Deninger's conjecture on $L$-function of elliptic curves at $s=3$

Abstract

I compute explicitly the regulator map on $K_4(X)$ for an arbitrary curve $X$ over a number field. Using this and Beilinson's theorem about regulators for modular curves ([B2]) I prove a formula expressing the value of the $L$-function $L(E,s)$ of a modular elliptic curve $E$ over $\Bbb Q$ at $s=3$ by the double Eisenstein-Kronecker series.