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Mean values of derivatives of modular \(L\)-series. (English) Zbl 0745.11032

Let \(E/\mathbb{Q}\) be a modular elliptic curve of conductor \(N\) and denote by \(L(E,s)\) its \(L\)-function. If \(D\) is a negative discriminant, let \(L_ D(E,s)\) be the \(L\)-function of \(E\) twisted with \(\left( {D\over\cdot} \right)\). The main theorem of the paper states that if the root number \(w\) of \(L(E,s)\) is 1, then there is a non-zero constant \(c\) such that \(\sum_ D L'_ D(E,1)\sim cY\;\log Y\), where the sum extends over all \(D\) satisfying \(0<-D\leq Y\), \(D\equiv1\bmod 4N\). The proof uses deeper methods and ideas from analytic number theory.
As a corollary one obtains that if \(w=1\) then there are infinitely many fundamental discriminants \(D<0\) such that i) for all \(p\mid N\), \(p\) splits in \(\mathbb{Q}(\sqrt D)\), ii) \(L_ D(E,s)\) has a simple zero at \(s=1\). By a theorem of V. A. Kolyvagin [Izv. Akad.Nauk SSSR, Ser. Mat. 52, No.6, 1154-1180 (1988; Zbl 0681.14016)], this implies that if \(E/\mathbb{Q}\) is a modular elliptic curve with \(L(E,1)\neq 0\), then both the Mordell- Weil group and the Tate-Shafarevich group of \(E/\mathbb{Q}\) are finite.
The fact that under the condition \(w=1\) there are infinitely many \(D<0\) satisfying i) and ii) was also proved independently by D. Bump, S. Friedberg and J. Hoffstein [Invent. Math. 102, No.3, 543-518 (1990; Zbl 0721.11023)] in a completely different way, using the theory of automorphic forms on \(GSp_ 4\).

MSC:

11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14H25 Arithmetic ground fields for curves
11G05 Elliptic curves over global fields
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