Quinary forms and paramodular forms

2021/12/07 by Neil Dummigan, Dummigan, Neil, Ariel Pacetti +5 · 3 citations
Mathematics · #11F33 #11F46 #11F55 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2112.03797

openalex publication_date 2021/12/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Rösner 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.

Cited by

• An orthogonal perspective on Gauss composition
• Computing Hilbert modular forms as orthogonal modular forms
• Hermitian modular forms and algebraic modular forms on SO(6)

Related

• A database of paramodular forms from quinary orthogonal modular forms
• Paramodular Cusp Forms
• Siegel modular forms (mod p) and algebraic modular forms
• Hecke eigenvalues of Siegel modular forms (mod p) and of algebraic modular forms
• Siegel Modular Forms and Theta Series attached to quaternion algebras II