Averages of twisted elliptic L-functions

Perelli, Alberto; Pomykała, Jacek

doi:10.4064/aa-80-2-149-163

Cited by 43 publications

(32 citation statements)

References 11 publications

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“…In the direction of this conjecture, we combine Theorem 1.1 with works by the second author and Skinner [OS98] and Perelli and Pomykala [PP97] to obtain the following result. COROLLARY 1.4.…”

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confidence: 98%

Heegner divisors,L-functions and harmonic weak Maass forms

2010

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Recent works, mostly related to Ramanujan's mock theta functions, make use of the fact that harmonic weak Maass forms can be combinatorial generating functions. Generalizing works of Waldspurger, Kohnen and Zagier, we prove that such forms also serve as "generating functions" for central values and derivatives of quadratic twists of weight 2 modular L-functions. To obtain these results, we construct differentials of the third kind with twisted Heegner divisor by suitably generalizing the Borcherds lift to harmonic weak Maass forms. The connection with periods, Fourier coefficients, derivatives of L-functions, and points in the Jacobian of modular curves is obtained by analyzing the properties of these differentials using works of Scholl, Waldschmidt, and Gross and Zagier.

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confidence: 98%

Heegner divisors,L-functions and harmonic weak Maass forms

2010

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“…Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. [1,3,7,9,10,11,13,15,17] and the references given there. This is of interest in various aspects such as the Birch-Swinnerton-Dyer conjecture, the Siegel zero (see [7]) and the theory of modular forms of half-integral weight (see [16,18] [12].…”

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confidence: 99%

Non-vanishing of class group $L$-functions at the central point

Blomer¹

2004

Annales De L’institut Fourier

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“…The first example illustrating this phenomenon for L(1, E d ) was given by James [9]. In his example the elliptic curve E is the −1 twist of the modular curve X 0 (14). Vatsal [16] has established that the elliptic curve E = X 0 (19) has a positive proportion of nonvanishing central derivatives L (1, E d ) = 0.…”

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confidence: 99%

“…For general elliptic curves Ono [12] has established that the number of d < X with L(1, E d ) = 0 is at least X/(log X) α for some α < 1. In the case of central derivatives the best known result is due to Perelli and Pomyka la [14] who show that there are at least X 1−ε such twists for any ε > 0. One should note that Ono in [12] employs the theorem of Waldspurger relating the central value L(1, E d ) with the dth Fourier coefficient of a half-integral weight modular form, and this is why he gets a stronger result in the case of central values.…”

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confidence: 99%