Selmer Groups in Twist Families of Elliptic Curves

Details Selmer Groups in Twist Families of Elliptic Curves Selmer Groups in Twist Families of Elliptic Curves

2012-03-13

arxiv.org/abs/1203.2791v1

Mathematics

Number Theory

to appear in Quaestiones Mathematicae. 16 pages

Scientific paper

The aim of this article is to give some numerical data related to the order of the Selmer groups in twist families of elliptic curves. To do this we assume the Birch and Swinnerton-Dyer conjecture is true and we use a celebrated theorem of Waldspurger to get a fast algorithm to compute $% L_{E}(1)$. Having an extensive amount of data we compare the distribution of the order of the Selmer groups by functions of type $\alpha \frac{(\log \log (X))^{1+\varepsilon}}{\log (X)}$ with $\varepsilon $ small. We discuss how the "best choice" of $\alpha $ is depending on the conductor of the chosen elliptic curves and the congruence classes of twist factors.

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