# CM Periods, CM Regulators, and Hypergeometric Functions, I

@article{Asakura2018CMPC,
title={CM Periods, CM Regulators, and Hypergeometric Functions, I},
author={Masanori Asakura and Noriyuki Otsubo},
year={2018},
volume={70},
pages={481 - 514}
}
• Published 27 March 2015
• Mathematics
Abstract We prove the Gross–Deligne conjecture on CM periods for motives associated with ${{H}^{2}}$ of certain surfaces fibered over the projective line. Then we prove for the same motives a formula which expresses the ${{K}_{1}}$ -regulators in terms of hypergeometric functions $_{3}{{F}_{2}}$ , and obtain a new example of non-trivial regulators.
11 Citations
CM periods, CM regulators and hypergeometric functions, II
• Mathematics
• 2018
We study periods and regulators of a certain class of fibrations of varieties whose relative $$H^1$$H1 has multiplication by a number field. Both are written in terms of values of hypergeometric
AN ALGEBRO-GEOMETRIC STUDY OF SPECIAL VALUES OF HYPERGEOMETRIC FUNCTIONS $_{3}F_{2}$
• Mathematics
Nagoya Mathematical Journal
• 2018
For a certain class of hypergeometric functions $_{3}F_{2}$ with rational parameters, we give a sufficient condition for the special value at $1$ to be expressed in terms of logarithms of algebraic
Regulators of $$K_2$$K2 of hypergeometric fibrations
We describe Beilinson regulators of hypergeometric fibrations in terms of generalized hypergeometric functions. Employing a formula of Rogers and Zudilin, we obtain the comparison of the regulator
Regulators of K_2 of Hypergeometric Fibrations
We discuss Beilinson's regulator on K_2 of certain fibrations of algebraic varieties which we call the hypergeomtric fibrations. The main result is to describe regulators via the hypergeometric
Explicit logarithmic formulas of special values of hypergeometric functions 3F2
• Mathematics
Communications in Contemporary Mathematics
• 2019
In [M. Asakura, N. Otsubo and T. Terasoma, An algebro-geometric study of special values of hypergeometric functions [Formula: see text], to appear in Nagoya Math. J.;
A FUNCTIONAL LOGARITHMIC FORMULA FOR THE HYPERGEOMETRIC FUNCTION $_{3}F_{2}$
• Mathematics
Nagoya Mathematical Journal
• 2018
We give a sufficient condition for the hypergeometric function $_{3}F_{2}$ to be a linear combination of the logarithm of algebraic functions.
Regulators of K_1 of Hypergeometric Fibrations
• Mathematics
• 2017
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
New p-adic hypergeometric functions concerning with syntomic regulators
We introduce new functions, which we call the p-adic hypergeometric functions of logarithmic type. We show the congruence relations that are similar to Dwork's. This implies that they are convergent
An algebro-geometric study of the unit arguments ${}_3F_2(1,1,q;a,b;1)$, I
• Mathematics
• 2016
Let $a$, $b$, $q$ be rational numbers such that none of $a$, $b$, $q$, $q-a$, $q-b$, $q-a-b$ is an integer. Then we prove, ${}_3F_2(1,1,q;a,b;1)$ is a $\overline{\mathbb{Q}}$-linear combination of
A functional logarithmic formula for hypergeometric functions 3F2
• Mathematics
• 2018
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

## References

SHOWING 1-10 OF 42 REFERENCES
On the periods of motives with complex multiplication and a conjecture of Gross-Deligne
• Mathematics
• 2002
We prove that the existence of an automorphism of finite order on a Q-variety X implies the existence of algebraic linear relations between the logarithm of certain periods of X and the logarithm of
On the regulator of Fermat motives and generalized hypergeometric functions
Abstract 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
Automorphic forms and the periods of abelian varieties
$m$ . The non-vanishing of the first cohomology group of a discrete subgroup of $SU(n, 1)$ . To describe our results, let $A$ be an abelian variety of dimension $g$ defined over $\overline{Q}$ ,
Hypergeometric Abelian Varieties
Abstract In this paper, we construct abelian varieties associated to Gauss’ and Appell–Lauricella hypergeometric series. Abelian varieties of this kind and the algebraic curves we define to construct
Periods of Hodge structures and special values of the gamma function
At the end of the 1970s, Gross and Deligne conjectured that periods of geometric Hodge structures with multiplication by an abelian number field are products of values of the gamma function at
On Special Values of Jacobi-Sum Hecke L-Functions
The Beilinson conjecture numerically is verified numerically for some cases and formulas for the values of L-functions at 0 appear analogous to the Chowla–Selberg formula for the periods of elliptic curves with complex multiplication.
ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC VARIETIES
This is the final project for Prof. Bhargav Bhatt’s Math 613 in the Winter of 2015 at the University of Michigan. The purpose of this paper is to prove a result of Grothendieck [Gro66] showing, for a
On the periods of abelian integrals and a formula of Chowla and Selberg
Given an imaginary quadratic field k of discriminant d , let E be an elliptic curve defined over Q, the algebraic closure of Q in C, which admits complex multiplication by some order in k. Let ~ be a
Determinant of period integrals
• Mathematics
• 1997
We prove a formula for the determinant of period integrals. Period integrals arise from comparison between Betti cohomologies and de Rham cohomologies. Our formula Theorem 1 in Section 4 expresses