The distribution of the Tamagawa ratio in the family of elliptic curves with a two-torsion pointReportar como inadecuado




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Research in the Mathematical Sciences

, 1:15

First Online: 02 December 2014Received: 31 July 2014Accepted: 09 October 2014

Abstract

In recent work, Bhargava and Shankar have shown that the average size of the 2-Selmer group of an elliptic curve over is exactly 3, and Bhargava and Ho have shown that the average size of the 2-Selmer group in the family of elliptic curves with a marked point is exactly 6. In contrast to these results, we show that the average size of the 2-Selmer group in the family of elliptic curves with a two-torsion point is unbounded. In particular, the existence of a two-torsion point implies the existence of rational isogeny. A fundamental quantity attached to a pair of isogenous curves is the Tamagawa ratio, which measures the relative sizes of the Selmer groups associated to the isogeny and its dual. Building on previous work in which we considered the Tamagawa ratio in quadratic twist families, we show that, in the family of all elliptic curves with a two-torsion point, the Tamagawa ratio is essentially governed by a normal distribution with mean zero and growing variance.

Electronic supplementary materialThe online version of this article doi:10.1186-s40687-014-0015-4 contains supplementary material, which is available to authorized users.

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Autor: Zev Klagsbrun - Robert J Lemke Oliver

Fuente: https://link.springer.com/







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