$L$-function of CM elliptic curves and generalized hypergeometric functions

TL;DR

This paper links special values of L-functions of certain CM elliptic curves to generalized hypergeometric functions by comparing motivic cohomology elements, providing a new perspective on their arithmetic properties.

Contribution

It introduces a novel method to express L-values of CM elliptic curves related to Fermat curves using generalized hypergeometric functions through motivic cohomology comparisons.

Findings

Expressed L-values in terms of hypergeometric functions.

Connected motivic cohomology elements with special function values.

Provided new insights into the arithmetic of CM elliptic curves.

Abstract

In this paper, we express special values of the $L$-functions of certain CM elliptic curves which are related to Fermat curves in terms of the special values of generalized hypergeometric functions by comparing Bloch's element with Ross's element in the motivic cohomology group.