Hironori Shiga

Algebraic Values of Schwarz Triangle Functions

We consider Schwarz maps for triangles whose angles are rather general rational multiples of π. Under which conditions can they have algebraic values at algebraic arguments? The answer is based mainly on considerations of complex multiplication of certain Prym varieties in Jacobians of hypergeometric curves. The paper can serve as an introduction to transcendence techniques for hypergeometric functions, but contains also new results and examples.

To the Hilbert class field from the hypergeometric modular function

Abstract In this article we make an explicit approach to the problem: “For a given CM field M , construct its maximal unramified abelian extension C ( M ) by the adjunction of special values of certain modular functions” in some restricted cases with [ M : Q ] ≥ 4 . We make our argument based on Shimuras main result on the complex multiplication theory of his article in 1967. His main result treats CM fields embedded in a quaternion algebra B over a totally real number field F . We determine the modular function which gives the canonical model for all B s coming from arithmetic triangle groups. That is our main theorem. As its application, we make an explicit case-study for B corresponding to the arithmetic triangle group Δ ( 3 , 3 , 5 ) . By using the modular function of K. Koike obtained in 2003, we show several examples of the Hilbert class fields as an application of our theorem to this triangle group.

Some Classical Problems in Number Theory via the Theory of K3 Surfaces

The theory of the elliptic modular function plays an important role in many situations in number theory. The elliptic modular function is obtained as a one-to-one correspondence between the parameter space of the family of elliptic curves (given by the Weierstrass normal form) and its period domain (i.e., the complex upper half plane). The K3 surface is considered to be a two-dimensional counterpart of the elliptic curve. So, if we consider a family of algebraic K3 surfaces with some normal form, we can obtain its modular function. We call it a K3 modular function (see [18, 19], some mathematical physicists call it a mirror map for K3 surfaces).

Appendix: Periods on the Kummer surface

Let ω1, …,ω4 be points on (z, w)-space C2 those are independent over R. Let T be a complex torus defined by

Criteria for complex multiplication and transcendence properties of automorphic functions.

{C^2}/\left( {Z{\omega _1} + \ldots + Z{\omega _4}} \right).

Triangle Fuchsian differential equations with apparent singularities

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