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The distribution of $\ell^{\infty}$-Selmer groups in degree $\ell$ twist families II
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abstract
We continue the investigation of the distribution of $\ell^{\infty}$-Selmer groups in degree $\ell$ twist families of Galois modules over number fields begun in the previous paper. Building off the work on higher Selmer groups in that part, we find conditions under which we can compute the distribution of the $\ell^{\infty}$-Selmer groups for a given degree $\ell$ twist family. Along the way, we show that the average rank in the quadratic twist family of any given abelian variety over a number field is bounded.
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Infinitely many hyperelliptic curves of small genus and small fixed rank, and of any genus and rank two
For any number field K and genus g ≥ 2, there are infinitely many non-isomorphic hyperelliptic curves over K with Jacobian rank 0, 1, or 2 over K; explicit higher-rank ranges are given for small genera over Q.
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