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In the introduction (p. 14) of this paper on FLT the authors say that a numerical criterion found by Wiles as part of his proof of FLT "seems to be very close to a special case of the Bloch-Kato conjecture".

Can someone explain how this numerical criterion is related to a (which ?) special case of the Bloch-Kato conjecture (which is now a theorem) ?

The numerical criterion is Theorem 5.3 (p. 139) in the linked paper.

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up vote 5 down vote accepted

The particular conjecture in question is about the power of $p$ defining the Selmer group attached to the adjoint of the $p$-adic Tate module of an elliptic curve. (In general, the Bloch--Kato conjecture deals with the order and/or rank of Selmer groups.)

This order is supposed to be equal to the algebraic part of a particular special value of the symmetric square $L$-function of the elliptic curve. (Note that the symmetric square and the adjoint agree up to a twist, and while it would be more logical to speak of the adjoint $L$-function at this point, the symmetric square $L$-function is more traditional; in any case, these $L$-functions would be the same up to the change of variables $s \mapsto s+1$.)

Now results of Hida and Ribet show that up to small prime factors, this algebraic part of the special value in question coincides with the congruence modulus of the modular form attached to the elliptic curve. On the other hand, Wiles's numerical criterion (or rather, the fact that it holds for the Hecke algebra in question) shows that the order of the adjoint Selmer group also coincides with this congruence modulus. Putting these two statements together gives the case of Bloch--Kato under consideration.

(Diamond--Flach--Guo have a paper carefully detailing all this.)

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Wow.. It seems that centuries have passed to see you again answering sir.. How are you ?. I hope that you are fine, and everything is going fine...+1 for another elegant answer... – Iyengar Jun 22 '12 at 8:22
Thanks for your answer. (I don't visit this site very often, whence I noticed your answer only by now.) – Ralph Aug 10 '12 at 23:12

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