Nonvanishing of central Hecke L-values and rank of certain elliptic curves

Abstract

Let D≡ 7 mod 8 be a positive squarefree integer, and let hD be the ideal class number of ED=\({\mathbb{Q}}\left( {\sqrt { - D} } \right)\). 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\(\sqrt D \) >(12/π)d2 (log∣d+M(k)), then the central L-value L(k+1, χD, d2k+1>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 \(\sqrt D \)>(12/π)d2 log d.