Moduli of Kummer varieties and its applications to number theory.

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The aim of this study is to clarify the structure of the modular variety, which is, in the case of genus 1, is the Zariski-closure of the set {(j(γ), j(nγ))|γ's are the points of the upper-half plane}. Here j(γ) is the j-invariant. When the genus is equal to 2, the invariants are known but rather complicated. So we take the Riemann theta constant θ_<mn>(γ), where m, n are half-integral vectors and γ is a point in the Siegel upper-half space, in place of the invariants. Classically the ratio of the θ_<mn>(γ)^2 is called the moduli of the abelian variety associated with γ.We consider the map Θ : S_g → IP^N(N = 2^g - 1), Θ(γ) = (… : Θ_<aO>(2γ) : …)(a ∈ 1/2Z^g/Z^g). The image Θ(γ) is essentially nothing but the moduli of γ.Let p be an odd prime. The Zariski closure of the set {(Θ(γ), Θ(pγ))|γ ∈ S_g} ⊂ IP^N × IP^N is called the modular variety of degree p and level (2,4).The defining eqations of the modular variety of degree 3 are known. In our study, we get the defining equations of modular variety of degree 7.In the course of the above study, we reach to hermitian theta functions which is closely connected to the moduli of Kummer surfaces. For these hermitian theta functions, we get the derivative formula, which is an analogue of the well-known Jacobi's derivative formula and Rosenhain's one.After that we begin to study hermitian modular forms and hermitian Jacobi forms and give an analogue of Saito-Kurokawa conjecture for hermitian Jacobi forms of degree 1.Our original study is to investigate modular varieties of genus g 【greater than or equal】 2. We have some small results about the structure of these varieties, and will try to study these objects more deeply.