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On the order of vanishing of modular L-functions at the critical point. (English) Zbl 0719.11029

Let $f\left(z\right)={\sum }_{n=1}^{\infty }{a}_{n}{e}^{2\pi inz}$ be a Hecke eigenform, newform of weight 2 for ${{\Gamma }}_{0}\left(N\right)$. Its L-series $L\left(s\right)={\sum }_{n=1}^{\infty }{a}_{n}{n}^{-s}$ is also the L-function of an elliptic curve E over $ℚ$. A condition for the finiteness of the group of rational points of E is ${L}^{\text{'}}\left(1,{\chi }_{d}\right)\ne 0$ for a certain quadratic character ${\chi }_{d}=\left(\frac{-d}{y}\right)·$

In § 2 of [D. Bump, S. Friedberg and J. Hoffstein, Bull. Am. Math. Soc., New Ser. 21, 89-93 (1989; Zbl 0699.10038)] the theorem is announced that ${L}^{\text{'}}\left(1,{\chi }_{d}\right)\ne 0$ holds for infinitely many ${\chi }_{d}$ associated to imaginary quadratic number fields. Their proof goes along the same lines as in their earlier paper [Ann. Mat., II. Ser. 131, 53-127 (1990; Zbl 0699.10039)]. The same result follows from the main theorem in V. K. Murty [Proc. Conf. on Automorphic Forms and Analytic Number Theory, Montréal, June 1989, 89-113 (1990)].

In the paper under review a more quantitative statement is proved: ${L}^{\text{'}}\left(1,{\chi }_{d}\right)\ne 0$ for at least ${Y}^{2/3-ϵ}$ primitive quadratic characters with $d, for Y large enough. This follows from the estimates

$\sum _{d\le Y}{|{L}^{\text{'}}\left(1,{\chi }_{d}\right)|}^{4}\ll {Y}^{2+ϵ},\phantom{\rule{1.em}{0ex}}\sum _{d}{L}^{\text{'}}\left(1,{\chi }_{d}\right)F\left(d/Y\right)={\alpha }_{F}YlogY+{\beta }_{F}Y+O\left({Y}^{13/14+ϵ}\right)$

with ${\alpha }_{F}\ne 0$; the test function F is smooth with compact support, and d runs over a set of squarefree numbers.

The proof uses an integral representation for the L-series, the symmetric square L-series associated to f, the large sieve inequality, and other techniques of analytic number theory. It is quite different from the proof of Bump, Friedberg and Hoffstein, which uses automorphic forms more heavily.

##### MSC:
 11F67 Special values of automorphic $L$-series, etc 11F11 Holomorphic modular forms of integral weight 11F66 Langlands L-functions; one variable Dirichlet series, etc. 11G05 Elliptic curves over global fields 11M41 Other Dirichlet series and zeta functions 11N36 Applications of sieve methods