) from automorphic representations of \(G^B\) to automorphic representations on \(G\), which is injective, matches \(L\)-functions and enjoys other properties compatible with the principle of functoriality; in particular, the image of the Jacquet-Langlands correspondence is to be contained in the space of holomorphic automorphic representations. If one admits this conjecture, then the algorithm above provides a way to produce examples of cuspidal Hilbert-Siegel modular forms of genus 2 over \(F\) and allows them to compute the \(L\)-factors of the corresponding automorphic representations for arbitrary finite primes \(p\) of \(F\).

In fact, they are also able to use these calculations to provide evidence for the Jacquet-Langlands correspondence itself by comparing the Euler factors found with those of known Hilbert-Siegel modular forms obtained by lifting. This is done in the final section of the paper where they observe that some of the Euler factors computed match those of lifts of Hilbert modular forms, for the primes computed. Although this does not definitively establish that these Hilbert-Siegel modular forms are indeed lifts, in principle one can establish equality in this way, using an analogue of the Sturm bound.

A short historical survey on the computation of Siegel modular forms and a presentation of numerical results for the quadratic field \(\mathbb Q(\sqrt 5)\) are given in the last section.

In this paper the authors develop an algorithm which computes automorphic forms on \(G^B\) in the following sense: given an ideal \(N\) in \(\mathcal O_F\) and an integer \(k\) greater than 3, the algorithm returns the Hecke eigensystems of all automorphic forms \(f\) of level \(N\) and parallel weight \(k\). More precisely, given a prime \(\mathfrak p\) in \(\mathcal O_F\), the algorithm returns the Hecke eigenvalues of \(f\) at \(\mathfrak p\), and hence the Euler factor \(L_{\mathfrak p}(f, s)\), for each eigenform \(f\) of level \(N\) and parallel weight \(k\).

The algorithm is a generalization of the one developed by the second author in [Exp. Math. 14, No. 4, 457–466 (2005; Zbl 1152.11328)] to the genus 2 case. Although the algorithm is only described in the case of a real quadratic field in this paper, it should be clear that it can be adapted to any totally real number field of narrow class number one.

The Jacquet-Langlands correspondence predicts the existence of a transfer map \(JL: \Pi(G^B) \to\Pi(G)\) from automorphic representations of \(G^B\) to automorphic representations on \(G\), which is injective, matches \(L\)-functions and enjoys other properties compatible with the principle of functoriality; in particular, the image of the Jacquet-Langlands correspondence is to be contained in the space of holomorphic automorphic representations. If one admits this conjecture, then the algorithm above provides a way to produce examples of cuspidal Hilbert-Siegel modular forms of genus 2 over \(F\) and allows them to compute the \(L\)-factors of the corresponding automorphic representations for arbitrary finite primes \(p\) of \(F\).

In fact, they are also able to use these calculations to provide evidence for the Jacquet-Langlands correspondence itself by comparing the Euler factors found with those of known Hilbert-Siegel modular forms obtained by lifting. This is done in the final section of the paper where they observe that some of the Euler factors computed match those of lifts of Hilbert modular forms, for the primes computed. Although this does not definitively establish that these Hilbert-Siegel modular forms are indeed lifts, in principle one can establish equality in this way, using an analogue of the Sturm bound.

A short historical survey on the computation of Siegel modular forms and a presentation of numerical results for the quadratic field \(\mathbb Q(\sqrt 5)\) are given in the last section.

Reviewer: Olaf Ninnemann (Berlin)

### MSC:

11F41 | Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces |

11F70 | Representation-theoretic methods; automorphic representations over local and global fields |