Theta relations with real multiplication by sqrt(3)

TL;DR

This paper computes new algebraic theta relations that characterize abelian surfaces with real multiplication by sqrt(3), providing explicit descriptions in Mumford's moduli space and proposing a method that may extend to higher discriminants.

Contribution

It introduces a novel algebraic method for deriving theta relations for abelian surfaces with real multiplication by sqrt(3), expanding the understanding of their moduli space.

Findings

Derived explicit theta relations for sqrt(3) real multiplication

Described the locus in Mumford's moduli space using canonical coordinates

Proposed a method potentially applicable to higher discriminants

Abstract

In this article we compute new theta relations which define the moduli space of abelian surfaces with real multiplication by square root three. We give the locus of square root three abelian surfaces in terms of the canonical coordinates on Mumford's moduli space of abelian surfaces with theta structure. The method that we use to compute the real multiplication theta relations is of purely algebraic nature. It differs from Runge's method and recent work by Gruenewald. We expect that our method generalizes to higher discriminants.