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Zbl 0719.11029
Iwaniec, H.
On the order of vanishing of modular L-functions at the critical point. (English)
[J] Sémin. Théor. Nombres Bordx., Sér. II 2, No.2, 365-376 (1990). ISSN 0989-5558

Let $f(z)=\sum\sp{\infty}\sb{n=1}a\sb ne\sp{2\pi inz}$ be a Hecke eigenform, newform of weight 2 for $\Gamma\sb 0(N)$. Its L-series $L(s)=\sum\sp{\infty}\sb{n=1}a\sb nn\sp{-s}$ is also the L-function of an elliptic curve E over ${\bbfQ}$. A condition for the finiteness of the group of rational points of E is $L'(1,\chi\sb d)\ne 0$ for a certain quadratic character $\chi\sb d=(\frac{-d}{y}).$ \par In {\S} 2 of [{\it D. Bump}, {\it S. Friedberg} and {\it J. Hoffstein}, Bull. Am. Math. Soc., New Ser. 21, 89-93 (1989; Zbl 0699.10038)] the theorem is announced that $L'(1,\chi\sb d)\ne 0$ holds for infinitely many $\chi\sb d$ 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 {\it V. K. Murty} [Proc. Conf. on Automorphic Forms and Analytic Number Theory, Montréal, June 1989, 89-113 (1990)]. \par In the paper under review a more quantitative statement is proved: $L'(1,\chi\sb d)\ne 0$ for at least $Y\sp{2/3-\epsilon}$ primitive quadratic characters with $d<Y$, for Y large enough. This follows from the estimates $$ \sum\sb{d\le Y}\vert L'(1,\chi\sb d)\vert\sp 4\ll Y\sp{2+\epsilon},\quad \sum\sb{d}L'(1,\chi\sb d)F(d/Y)=\alpha\sb FY \log Y+\beta\sb FY+O(Y\sp{13/14+\epsilon}) $$ with $\alpha\sb F\ne 0$; the test function F is smooth with compact support, and d runs over a set of squarefree numbers. \par 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.
[R.W.Bruggeman (Utrecht)]

MSC 2000:: *11F67 Special values of automorphic L-series, etc
11F11 Modular forms, one variable
11F66 Dirichlet series and functional equations related to modular forms
11G05 Elliptic curves over global fields
11M41 Other Dirichlet series and zeta functions
11N36 Appl. of sieve methods

Keywords: twisted L-function; vanishing theorem; cusp form; elliptic curve; integral representation for the L-series; symmetric square L-series; large sieve inequality

Citations: Zbl 0699.10038; Zbl 0699.10039

Cited in: Zbl 1244.11067 Zbl 1012.11052 Zbl 0905.11024 Zbl 0893.11034 Zbl 0889.11016

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