On simple zeros of certain L-series. (English) Zbl 0712.14010

Murty, M. Ram

Number theory, Proc. 1st Conf. Can. Number Theory Assoc., Banff/Alberta (Can.) 1988, 427-439 (1990).

[For the entire collection see Zbl 0689.00005.]
Kolyvagin proved recently when E is a modular elliptic curve over \({\mathbb{Q}}\) with Mordell-Weil rank 0, the Tate-Shafarevich group is finite provided there exists a “quadratic twist” of L-series for E having a simple zero at \(s=1.\)
In the present paper, the author shows that such a “quadratic twist” always exists under the assumption that the generalized Riemann hypothesis is true. Further, the author considers the \(\chi\)-twist of l with any character \(\chi\) mod q and shows that for the classical Dirichlet series \(L(s,\chi),\) \(L(1/2,\chi)\neq 0\) for at least \(\phi(q)\) characters \(\chi\) (mod q). The key of his proof is ‘Weil’s explicit formula’.

Reviewer: K.Katayama

MSC:

14G10	Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14G25	Global ground fields in algebraic geometry

Keywords:

finiteness of Tate-Shafarevich group; quadratic twist of L-series; generalized Riemann hypothesis; Weil’s explicit formula