Moduli Spaces and Special Functions

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The head of investigator MATSUMOTO Keiji constructed period maps and automorphic forms derived from the inverses of period maps for certain families of algebraic varieties by using Prym varieties of algebraic curves.In fact, it was shown that the period map for the family of smooth cubic surfaces could be expressed in terms of periods of the Prym varieties for curves of genus 10. Automorphic forms on the 4-dimensional complex ball giving the inverse of this period map were expressed by theta constants associated to the Prym varieties.For the family of the 4-fold coverings of the complex projective line branching at eight points, the period map from this family to the 5-dimensional complex ball was constructed by using the Prym varieties of these curves. Automorphic forms on the 5-dimensional complex ball giving the inverse of this period map were expressed by theta constants associated to the Prym varieties.SAITO Mutsumi showed that the ring of differential operators on affine tone varieties and the algebra of symmetries of the system of A-hypergeometric differential equations were anti-isomorphic, and classified systems of A-hypergeometric differential equations combinatonally under these symmetries. He studied the condition that the graded ring gr(D(R_A)) was finitely generated, and gave the composition factors of the ring R_A of functions on any tone variety as a D(R_A)-module.SHIMADA Ichiro showed that if the singularity of each singular fiber was not bad for an algebraic fiber space, the boundary homomorphism from the second homotopy group of the base space to the fundamental group of any general fiber could be constructed. He showed that the fundamental group of the complement of a resultant hypersurface was commutative. He also showed that any supersingular K3 surface could be expressed as a branched double cover of the projective plane.