Theta relations with real multiplication by sqrt(3)

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.