Algebraic varieties (awaiting review)

A much deeper occurrence of modular forms comes from the theory of algebraic varieties over $\Q$, or more generally over a number field: to such an object one can associate in a natural way one or several Dirichlet series defined as Euler products involving the number of points of the variety over all finite fields. It is conjectured, and far from proved, that all these Dirichlet series satisfy similar properties to that of the Riemann zeta function $\zeta(s)$ and generalizations: meromorphic continuation to the whole complex plane with known poles, a functional equation when $s\mapsto k-s$ for suitable $k$, and so forth. The existence of this functional equation together with suitable regularity conditions is closely linked to the fact that the function $\sum_{n\ge1}a(n)q^n$ has a modularity property. In the specific case of elliptic curves defined over $\Q$, this has been proved, giving another much deeper source of modular forms (of weight $2$ and trivial multiplier system) linked to elliptic curves.

Much more recently, Brumer‒Kramer have conjectured a similar connection between isogeny classes of abelian surfaces defined over $\Q$ with endomorphism ring reduced to $\Z$ and certain modular forms on a subgroup of $\Sp_4(\Q)$ called the paramodular group.