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Consequences of the Gross–Zagier formulae: Stability of average L-values, subconvexity, and non-vanishing mod p

Michel, Philippe and Ramakrishnan, Dinakar (2012) Consequences of the Gross–Zagier formulae: Stability of average L-values, subconvexity, and non-vanishing mod p. In: Number Theory, Analysis and Geometry: In Memory of Serge Lang. Springer , New York, pp. 437-459. ISBN 978-1-4614-1259-5.

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Applying the celebrated results of Gross and Zagier for central values of L-series of holomorphic forms of prime level, we deduce an exact average formula for suitable twists of such L-values, with a relation to the class number of associated imaginary quadratic fieds, thereby strengthening a result of Duke. We also obtain a stability result, as well as subconvexity (in this setting), and certain non-vanishing assertions.

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Additional Information:© 2012 Springer. In memory of Serge Lang. Serge Lang always conveyed infectious excitement about mathematics to anyone he came into contact with, and he will be missed. He was quite interested in the values of L-functions and in the divisibility properties of arithmetic invariants, and it is a pleasure to dedicate this article to him. We would like to thank B. Gross for discussions concerning the occurrence of a factorial in Theorems 5.5, 5.6 of [11], and also M. Flach, H. Hida, K. Prasanna, V. Vatsal, and D. Whitehouse for helpful conversations concerning parts of the paper. Thanks are also due to Paul Nelson and the anonymous referee for reading the paper carefully and finding a number of typos and small inaccuracies. The first author would like to thank Caltech for its hospitality during the preparation of this work, and acknowledge the partial support he received from the ERC advanced research grant no. 228304. The second author was partially supported by the National Science Foundation through the grants DMS-0402044 and DMS-0701089.
Funding AgencyGrant Number
European Research Council (ERC)228304
Subject Keywords:Average L-values; Gross–Zagier formulae; Non-vanishing; Subconvexity; Stability
Classification Code:MSC (2010): 11F11; 11F12; 11F67
Record Number:CaltechAUTHORS:20120620-113738336
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:31987
Deposited By: Ruth Sustaita
Deposited On:20 Jun 2012 20:13
Last Modified:09 Nov 2021 20:02

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