Computing modular correspondences for abelian varieties

TL;DR

This paper generalizes classical modular polynomials to higher dimensions, providing algebraic equations for modular correspondences between abelian varieties with additional structures.

Contribution

It introduces a higher-dimensional analogue of modular polynomials and explicitly describes the algebraic equations for the associated modular correspondences.

Findings

Provides explicit algebraic equations for higher-dimensional modular correspondences

Extends classical elliptic curve modular polynomial concepts to abelian varieties

Facilitates computational approaches in higher-dimensional moduli spaces

Abstract

The aim of this paper is to give a higher dimensional equivalent of the classical modular polynomials $\Phi_\ell(X,Y)$. If $j$ is the $j$-invariant associated to an elliptic curve $E_k$ over a field $k$ then the roots of $\Phi_\ell(j,X)$ correspond to the $j$-invariants of the curves which are $\ell$-isogeneous to $E_k$. Denote by $X_0(N)$ the modular curve which parametrizes the set of elliptic curves together with a $N$-torsion subgroup. It is possible to interpret $\Phi_\ell(X,Y)$ as an equation cutting out the image of a certain modular correspondence $X_0(\ell) \to X_0(1) \times X_0(1)$ in the product $X_0(1) \times X_0(1)$. Let $g$ be a positive integer and $\overn \in \N^g$. We are interested in the moduli space that we denote by $\Mn$ of abelian varieties of dimension $g$ over a field $k$ together with an ample symmetric line bundle $\pol$ and a symmetric theta structure of type $\overn$. If $\ell$ is a prime and let $\overl=(\ell, ..., \ell)$, there exists a modular correspondence $\Mln \to \Mn \times \Mn$. We give a system of algebraic equations defining the image of this modular correspondence.