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