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?

  • $\begingroup$ Nice question! I fixed a very minor typo in the statement of Bloch--Kato. $\endgroup$ Oct 28, 2020 at 8:22

1 Answer 1


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


(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.

  • $\begingroup$ Excellent answer. Welcome on MA, Jack Sprat. $\endgroup$
    – Joël
    Oct 28, 2020 at 3:34

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.