Differentiating Siegel modular forms, and the moving slope of \mathcal A_g

2022/07/08 by Grushevsky, Samuel, Ibukiyama, Tomoyoshi, Mondello, Gabriele +1
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2207.04139

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≤ 4, and gives an explicit upper bound for the moving slope of \mathcal A₅ and a conjectural upper bound for the moving slope of \mathcal A₆.

Related

• Siegel modular forms mod p
• Siegel modular forms (mod p) and algebraic modular forms
• Vector-valued modular forms and the Gauss map
• The geometry of Siegel modular varieties
• Siegel modular forms of degree three and the cohomology of local systems