Tenth Algorithmic Number Theory Symposium ANTSX

Explicit 2descent and the average size of the 2Selmer group of Jacobians of odd hyperelliptic curves
Manjul Bhargava
Abstract: We show how to construct explicit models for elements in the 2Selmer 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 2Selmer 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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© 201112 Kiran S. Kedlaya (with thanks to Pierrick Gaudry and Emmanuel ThomÃ©)
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