On Zagier's conjecture for $L(E,2)$: a number field example

TL;DR

This paper provides a detailed example of Zagier's conjecture for a CM elliptic curve over a real quadratic field, linking L-values to elliptic dilogarithm evaluations via K-theory.

Contribution

It offers the first explicit example of Zagier's conjecture for a CM elliptic curve over a real quadratic field, illustrating the conjecture's application in this context.

Findings

Explicit computation of L(E,2) for the example curve

Connection established between L-values and elliptic dilogarithm values

Demonstration of K-theory's role in the conjecture

Abstract

We work out an example, for a CM elliptic curve E defined over a real quadratic field F, of Zagier's conjecture. This relates L(E,2) to values of the elliptic dilogarithm function at a divisor in the Jacobian of E which arises from K-theory.