Averages of twisted elliptic \(L\)-functions. (English) Zbl 0878.11022

The authors improve on the lower bound for the number of nonvanishing derivatives (at \(s=1\)) of twisted elliptic \(L\)-functions (theorem 1). They also prove the nontrivial estimate for the analytic rank of twisted elliptic curves over rationals (theorem 2). Results are based on the recent large sieve type estimates over fundamental discriminants obtained by D. R. Heath-Brown [Acta Arith. 72, 235-275 (1995; Zbl 0828.11040)]. The construction of the test function in the context of Weil explicit formulas is a crucial step in their proof of theorem 2 (see section 5). Such a construction of a new test function was used quite recently by Kowalski and Michel in their proof of an unconditional upper bound for the analytic rank of the new part of the Jacobian of the modular curve [Sur le rang de \(J_0(q)\) (preprint; 1997)].
Theorem 1 can be generalized for a wider class of \(L\)-functions and their derivatives [J. Pomykała, Nonvanishing on \(n\)th derivatives of twisted elliptic \(L\)-functions in the critical strip, J. Théor. Nombres Bordx. (to appear); A. Dabrowski and J. Pomykała, Nonvanishing of motivic \(L\)-functions (preprint; 1997)].


11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11M41 Other Dirichlet series and zeta functions
11N36 Applications of sieve methods


Zbl 0828.11040
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