The Beilinson Conjectures for CM Elliptic Curves via Hypergeometric Functions

TL;DR

This paper connects special values of L-functions for CM elliptic curves to hypergeometric functions, providing new proofs of Beilinson conjectures and relating them to regulator formulas.

Contribution

It expresses L-values of CM elliptic curves in terms of hypergeometric functions and offers a novel proof of Beilinson's conjectures for these curves.

Findings

L-values at s=2 are expressed via hypergeometric functions.

Comparison with Rogers-Zudilin results and Otsubo's regulator formulas.

Provides a new proof of Beilinson conjectures for CM elliptic curves.

Abstract

We consider certain CM elliptic curves which are related to Fermat curves, and express the values of $L$-functions at $s=2$ in terms of special values of generalized hypergeometric functions. We compare them and a similar result of Rogers-Zudilin with Otsubo's regulator formulas, and give a new proof of the Beilinson conjectures originally due to Bloch.