The Shimura-Taniyama-Weil conjecture asserts that if E is an elliptic curve over Q, then there is an integer N ≥ 1 and a weight-two cusp form f of level N, normalized so that a1(f) = 1, such that ap(E) = ap(f), for all primes p of good reduction for E. When this is the case, the curve E is said to be modular. See these notes: https://www.ams.org/notices/199911/comm-darmon.pdf

Can this be generalized to arbitrary curves E, with genus g >= 1? Which automorphic forms will replace weight-two cusp forms in this setting? Hasse-Weil zeta functions can be described for any algebraic variety, so that part is still valid.



1 Answer 1


Good question. Below isn't the best answer you could get, but it was too long for a comment.

Perhaps it is better for the understanding of these theories to have the point of view of the Galois representation (as in the survey you put in your question). First, a $p$-adic Galois representation $\rho$ is said to be modular if it is equivalent to a $p$-adic Galois representation $\rho_{f}$ attached to a cuspidal newform $f$.

Then the modularity of elliptic curves can be stated as "for some $p$, the Galois representation $\rho_{E,p}$ attached to the elliptic curve $E$ is modular". Note that this is equivalent to your statement "$a_{l}(E)=a_{l}(f)$" according to Carayol's theorem (and the fact that the trace of $\rho_{f,p}(\mathbf{Frob }_{l})$ is $a_{l}(f)$). That's what people try to generalize, and that's also the Langlands setting. Finally, in a perspective of generalization, it is good to keep in mind that $\rho_{E}$ is in fact $H^{1}_{et}(E,\mathbb{Q}_{ p})$ . In fact, for any projected smooth curve $X$, the space $H^{1}_{et}(X,\mathbb{Q}_{p})$ will always be a (nice i.e. de Rham) Galois representation.

Therefore, you can first ask if (a subrepresentation of) the $H^{1}_{et}(X,\mathbb{Q}_{p})$ is isomorphic to a representation from of the automorphic world.

But for a genus $g$ curve the space $H^{1}_{et}(X,\mathbb{Q}_{p})$ has dimension $2g$, so the Langlands philosophy tells you that 'it should be associated with a cuspidal representation of $\mathbf{Gl}_{2g}$ (or maybe if your representation have nice proprieties something a bit smaller like $\mathbf{GSpin}_{2g+1}$. Edit actualy this is always happening, see the comment of David Loeffler bellow). I believe that we cannot hope for a more precise conjecture, it is too general. But my opinion must be considered with reserve as this is really deep and complicated.

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    $\begingroup$ "if your representation have nice proprieties..." this Galois representation always has nice properties. $H^1$ of a curve is the Tate module of its Jacobian variety. The Jacobian of a curve is always principally polarized, so its Tate module carries a canonical symplectic form; thus the Galois representation always lands in $\operatorname{GSp}_{2g}$, and hence should correspond to an automorphic form on the Langlands dual of $\operatorname{GSp}_{2g}$, which is $\operatorname{GSpin}_{2g+1}$ (note $GSpin_3 = GSp_2\ (=GL_2)$, and $GSpin_5 = GSp_4$, but this does not generalise to higher $g$). $\endgroup$ Dec 10, 2022 at 9:28
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    $\begingroup$ Oooh yes I see your point, I'm going to edit. Thank you @David. $\endgroup$ Dec 10, 2022 at 11:38

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