Compositio Mathematica



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


TONGHAI YANG a1
a1 Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, U.S.A. e-mail: thyang@math.lsa.umich.edu

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Abstract

Let D[identical with] 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[identical with]1 mod 4 be a squarefree integer relatively prime to D. Then for any integer k[gt-or-equal, slanted]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$>0. Furthermore, for k[less-than-or-eq, slant]1, we can take M(k)=0. Finally, if D=p is a prime, and d>0, then the associated elliptic curve A(p)$^d$ has Mordell–Weil rank 0 (over its definition field) when $\sqrt{D}$>(12/π)d$^2$ log d.


Key Words: central Hecke L-value; elliptic curves; eigenfunction; nonvanishing..