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In the homepage for the CRM's special semester this year, I found the interesting statement that the modularity theorem (formerly the Taniyama-Shimura-Weil conjecture) is a special case of the Bloch-Kato conjecture for the symmetric square motive of an elliptic curve.

The Bloch-Kato conjecture, as I know about it, is the following statement: $$\text{ord}_{s=0}L(s,V)=\text{dim }H_{f}^{1}(K,V^{*}(1))-\text{dim }H^{0}(K,V^{*}(1))$$ where $H_{f}^{1}$ is the Bloch-Kato Selmer group. I do not know much about motives and I do not know what the symmetric square motive of an elliptic curve is. In the formulation of the Bloch-Kato conjecture above I am taking $V$ to be a geometric Galois representation. My question is, how do we view the modularity theorem as this special case of the Bloch-Kato conjecture?

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  • $\begingroup$ Nice question! I fixed a very minor typo in the statement of Bloch--Kato. $\endgroup$ – David Loeffler Oct 28 at 8:22
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That is not what the link says. To quote (emphasis mine):

... in which this conjecture was reduced to a special instance of the Bloch-Kato conjecture for the symmetric square motive of an elliptic curve.

That means something quite different. You could equally say that Wiles "reduced" the proof to the fact that $X(3)$ and $X(5)$ have genus zero, or that he "reduced" the proof to the Langlands-Tunnell theorem that (projective) $A_4$ and $S_4$ representations are automorphic. Shimura-Taniyama is no more a "special case" of these claims than it is of the Bloch-Kato conjecture.

The more relevant thing to say is that one (inductive) step in Wiles' argument required comparing the size of a certain congruence module (measuring congruences between one cuspform and other forms of a fixed level) and a relative tangent space (measuring congruences between one Galois representation and a certain prescribed family of deformations of that representation). Hida had already shown that the former quantity could be interpreted in terms of the special value of the adjoint L-function. So if one knew that the value of this adjoint L-function (divided by the correct period to obtain an integer) was divisible by the same power of $p$ as the order of the relative tangent space (which could be interpreted in terms of a Bloch-Kato Selmer group, then the inductive step would hold. This desired equality can indeed be interpreted as a special case of the Bloch-Kato conjecture, although not formulated for $\mathbf{Q}_p$ representations as you have done but in the more precise form by Bloch and Kato for $p$-adic lattices in Galois representations coming from motives $M$. Namely, in the context of your equation, the L-value on the LHS trivially doesn't vanish as it lies on the edge of the critical strip. So one can hope (given the motive in question) to define a suitable period $\Omega$ such that $L(s,M)/\Omega \in \mathbf{Q}^{\times}$, and then (after taking into account local fudge-factors) interpret the resulting integer (or at least the $p$-power part) as the order of a Bloch-Selmer group $H^1_f(\mathbf{Q}, T \otimes \mathbf{Q}_p/\mathbf{Z}_p)$, where $T$ is a $\mathbf{Z}_p$-lattice inside a Galois representation associated to $M$.

In this particular case, you can take the variety $E \times E$. Then the motive $M$ is a suitable piece of this. Then one is interested in the special value $L(M,2)$. The corresponding lattice $T$ can then be found inside

$$H^2(E \times E,\mathbf{Z}_p(1))$$

Precisely, there is a lattice corresponding to the Tate module of $E$ at $p$. Denote the dual of this lattice by $\rho$. The lattice $\rho$ is isomorphic to $H^1(E,\mathbf{Z}_p)$. By the Kunneth formula, one then finds a copy of $\rho^{\otimes 2}$ inside $H^2(E \times E,\mathbf{Z}_p)$, and hence a copy of $\rho^{\otimes 2}(1)$ inside the group above. Since $\rho$ has dimension $2$, there is a decomposition (let $p$ be odd) $\rho^{\otimes 2} \simeq \mathrm{Sym}^2(\rho) \oplus \mathbf{Z}_p(-1)$, and then $T$ is identified with $\mathrm{Sym}^2(\rho)(1)$. (Apologies if I have got the twisting wrong, it is irritating to keep straight).

It might also be worth mentioning that Wiles famously didn't use any results towards the Bloch-Kato conjecture, but rather proved what he needed by himself and thus deduced some special cases of the Bloch-Kato conjecture, in particular an automorphic formula for the order of the group

$$H^1_f(\mathbf{Q},\mathrm{Sym}^2(\rho)(1)).$$

(The Galois representation can also be identified the trace zero matrices $\mathrm{ad}^0(\rho)$ in the [lattice of the] adjoint representation.) Note that Wiles' formula manifestly implies that this group is finite. But even the finiteness of this formula was unknown for a general elliptic curve before the work of Wiles. (Although it was known in some cases by Flach for modular elliptic curves; not so useful for proving modularity.) Many modern modularity proofs (which use the same underlying mechanism that Wiles did) thus also end up resulting in some results towards Bloch-Kato conjecture for adjoint representations.

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  • $\begingroup$ Excellent answer. Welcome on MA, Jack Sprat. $\endgroup$ – Joël Oct 28 at 3:34

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