# Can Taniyama-Shimura conjecture be generalized to curves of higher genus (within Langlands framework)?

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.

Thanks!

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.
• "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 $g$). Dec 10, 2022 at 9:28