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

Theta relations with real multiplication by sqrt(3)