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Let $E$ be an elliptic curve, and $\text{Sha}(E)$ its Shafarevich-Tate group which measures the failure of the local-to-global principle for the curve. It is conjectured that $\text{Sha}(E)$ is a finite group for every elliptic curve $E/\mathbb{Q}$, and it is known that for any prime $p$ the $p^\infty$-part of $\text{Sha}(E)$ is finite.

My question is: does there exist an infinite family of elliptic curves $E$ such that $\text{Sha}_2(E), \text{Sha}_3(E), \text{Sha}_5(E)$ all tend to infinity in size?

A similar, but perhaps more accessible question, is to ask the same thing of the Selmer groups instead.

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    $\begingroup$ It is only known in general that the $n$ torsion part is finite. The primary parts are known to be finite only if the analytic rank is zero or one (over the rationales). $\endgroup$
    – Chris Wuthrich
    Commented Sep 15, 2022 at 7:01
  • $\begingroup$ I ave no troubles believing that such a family exists. But I wonder what you want to do with it? $\endgroup$
    – Chris Wuthrich
    Commented Sep 15, 2022 at 7:02
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    $\begingroup$ @ChrisWuthrich I want to think about the problem of bounding the rank uniformly in large families, and one approach is to think about families where at least one of 2, 3, or 5 Selmer is bounded in size. These particular primes arose because we have a good understanding of the co-regular spaces parametrizing the Selmer elements. $\endgroup$
    – Stanley Yao Xiao
    Commented Sep 15, 2022 at 18:32
  • $\begingroup$ What do you mean by "family"? And "large family"? $\endgroup$
    – Chris Wuthrich
    Commented Sep 15, 2022 at 20:29
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    $\begingroup$ Delaunay heuristics will say that the proportion of elliptic curves with sha of size $(2\cdot 3\cdot 5)^{2k}$ is positive for all $k$, right? So you expect it to exist in that sense, no? But explicitly constructing one seems hard to me. A cyclic isogeny of degree 30 might have helped but they don't exist over $\mathbb{Q}$. $\endgroup$
    – Chris Wuthrich
    Commented Sep 16, 2022 at 7:48

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