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.

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- Last edited by Andrew Sutherland on 2019-07-31 15:29:29

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