The modularity of an abelian variety
Jae-Hyun Yang
TL;DR
The paper proposes a structural generalization of modularity from elliptic curves to higher-dimensional simple abelian varieties over $\mathbb Q$ by using Siegel modular forms of genus $g$ and their $L$-functions. It defines a modularity criterion for $A$ through a weight $g+1$ Siegel Hecke eigenform $F$ and discusses spinor and standard zeta functions $Z_F(s)$, $D_F(s)$, and $L(A,s)$, together with conjectural equivalences MAV1–MAV4. It also provides a rigorous Hecke-algebra framework, including $p$-adic Satake parameters $\alpha_0,\dots,\alpha_g$, which control eigenvalues and local $L$-factors, supporting a program to attach Galois representations to Siegel eigenforms. The work outlines pathways toward a Langlands-type correspondence for abelian varieties and sets open problems for constructing modularity proofs beyond dimension one.
Abstract
We introduce the concept of the modularity of an abelian variety defined over the rational number field extending the modularity of an elliptic curve. We discuss the modularity of an abelian variety over the rational number field. We conjecture that a simple abelian variety over the rational number field is modular.