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

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

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 \(L'(1,\chi_ d)\neq 0\) 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: \(L'(1,\chi_ d)\neq 0\) for at least \(Y^{2/3-\epsilon}\) primitive quadratic characters with \(d<Y\), for Y large enough. This follows from the estimates \[ \sum_{d\leq Y}| L'(1,\chi_ d)|^ 4\ll Y^{2+\epsilon},\quad \sum_{d}L'(1,\chi_ d)F(d/Y)=\alpha_ FY \log Y+\beta_ FY+O(Y^{13/14+\epsilon}) \] with \(\alpha_ F\neq 0\); 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.

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 \(L'(1,\chi_ d)\neq 0\) 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: \(L'(1,\chi_ d)\neq 0\) for at least \(Y^{2/3-\epsilon}\) primitive quadratic characters with \(d<Y\), for Y large enough. This follows from the estimates \[ \sum_{d\leq Y}| L'(1,\chi_ d)|^ 4\ll Y^{2+\epsilon},\quad \sum_{d}L'(1,\chi_ d)F(d/Y)=\alpha_ FY \log Y+\beta_ FY+O(Y^{13/14+\epsilon}) \] with \(\alpha_ F\neq 0\); 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.

Reviewer: R.W.Bruggeman (Utrecht)

### MSC:

11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |

11F11 | Holomorphic modular forms of integral weight |

11F66 | Langlands \(L\)-functions; one variable Dirichlet series and functional equations |

11G05 | Elliptic curves over global fields |

11M41 | Other Dirichlet series and zeta functions |

11N36 | Applications 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
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\textit{H. Iwaniec}, Sémin. Théor. Nombres Bordx., Sér. II 2, No. 2, 365--376 (1990; Zbl 0719.11029)

### References:

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[2] | Bump, D., Friedberg, S. and Hoffstein, J., Eisenstein series on the metaplectic group and non-vanishing theorems for automorphic L-functions and their derivatives. Ann. Math.131 (1990), 53-127. · Zbl 0699.10039 |

[3] | Deshouillers, J.-M. and Iwaniec, H., The non-vanishing of Rankin-Selberg zeta-functions at special points, AMS Contemporary Mathematics Vol.53 (1986), 51-95. · Zbl 0595.10025 |

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[5] | Murty, K. and Murty, R., Mean values of derivatives of modular L-series. (to appear in Ann. Math.). (See also Murty, K., Non-vanishing of L-functions and their derivatives in Automorphic Forms and Analytic Number Theory, (edited by R. Murty), CRM Publications, Montreal1990, 89-113). · Zbl 0745.11033 |

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[7] | Rohrlich, D., On L-functions of elliptic curves and anticyclotomic towers. Invent. Math.75 (1984), 383-408. · Zbl 0565.14008 |

[8] | Rohrlich, D., On L-functions of elliptic curves and cyclotomic towers. Invent. Math.75 (1984), 409-423. · Zbl 0565.14006 |

[9] | Roiirlich, D., L-functions and division towers. Math. Ann.281(1988), 611-632. · Zbl 0656.14013 |

[10] | Rohrlich, D., Non-vanishing of L-functions for GL(2). Invent. Math.97 (1989), 381-403. · Zbl 0677.10020 |

[11] | Rohrlich, D., The vanishing of certain Rankin-Selberg convolutions, in Automorphic Forms and Analytic Number Theory. (edited by R. Murty) CRM Publications, Montreal1990, 123-133. · Zbl 0737.11014 |

[12] | Shimura, G., On modular forms of half-integral weight. Ann. Math.97 (1973), 440-481. · Zbl 0266.10022 |

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