Shimura correspondence for level p2 and the central values of L-series, II

Given a Hecke eigenform f of weight 2 and square-free level N, by the work of Kohnen, there is a unique weight 3/2 modular form of level 4N mapping to f under the Shimura correspondence. Furthermore, by the work of Waldspurger the Fourier coefficients of such a form are related to the quadratic twis...

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Detalles Bibliográficos
Publicado:	2014
Materias:	L-series special values Shimura correspondence
Acceso en línea:	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_17930421_v10_n7_p1595_Pacetti http://hdl.handle.net/20.500.12110/paper_17930421_v10_n7_p1595_Pacetti
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

id	paper:paper_17930421_v10_n7_p1595_Pacetti
record_format	dspace
spelling	paper:paper_17930421_v10_n7_p1595_Pacetti2025-07-30T19:04:44Z Shimura correspondence for level p2 and the central values of L-series, II L-series special values Shimura correspondence Given a Hecke eigenform f of weight 2 and square-free level N, by the work of Kohnen, there is a unique weight 3/2 modular form of level 4N mapping to f under the Shimura correspondence. Furthermore, by the work of Waldspurger the Fourier coefficients of such a form are related to the quadratic twists of the form f. Gross gave a construction of the half integral weight form when N is prime, and such construction was later generalized to square-free levels. However, in the non-square free case, the situation is more complicated since the natural construction is vacuous. The problem being that there are too many special points so that there is cancellation while trying to encode the information as a linear combination of theta series. In this paper, we concentrate in the case of level p2, for p > 2 a prime number, and show how the set of special points can be split into subsets (indexed by bilateral ideals for an order of reduced discriminant p2) which gives two weight 3/2 modular forms mapping to f under the Shimura correspondence. Moreover, the splitting has a geometric interpretation which allows to prove that the forms are indeed a linear combination of theta series associated to ternary quadratic forms. Once such interpretation is given, we extend the method of Gross-Zagier to the case where the level and the discriminant are not prime to each other to prove a Gross-type formula in this situation. © 2014 World Scientific Publishing Company. 2014 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_17930421_v10_n7_p1595_Pacetti http://hdl.handle.net/20.500.12110/paper_17930421_v10_n7_p1595_Pacetti
institution	Universidad de Buenos Aires
institution_str	I-28
repository_str	R-134
collection	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic	L-series special values Shimura correspondence
spellingShingle	L-series special values Shimura correspondence Shimura correspondence for level p2 and the central values of L-series, II
topic_facet	L-series special values Shimura correspondence
description	Given a Hecke eigenform f of weight 2 and square-free level N, by the work of Kohnen, there is a unique weight 3/2 modular form of level 4N mapping to f under the Shimura correspondence. Furthermore, by the work of Waldspurger the Fourier coefficients of such a form are related to the quadratic twists of the form f. Gross gave a construction of the half integral weight form when N is prime, and such construction was later generalized to square-free levels. However, in the non-square free case, the situation is more complicated since the natural construction is vacuous. The problem being that there are too many special points so that there is cancellation while trying to encode the information as a linear combination of theta series. In this paper, we concentrate in the case of level p2, for p > 2 a prime number, and show how the set of special points can be split into subsets (indexed by bilateral ideals for an order of reduced discriminant p2) which gives two weight 3/2 modular forms mapping to f under the Shimura correspondence. Moreover, the splitting has a geometric interpretation which allows to prove that the forms are indeed a linear combination of theta series associated to ternary quadratic forms. Once such interpretation is given, we extend the method of Gross-Zagier to the case where the level and the discriminant are not prime to each other to prove a Gross-type formula in this situation. © 2014 World Scientific Publishing Company.
title	Shimura correspondence for level p2 and the central values of L-series, II
title_short	Shimura correspondence for level p2 and the central values of L-series, II
title_full	Shimura correspondence for level p2 and the central values of L-series, II
title_fullStr	Shimura correspondence for level p2 and the central values of L-series, II
title_full_unstemmed	Shimura correspondence for level p2 and the central values of L-series, II
title_sort	shimura correspondence for level p2 and the central values of l-series, ii
publishDate	2014
url	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_17930421_v10_n7_p1595_Pacetti http://hdl.handle.net/20.500.12110/paper_17930421_v10_n7_p1595_Pacetti
_version_	1840324905832284160

Shimura correspondence for level p2 and the central values of L-series, II

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