Tenth Algorithmic Number Theory Symposium ANTS-X University of California, San Diego July 9 – 13, 2012 Tenth Algorithmic Number Theory Symposium (ANTS-X) July 9 – 13, 2012

Explicit 2-descent and the average size of the 2-Selmer group of Jacobians of odd hyperelliptic curves

Manjul Bhargava

Abstract: We show how to construct explicit models for elements in the 2-Selmer groups of Jacobians of odd hyperelliptic curves, through a study of the rational (and integral) orbits of a certain natural representation of the odd split orthogonal group. As a consequence of this study, we show that the average size of the 2-Selmer group of the Jacobians of odd hyperelliptic curves over Q (of any given genus) is 3. This implies that the average rank of the Jacobians of such hyperelliptic curves is bounded (by 3/2). Via Chabauty methods, the result also then implies a uniform bound on the number of rational points on the majority of these curves. This is joint work with Dick Gross.

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