The classical setting of modularity is that one can associate elliptic modular forms (or automorphic representations of GL(2)/$\mathbb Q$) to elliptic curves over $\mathbb Q$. This has far-reaching consequences to elementary problems in arithmetic: Fermat's last theorem, the congruence number problem, sums of 2 cubes, ... Conjectures and some results are also known over other number fields.

Now there are various conjectural generalizations of modularity to higher dimensions that are part of Langlands' philosphy. For instance the paramodular conjecture, asserting abelian surfaces are associated to certain kinds of Siegel modular forms (cf. Modularity of higher dimensional abelian varieties, Langlands in dimension 2: the Yoshida conjecture). Certainly knowing modularity in higher rank would be phenomenal, and is inherently interesting. But...

Question: Are there appealing "elementary" applications, or suspected applications (analogous to connections with elliptic curves) of knowing modularity (automorphy) in higher rank, i.e., that certain motives (beyond GL(2) type) are associated to automorphic representations?

The issue, to my naive mind, is that elliptic curves are given by rather simple equations that naturally arise in classical number theory, whereas abelian surfaces are not described such simple equations. However, philosophically, I feel that there is a lot of arithmetic in these conjectures, which should manifest itself in concrete (and hopefully easily to explain) ways---the question is how?

(Note Langlands also brings up this issues at the end of his introduction to his 2010 Notre Dame talk Is there beauty in mathematical theories?, writing "[a] very hard question is what serious, concrete number-theoretical results does the theory suggest or, if not suggest, at least entail. A few mathematicians have begun, implicitly or explicitly, to reflect on this. I have not.")


A quick fix for the lack of nice equations defining abelian surfaces is to write down equations for the corresponding genus $2$ curves. Something similar works in higher genus, but there most abelian varieties won't be Jacobians of curves. However, this is no problem as the modularity conjecture is hard enough for curves already.

I think many would say that the most important concrete consequences of modularity are statistical statements that follow from the existence of $L$-functions. For instance, the following statements should follow from the $GL_n$ modularity of curves:

$$ \sum_{ p < X} \frac{ p + 1 - |C(\mathbb F_p)|}{\sqrt{p}} = o (\pi(X))$$

$$\sum_{ p < X} \frac{ (p + 1 - |C_1(\mathbb F_p)|) (p + 1 - |C_2(\mathbb F_p)|) }{p} = \operatorname{rank} (\operatorname{NS}( C_1 \times C_2) )\pi(X) - 2\pi(X) + o (\pi(X)) $$

where $\operatorname{NS}$ is the Neron-Severi group of divisors defined over $\mathbb Q$ modulo numerical equivalence.

The first from the ordinary $L$-function and the second from the Rankin-Selberg $L$-function plus the Tate conjecture for abelian varieties due to Faltings.

These will all fit into the framework of the generalized Sato-Tate conjecture, which would follow from super-general modularity statements.


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