PARABOLIC POINTS AND ZETA-FUNCTIONS OF MODULAR CURVES

Abstract

In this paper we obtain explicit formulas for the values at the center of the critical strip of Dirichlet series connected with weight 2 parabolic forms of the group Γ0(N). In particular, these formulas allow us to verify the Birch-Swinnerton-Dyer conjecture on the order of a zero for uniformizable elliptic curves over certain Γ-extensions. We also give applications to noncommutative reciprocity laws.