On Zagier's conjecture for base extensions of elliptic curves

TL;DR

This paper proves a weak version of Zagier's conjecture for the L-value at 2 of elliptic curves over base extensions of Q, utilizing Beilinson's theorem on modular curves.

Contribution

It introduces a novel approach to Zagier's conjecture for elliptic curves over abelian extensions using Beilinson's theorem.

Findings

Proves a weak form of Zagier's conjecture for E/F at s=2

Utilizes Beilinson's theorem on modular curves

Establishes connections between L-values and motivic cohomology

Abstract

Let E be an elliptic curve over Q, and let F be a finite abelian extension of Q. Using Beilinson's theorem on a suitable modular curve, we prove a weak version of Zagier's conjecture for L(E/F,2), where E/F is the base extension of E to F.