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If $E$ is an elliptic curve over $K$, is there any effective estimate for the discriminant of the extension $L/K$ for which the $p$-part of the Selmer or Tate-Shafarevich groups become large?

I will add this edit to make the question more precise:

while going up any tower of number fields, do we fully understand the growth of $Sel_p$?

Given $E/K$ is it possible/ is it known that there will always exist an extension $M/K$ such that $Sel_p(E/M)$ is large (say $> p^m$ for any integer $m$, and $M$ depends on $m$ somehow)? In that case, what is the effective bound on $M$?

Kloosterman-Schaeffer (2003) prove that for $p>3$, there are infinitely many elliptic curves defined over $K$ with Selmer group large when $[K:\mathbb{Q}]\leq g+1$ ($g=$ genus of $X_0(p)$). Are there known effective bounds? If one can't give an effective bound in terms of the degree, is it possible to give one in terms of the discriminant?

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    $\begingroup$ It is not clear to me what you are asking. What do you mean by "large"? Are you interested in extension in which the $p$-primary part (or the $p$-torsion part) of theTate-Shafarevich grows? "of the extension"? There will be some where it grows and lots where it does not, but not a single one. So do you want a bound below which Sha does not become larger? What is that allowed to depend on? Is the degree or the Galois group of $L/K$ fixed?... $\endgroup$ – Chris Wuthrich May 23 '18 at 19:15
  • $\begingroup$ while going up any tower of number fields, do we fully understand the growth of Sel_p? Given E/K is it possible/ known that there will always exist M/K such that Sel_p(E/M) is large (say> p^m for any integer m, and M depends on m somehow)? What is the effective bound on M? Kloosterman-Schaeffer (2003) prove that for p>3, there are infinitely many elliptic curves defined over K with Selmer group large when [K:\Q]\leq g+1 (g= genus of X_0(p)). Are there known effective bounds? If one can't give an effective bound in terms of the degree, is it possible to give one in terms of the discriminant? $\endgroup$ – debanjana May 25 '18 at 23:27
  • $\begingroup$ Please edit your question to make it precise. $\endgroup$ – Chris Wuthrich May 26 '18 at 12:21
  • $\begingroup$ Going through more literature I realise that my question was in the spirit of what was proven by Bartel (in 2010?). He shows that for prime $p$ and quadratic extension $M\neq \mathbb{Q}(\sqrt{p})$ for $p\equiv 1\mod 4$ and given positive integer $d$, there exists $F/\mathbb{Q}$, Galois of degree $D_{2p}$, and $E/\mathbb{Q}$ such that $F$ contains $M$ and $Sel_p(E/F)\geq p^d$. But he remarks that it's impossible to push his construction further to lower the degree of the extension. $\endgroup$ – debanjana Jul 17 '18 at 14:18

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