Quinary forms and paramodular forms

TL;DR

This paper establishes a detailed relationship between algebraic modular forms for a specific unitary group and forms from quinary lattices, enabling effective computation of Hecke eigenvalues and revealing new congruences between modular forms.

Contribution

It connects algebraic modular forms with quinary lattice forms, proving conjectures and providing tools for computing Hecke eigenvalues and discovering new congruences.

Findings

Proved conjectures of Ibukiyama on Jacquet-Langlands correspondences.

Developed an effective method for computing Hecke eigenvalues for Siegel modular forms.

Discovered new congruences between modular forms of different degrees.

Abstract

We work out the exact relationship between algebraic modular forms for a two-by-two general unitary group over a definite quaternion algebra, and those arising from genera of positive-definite quinary lattices, relating stabilisers of local lattices with specific open compact subgroups, paramodular at split places, and with Atkin-Lehner operators. Combining this with the recent work of R\"osner and Weissauer, proving conjectures of Ibukiyama on Jacquet-Langlands type correspondences (mildly generalised here), provides an effective tool for computing Hecke eigenvalues for Siegel modular forms of degree two and paramodular level. It also enables us to prove examples of congruences of Hecke eigenvalues connecting Siegel modular forms of degrees two and one. These include some of a type conjectured by Harder at level one, supported by computations of Fretwell at higher levels, and a subtly different congruence discovered experimentally by Buzzard and Golyshev.