大坪 紀之

We give a precise description of the homology group of the Fermat curve as a cyclic module over a group ring. As an application, we prove the freeness of the profinite homology of the Fermat tower. This allows us to define measures, an equivalent of Anderson's adelic beta functions, in a manner similar to Ihara's definition of l-adic universal power series for Jacobi sums. We give a simple proof of the interpolation property using a motivic decomposition of the Fermat curve.

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 formulas 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 to the L-values at 1 by the Birch-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 kth 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 regulator 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.

We compare two calculations due to Bloch and the author of the regulator of an elliptic curve with complex multiplication which is a quotient of a Fermat curve, and express the special value of its L-function at s = 0 in terms of special values of generalized hypergeometric functions. (C) 2010 Elsevier Inc. All rights reserved.

We show the finiteness of the p-primary torsion part of the Chow group of zero-cycles on the Fermat quartic surface F over Q and Q(root-1) for p X 6. Proving this, we also compute the (positive) Z(p)-corank of the Selmer groups associated to H-2(F)(2), and construct integral elements in K-1, which are in correspondence with conjectures of Beilinson and Bloch-Kato.

Conjecures of Beilinson and Bloch-Kato describe the order of zeros of L-functions of motives in terms of motivic cohomology groups and Selmer groups. We restrict our attention to the parts generated by the cycle classes, and give modest evidence for the conjectures.