Nonvanishing of Central Hecke L-values and Rank of Certain Elliptic Curves

Let D ≡ 7 mod 8 be a positive squarefree integer, and let hD be the ideal class number of ED = Q( √−D). Let d ≡ 1 mod 4 be a squarefree integer relatively prime to D. Then for any integer k ≥ 0 there is a constant M = M(k), independent of the pair (D, d), such that if (−1)k = sign(d), (2k + 1, hD) = 1, and √ D > 12 π d(log |d|+ M(k)), then the central L-value L(k + 1, χ D,d ) > 0. Furthermore, for k ≤ 1, we can take M(k) = 0. Finally, If D = p is a prime, and d > 0, then the associated elliptic curve A(p)d has Mordell-Weil rank 0 (over its definition field) when √ D > 12 π d2 log d. 0. Introduction. Let D ≡ 3 mod 4 be a positive squarefree integer, and let d ≡ 1 mod 4 be a squarefree integer relatively prime to D. We consider Hecke characters χ of ED = Q( √−D) of conductor d√−DO satisfying (1) χ(Ā) = χ(A) for every ideal of ED relatively prime to the conductor, and (2) χ(αO) = ±α for every principal ideal relatively prime to the conductor. Here O is the ring of integers of ED. There are hD such Hecke characters for each pair (D, d), differing from each other by an ideal class character of ED, where hD is the ideal class number of ED. We denote such a Hecke character by χD,d. These Hecke characters were studied by Rohrlich ([Roh2-3]), who also allowed D or d to be even. In particular, he proved, that for almost all pairs (D, d) such that D > |d|39+2 and the root number of χD,d is one, the central L-value L(1, χD,d) 6= 0. Here 2 is any positive number. He and Montgomery ([MR]) also proved a more definite result asserting that L(1, χD,1) 6= 0 if and only if the root number of χD,1 is one. Rodriquez Villegas further gave a nice formula in [RV1-2] for the central L-value L(1, χD,1) for D ≡ 7 mod 8. From this 1991 Mathematics Subject Classification. 11G05 11M20 14H52.