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It is well-known that in his celebrated proof of Fermat's Last Theorem, Wiles made a crucial use of the result of Langlands and Tunnell to deduce modularity of the Galois representation on the 3-torsion points of a rational elliptic curve, provided this representation is irreducible. And the 3-5 trick completed the last steps of the argument.

These small primes, which in fact allowed his line of argument ''to get started'' thus played a crucial role in the proof, and I believe that small primes have played an analogous role as stepping stones in most subsequent developments basing on the kind of techniques pioneered by Wiles (e.g. the proof of Serre's modularity conjecture by Khare-Wintenberger).

One could imagine small primes playing a similar role in an eventual proof of some of the outstanding conjectures in arithmetic geometry. So I wonder whether there are (small) values of a prime number $p$ for which one knows that

  • the $p$-primary part of the Tate-Shafarevich group of an elliptic curve group is finite?
  • the $p$-adic cycle class map of a smooth projective variety is surjective?
  • the $p$-adic regulator of a number field is non-zero?

beyond what one knows in general for an unspecified value of $p$.

But given the above motivation, maybe one should first ask to more knowledgeable mathematicians whether

  • do $p$-adic deformation techniques seem well-suited to the study of any of the above three conjectures?
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As for Sha, I am reasonably confident in saying that nobody knows what the 2-torsion of Sha has to do with the 3-torsion. Of course, for any given elliptic curve, one can often prove that the 2-primary part of sha is finite. But it is not known, whether the 2-primary (say) part is finite for all elliptic curves over $\mathbb{Q}$. – Alex B. Aug 6 '11 at 12:33

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