The effect of torsion on the distribution of Sh{cyrillic} among quadratic twists of an elliptic curve

Let E be an elliptic curve of rank zero defined over Q{double-struck} and l an odd prime number. For E of prime conductor N, in Quattrini (2006) [Qua06], we remarked that when l||E(Q{double-struck})Tor|, there is a congruence modulo l among a modular form of weight 3/2 corresponding to E and an Eise...

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Detalles Bibliográficos
Autor principal:	Quattrini, Patricia L.
Publicado:	2011
Materias:	Congruences of modular forms Distribution of Sh{cyrillic} Torsion points of elliptic curves
Acceso en línea:	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0022314X_v131_n2_p195_Quattrini http://hdl.handle.net/20.500.12110/paper_0022314X_v131_n2_p195_Quattrini
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

id	paper:paper_0022314X_v131_n2_p195_Quattrini
record_format	dspace
spelling	paper:paper_0022314X_v131_n2_p195_Quattrini2023-06-08T14:49:18Z The effect of torsion on the distribution of Sh{cyrillic} among quadratic twists of an elliptic curve Quattrini, Patricia L. Congruences of modular forms Distribution of Sh{cyrillic} Torsion points of elliptic curves Let E be an elliptic curve of rank zero defined over Q{double-struck} and l an odd prime number. For E of prime conductor N, in Quattrini (2006) [Qua06], we remarked that when l\|\|E(Q{double-struck})Tor\|, there is a congruence modulo l among a modular form of weight 3/2 corresponding to E and an Eisenstein series. In this work we relate this congruence in weight 3/2, to a well-known one occurring in weight 2, which arises when E(Q{double-struck}) has an l torsion point. For N prime, we prove that this last congruence can be lifted to one involving eigenvectors of Brandt matrices Bp(N) in the quaternion algebra ramified at N and infinity. From this follows the congruence in weight 3/2. For N square free we conjecture on the coefficients of a weight 3/2 Cohen-Eisenstein series of level N, which we expect to be congruent to the weight 3/2 modular form corresponding to E. © 2010 Elsevier Inc. Fil:Quattrini, P.L. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2011 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0022314X_v131_n2_p195_Quattrini http://hdl.handle.net/20.500.12110/paper_0022314X_v131_n2_p195_Quattrini
institution	Universidad de Buenos Aires
institution_str	I-28
repository_str	R-134
collection	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic	Congruences of modular forms Distribution of Sh{cyrillic} Torsion points of elliptic curves
spellingShingle	Congruences of modular forms Distribution of Sh{cyrillic} Torsion points of elliptic curves Quattrini, Patricia L. The effect of torsion on the distribution of Sh{cyrillic} among quadratic twists of an elliptic curve
topic_facet	Congruences of modular forms Distribution of Sh{cyrillic} Torsion points of elliptic curves
description	Let E be an elliptic curve of rank zero defined over Q{double-struck} and l an odd prime number. For E of prime conductor N, in Quattrini (2006) [Qua06], we remarked that when l\|\|E(Q{double-struck})Tor\|, there is a congruence modulo l among a modular form of weight 3/2 corresponding to E and an Eisenstein series. In this work we relate this congruence in weight 3/2, to a well-known one occurring in weight 2, which arises when E(Q{double-struck}) has an l torsion point. For N prime, we prove that this last congruence can be lifted to one involving eigenvectors of Brandt matrices Bp(N) in the quaternion algebra ramified at N and infinity. From this follows the congruence in weight 3/2. For N square free we conjecture on the coefficients of a weight 3/2 Cohen-Eisenstein series of level N, which we expect to be congruent to the weight 3/2 modular form corresponding to E. © 2010 Elsevier Inc.
author	Quattrini, Patricia L.
author_facet	Quattrini, Patricia L.
author_sort	Quattrini, Patricia L.
title	The effect of torsion on the distribution of Sh{cyrillic} among quadratic twists of an elliptic curve
title_short	The effect of torsion on the distribution of Sh{cyrillic} among quadratic twists of an elliptic curve
title_full	The effect of torsion on the distribution of Sh{cyrillic} among quadratic twists of an elliptic curve
title_fullStr	The effect of torsion on the distribution of Sh{cyrillic} among quadratic twists of an elliptic curve
title_full_unstemmed	The effect of torsion on the distribution of Sh{cyrillic} among quadratic twists of an elliptic curve
title_sort	effect of torsion on the distribution of sh{cyrillic} among quadratic twists of an elliptic curve
publishDate	2011
url	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0022314X_v131_n2_p195_Quattrini http://hdl.handle.net/20.500.12110/paper_0022314X_v131_n2_p195_Quattrini
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The effect of torsion on the distribution of Sh{cyrillic} among quadratic twists of an elliptic curve

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