Superspecial Abelian Varieties and the Eichler Basis Problem for Hilbert Modular Forms

TL;DR

This paper constructs Hilbert modular newforms of weight two using theta series from superspecial abelian varieties with real multiplication, providing a geometric perspective on the Eichler Basis Problem.

Contribution

It offers a novel geometric construction of Hilbert modular forms via superspecial abelian varieties, connecting isogeny modules to classical theta series.

Findings

Construction of newforms from superspecial abelian varieties

Geometric reinterpretation of the Eichler Basis Problem

Explicit link between isogenies and modular forms

Abstract

Let $p$ be an unramified prime in a totally real field $L$ such that $h^+(L)=1$. Our main result shows that Hilbert modular newforms of parallel weight two for $\Gamma_0(p)$ can be constructed naturally, via classical theta series, from modules of isogenies of superspecial abelian varieties with real multiplication on a Hilbert moduli space. This can be viewed as a geometric reinterpretation of the Eichler Basis Problem for Hilbert modular forms.