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Asymptotics of the Tate-Shafarevich Group

  • Author(s): Manber, Shelly
  • Advisor(s): Yuan, Xinyi
  • et al.
Abstract

For any elliptic curve $E$ with everywhere good or split multiplicative reduction over a finite extension of $\mathbb{F}_q(t)$ with $q=p^e$, $p\ne 2$, we consider the family of elliptic curves $\{E^{(p^n)}\}_{n\in \mathbb{Z}_+}$ consisting of $p^n$-power frobenius automorphism twists of $E$. Assuming BSD for function fields, we compute the asymptotics of the order of the Tate-Shafarevich group of $E^{(p^n)}$ as $n$ approaches infinity. We prove that there is a twist $E^{(p^k)}$ such that for any $n$ greater than $k$, we can explicitly compute the order of the Tate-Shafarevich group of $E^{(p^n)}$ as a power of $p$ (given in terms of $n$) times the order of the Tate-Shafarevich group of $E^{(p^k)}$. In particular, as $n$ approaches infinity, the size of the Tate-Shafarevich group of $E^{(p^n)}$ grows on the order of $\mathcal{O}(q^{\delta p^n})$ for some constant $\delta$ determined by $E^{(p^k)}$. More precisely, the size of the Tate-Shafarevich group of $E^{(p^n)}$ grows as $\mathcal{\theta}(p^{e\delta p^n-n(m+r)})$ where $r$ is the rank of $E^{(p^k)}$ and $m$ is the number of places of bad reduction of $E^{(p^k)}$.

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