Periods of modular GL2-type abelian varieties and p-adic integration

TL;DR

This paper generalizes the construction of p-adic lattices associated with modular forms over number fields, extending from elliptic curves to higher-dimensional abelian varieties, and provides numerical evidence for abelian surfaces.

Contribution

It extends the Guitart-Masdeu-Sengun p-adic lattice construction to modular forms over number fields with higher degree, linking to abelian varieties of corresponding dimension.

Findings

Numerical evidence supports the conjectural p-adic lattice construction for abelian surfaces.

The construction generalizes from elliptic curves to higher-dimensional abelian varieties.

The work suggests a deep connection between modular forms and p-adic geometry in higher dimensions.

Abstract

Let F be a number field and N an integral ideal in its ring of integers. Let f be a modular newform over F of level Gamma0(N) with rational Fourier coefficients. Under certain additional conditions, Guitart-Masdeu-Sengun constructed a p-adic lattice which is conjectured to be the Tate lattice of an elliptic curve E whose L-function equals that of f. The aim of this note is to generalize this construction when the Hecke eigenvalues of f generate a number field of degree d >= 1, in which case the geometric object associated to f is expected to be, in general, an abelian variety A of dimension d. We also provide numerical evidence supporting the conjectural construction in the case of abelian surfaces.