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 h$_D$ be the ideal class number of E$_D$=Q($\sqrt{−D}$). Let d≡1 mod 4 be a squarefree integer relatively prime to D. Then for any integer k[ges ]0 there is a constant M=M(k), independent of the pair (D,D), such that if (−1)$^k$=sign (d), (2k+1,h$_D$)=1, and$\sqrt{D}$>(12/π)d$^2$(log|d+M(k)), then the central L-value L(k+1, χ$^2k+1$$_D, d$>0. Furthermore, for k[les ]1, we can take M(k)=0. Finally, if D=p is a prime, and d>0, then the associated elliptic curve A(p)$^d$ has Mordell–Weil rank 0 (over its definition field) when $\sqrt{D}$>(12/π)d$^2$ log d.