Stephen D. Miller and Tonghai Yang.
February 29, 2000
Math. Res. Letters, 7 (2000), pp. 263-278.
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Part of the introduction (adapted for the web):
Let L(s,c) denote the Hecke L-function of c, and L(s,c) its completion; L(s,c) satisfies the functional equation L(s,c) = W(c)L(2-s,c), where W(c) = ±1 is the root number. If c is a canonical Hecke character with W(c) = 1, then the central value L(1,c) ¹ 0 by a theorem of Montgomery and Rohrlich. Of course, it automatically vanishes when W(c) = -1 by the functional equation. The main result of this paper is
Let c be a canonical Hecke character whose root number W(c) = -1. Then the central derivative L¢(1,c) ¹ 0.
We also prove that L¢(1,c) ¹ 0 when c is a small quadratic twist of a canonical character with W(c) = -1.
When D = p is a prime, canonical Hecke characters are closely connected with the elliptic curves A(p) extensively studied by Gross. These curves are defined over F = Q( j( (1+Ö-p)/2 ) ), where j is the usual modular j-function, and have complex multiplication by O. Combining and the above result of Montgomery-Rohrlich with Gross-Zagier and Kolyvagin-Logachev, one has
Let p > 3 be a prime congruent to 3 modulo 4. Then
(a) The Mordell-Weil rank of A(p) is
(b) The Shafarevich-Tate group Sha(A(p)/F) is