I already asked this question some days ago on http://math.stackexchange.com/questions/158747/bloch-kato-conjecture-and-wiles-numerical-criterion but didn't receive any response.

In the introduction (p. 14) of their paper on Wiles' proof of FLT, Darmon, Diamond and Taylor state that a numerical criterion of Wiles "seems to be very close to a special case of the Bloch-Kato conjecture".

This numerical criterion is a statement in commutative algebra (Theorem 5.3 on p. 139 in the paper): A surjective morphism $\phi: R \to T$ of augmented complete intersections over the ring of integers $\mathcal{O}$ of a finite extension over $\mathbb{Q}_p$ is an isomorphism iff $$|\ker \pi_R/(\ker \pi_R)^2| \le |\mathcal{O}/\pi_T(\text{Ann}_T(\ker \pi_T))|,$$ where $\pi_R: R \to \mathcal{O}$ is the augmentation.

**Question:** What is the relation between Wiles' numerical criterion and the Bloch-Kato conjecture ?

In order to be more explicit, let $K$ be a number field and let $V$ be a geometric representation of the absolute Galois group $G_K$ (in essential $V$ is a finite dimensional vector space over $\mathbb{Q}_p$ with a continuous action of $G_K$). Then the Bloch-Kato conjecture can be formulated as $$\dim H_f^1(G_K,V^\ast(1))-\dim H^0(G_K,V^\ast(1))=\operatorname{ord}_{s=0}L(V,s)$$ where $H^1_f$ denotes the Selmer group, $V^\ast$ is the dual representation and $L$ the $L$-function.

Then the question is for which $K, V$ and in what way this formula is close to the numerical criterion described above ?