Superspecial Abelian Varieties and the Eichler Basis Problem for Hilbert 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.

to appear in J.N.T