Value at 2 of the L-function of an elliptic curve

Abstract

We study the special value at 2 of L-functions of modular forms of weight 2 on congruence subgroups of the modular group. We prove an explicit version of Beilinson's theorem for the modular curve X_1(N). When N is prime, we deduce that the target space of Beilinson's regulator map is generated by the images of Milnor symbols associated to modular units of X_1(N). We also suggest a reformulation of Zagier's conjecture on L(E,2) for the jacobian J_1(N) of X_1(N), where E is an elliptic curve of conductor N. In this direction we define an analogue of the elliptic dilogarithm for any jacobian J : it is a function R_J from the complex points of J to a finite-dimensional vector space. In the case J=J_1(N), we establish a link between the aforementioned L-values and the function R_J evaluated at \Q-rational points of the cuspidal subgroup of J.