Non-vanishing of the Central Derivative of Canonical Hecke L-functions

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 K = Q(Ö-D) be an imaginary quadratic field of discriminant -D < -4, O its ring of integers, and h its ideal class number. A Hecke character c of K of conductor f is a called "canonical" if
 c( a ) = c(a) for each ideal a relatively prime to f.
(1.1)
 c(aO) = ±a for principal ideals aO relatively prime to f.
(1.2)
 The conductor f is divisible only by primes dividing D.
(1.3)
Every Hecke character of K satisfying (1.1) and (1.2) is actually a quadratic twist of a canonical Hecke character (see Section 2 for a precise description of these characters and which fields have them).

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

Theorem
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

Corollary
Let p > 3 be a prime congruent to 3 modulo 4. Then

(a)     The Mordell-Weil rank of A(p) is

rankZA(p)(F) =  ì
í
î
 h,
 p º 3 (mod 8)
 0,
 p º 7 (mod 8).

(b)     The Shafarevich-Tate group Sha(A(p)/F) is finite.