Suppose $E$ is an elliptic curve over $\QQ$ with no nontrivial rational $2$-torsion point. Given a nonzero integer $d$, take $E^d$ to be the quadratic twist of $E$ coming from the field $\QQ(\sqrt{d})$. For every nonnegative integer $r$, we will determine the natural density of $d$ such that $E^d$ has $2$-Selmer rank $r$. We will also give a generalization of this result to abelian varieties defined over number fields. These results fit into the following general framework: take $\ell$ to be a rational prime, take $F$ to be number field, take $\zeta$ to be a primitive $\ell^{th}$ root of unity, and take $N$ to be an $\ell$-divisible $\Z_{\ell}[\zeta]$-module with an action of the absolute Galois group $G_F$ of $F$. Given a homomorphism $\chi$ from $G_F$ to $\langle \zeta \rangle$, we can define a twist $N^{\chi}$, and we can define a $(1 - \zeta)$-Selmer group for each of these twists. Under some hypotheses, we determine the distribution of $(1 - \zeta)$-Selmer ranks in this family of degree $\ell$ twists. To give a non-geometric example, this framework allows us to determine some aspects of the distribution of $\ell$-primary part of the class groups in families of degree $\ell$ extensions. One of the main goals of this dissertation is to prove these results in a way that streamlines the approach to finding the distribution of higher Selmer groups developed by the author.Along the way, we generalize the Cassels-Tate pairing to a certain pairing between Selmer groups defined from finite Galois modules. On the analytic side, we give a general bilinear character sum estimate that is suitable both for the base-case work of this dissertation and as a replacement for the Chebotarev density theorem in future work on higher Selmer groups.
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$\ell^{\infty}$-Selmer Groups in Degree $\ell$ Twist Families
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