On the Central Derivative of Hecke L-Series

Abstract

The well-known Birch and Swinnerton-Dyer conjecture gives a deep connection between the leading coefficient of the L-series and the arithmetic properties of an abelian variety. Both are very important and subtle. This paper is part of an effort to compute the analytic side explicitly in a special case. Indeed, we are interested in the central derivative of certain algebraic Hecke L-series, related to CM abelian varieties or more precisely pieces of it (CM motives). The result, together with the Gross-Zagier formula proved by Zhang ([Zh]), would also give a new way to compute the height of certain Heegner cycles on a Kuga-Sato variety. Let p ≡ 3 mod 4 be a prime number such that p > 3. Let k ≥ 0 be an integer. Let E = Q( √−p) and view it as a subfield of C such that √−p = i√p. Let hp be the ideal class number of E. A canonical Hecke character of E of weight 2k + 1 is a Hecke character μ satisfying