Hidekazu Furusho

p-adic multiple zeta values II. Tannakian interpretations

We establish a Tannakian formalism of p-adic multiple polylogarithms and p-adic multiple zeta values introduced in our previous paper via a comparison isomorphism between a de Rham fundamental torsor and a rigid fundamental torsor of the projective line minus three points and also discuss its Hodge and étale analogues. As an application we give a way to erase log poles of p-adic multiple polylogarithms and introduce overconvergent p-adic multiple polylogarithms which might be p-adic multiple analogue of Zagiers single-valued complex polylogarithms.

Regularization and generalized double shuffle relations for

We will introduce a regularization for p-adic multiple zeta values and show that the generalized double shue relations hold. This settles a question raised by Deligne, given as a project in the Arizona Winter School 2002. Our approach is to use the theory of Coleman functions on the moduli space of genus zero curves with marked points and its compactication. The main ingredients are the analytic continuation of Coleman functions to the normal bundle of divisors at innit y and denition of a special tangential base point on the moduli space.

p

We construct p-adic multiple L-functions in several variables, which are generalizations of the classical Kubota–Leopoldt p-adic L-functions, by using a specific p-adic measure. Our construction is from the p-adic analytic side of view, and we establish various fundamental properties of these functions. (a) Evaluation at non-positive integers: We establish their intimate connection with the complex multiple zeta-functions by showing that the special values of the p-adic multiple L-functions at non-positive integers are expressed by the twisted multiple Bernoulli numbers, which are the special values of the complex multiple zeta-functions at non-positive integers. (b) Multiple Kummer congruences: We extend Kummer congruences for Bernoulli numbers to congruences for the twisted multiple Bernoulli numbers. (c) Functional relations with a parity condition: We extend the vanishing property of the Kubota–Leopoldt p-adic L-functions with odd characters to our p-adic multiple L-functions. (d) Evaluation at positive integers: We establish their close relationship with the p-adic twisted multiple star polylogarithms by showing that the special values of the p-adic multiple L-functions at positive integers are described by those of the p-adic twisted multiple star polylogarithms at roots of unity, which generalizes the result of Coleman in the single variable case.

-adic multiple zeta values

Consider an elliptic curve defined over an imaginary quadratic field K with good reduction at the primes above p ≥ 5 and with complex multiplication by the full ring of integers of K. In this paper, we construct p -adic analogues of the Eisenstein-Kronecker series for such an elliptic curve as Coleman functions on the elliptic curve. We then prove p -adic analogues of the first and second Kronecker limit formulas by using the distribution relation of the Kronecker theta function.

Fundamentals of p -adic multiple L -functions and evaluation of their special values

This is an expose of the theory of p‐adic multiple zeta value developed by the author. This theory is almost parallel to the story of the complex case. p‐adic multiple polylogarithm, which is a p‐adic function, is constructed by Coleman’s p‐adic iterated integration theory. p‐adic multiple zeta value is defined to be its special value at 1. Many nice properties of p‐adic multiple zeta value is shown. p‐adic KZ equation is also introduced and the p‐adic Drinfel’d associator is constructed from two fundamental solutions of the p‐adic KZ equation. A relation of p‐adic multiple zeta value with the p‐adic Drinfel’d associator as well as its relation with the fundamental group of the projective line minus three points is established.