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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2011
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| 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 |
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| Sumario: | 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. |
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