Differentiating Siegel modular forms, and the moving slope of ${\mathcal A}_g$

TL;DR

This paper investigates the geometry of the moduli space of abelian varieties by constructing a new differential operator on Siegel modular forms, providing bounds on the moving slope of these spaces.

Contribution

It introduces a novel non-linear holomorphic differential operator on Siegel modular forms and applies it to derive bounds on the moving slope of ${ m A}_g$ for various g.

Findings

Recovered known divisors for g ≤ 4

Provided explicit upper bounds for g=5

Proposed conjectural bounds for g=6

Abstract

We study the cone of moving divisors on the moduli space ${\mathcal A}_g$ of principally polarized abelian varieties. Partly motivated by the generalized Rankin-Cohen bracket, we construct a non-linear holomorphic differential operator that sends Siegel modular forms to Siegel modular forms, and we apply it to produce new modular forms. Our construction recovers the known divisors of minimal moving slope on ${\mathcal A}_g$ for $g\leq 4$, and gives an explicit upper bound for the moving slope of ${\mathcal A}_5$ and a conjectural upper bound for the moving slope of ${\mathcal A}_6$.