Gauss-Manin connection in disguise: Genus two curves

TL;DR

This paper explores an algebra of meromorphic functions on the genus two Siegel domain, incorporating modular forms and their derivatives, using moduli of genus two curves and Gauss-Manin connections.

Contribution

It introduces a new algebraic framework that unifies modular forms and their derivatives on genus two curves via Gauss-Manin connections.

Findings

Contains Siegel modular forms for an index six subgroup

Algebra is closed under three canonical derivatives

Links moduli of genus two curves with modular forms

Abstract

We describe an algebra of meromorphic functions on the Siegel domain of genus two which contains Siegel modular forms for an arithmetic index six subgroup of the symplectic group and it is closed under three canonical derivations of the Siegel domain. The main ingredients of our study are the moduli of enhanced genus two curves, Gauss-Manin connection and the modular vector fields living on such moduli spaces.