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We define the adelic hypergeometric function of special Gaussian type by means of a tower of hypergeometric curves. This function takes values in an adelic completed group ring and interpolates all the hypergeometric functions of the same type over all finite fields. It specializes at the unit argument to the adelic beta function of Ihara and Anderson. We prove some transformation formulas and a summation formula for the adelic hypergeometric function, which are known classically for complex hypergeometric functions.

We study the Gauss and Jacobi sums from a viewpoint of motives. We exhibit isomorphisms between Chow motives arising from the Artin-Schreier curve and the Fermat varieties over a finite field, that can be regarded as (and yield a new proof of) classically known relations among Gauss and Jacobi sums such as Davenport-Hasse's multiplication formula. As a key step, we define motivic analogues of the Gauss and Jacobi sums as algebraic correspondences, and show that they represent the Frobenius endomorphisms of such motives. This generalizes Coleman's result for curves. These results are applied to investigate the group of invertible Chow motives with coefficients in a cyclotomic field., Comment: 30 pages. Remark 6.7 was added

Many product formulas are known classically for generalized hypergeometric functions over the complex numbers. In this paper, we establish some analogous formulas for generalized hypergeometric functions over finite fields.

We give a definition of generalized hypergeometric functions over finite fields using modified Gauss sums, which enables us to find clear analogy with classical hypergeometric functions over the complex numbers. We study their fundamental properties and prove summation formulas, transformation formulas and product formulas. An application to zeta functions of K3-surfaces is given. In the appendix, we give an elementary proof of the Davenport-Hasse multiplication formula for Gauss sums.

We give a new method to prove in a uniform and easy way various transformation formulas for Gauss hypergeometric functions. The key is Jacobi's canonical form of the hypergeometric differential equation. Analogy for $q$-hypergeometric functions is also studied., Comment: 18 pages, formerly entitled "A canonical form of the hypergeometric equation and transformation formulas"

We give a sufficient condition for that the hypergeometric function 3F2 is a linear combination of the logarithmic function. The proof is based on the regulator formula which we proved in another preprint, arXiv:1709.04144., Comment: 12pages

We study a deformation of what we call hypergeometric fibrations. Its periods and K_1-regulators are described in terms of hypergeometric functions 3F2 in a variable given by the deformation parameter., Comment: 26 pages

For certain class of hypergeometric functions ${}_3F_2$ with rational parameters, we give a sufficient condition for the special value at $1$ to be expressed in terms of logarithms of algebraic numbers. We give two proofs, both of which are algebro-geometric and related to higher regulators., Comment: 11 pages. The title was changed from "An algebro-geometric study of the unit arguments ${}_3F_2(1,1,q;a,b;1)$, I"

We study periods and regulators of a certain class of fibrations of varieties whose relative $H^1$ has multiplication by a number field. Both are written in terms of values of hypergeometric functions ${}_3F_2$ and the former reduces to values of the gamma function, which provide examples of the conjecture of Gross-Deligne., Comment: 25 pages

For motives associated with Fermat curves, there are elements in motivic cohomology whose regulators are written in terms of special values of generalized hypergeometric functions. Using them, we verify the Beilinson conjecture numerically for some cases and find formulae for the values of L-functions at 0. These appear analogous to the Chowla-Selberg formula for the periods of elliptic curves with complex multiplication, which are related with the L-values at 1 by the Birch and Swinnerton-Dyer conjecture.

We study the Abel-Jacobi image of the Ceresa cycle W_k-W_k^-, where W_k is the image of the k-th symmetric product of a curve X on its Jacobian variety. For the Fermat curve of degree N, we express it in terms of special values of generalized hypergeometric functions and give a criterion for the non-vanishing of W_k-W_k^- modulo algebraic equivalence, which is verified numerically for some N and k.

We calculate the Beilinson regulators of motives associated to Fermat curves and express them by special values of generalized hypergeometric functions. As a result, we obtain surjectivity results of the regulator, which support the Beilinson conjecture on special values of L-functions.