# Bloch-Kato conjecture and Wiles' numerical criterion

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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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 $s \mapsto s+1$.)