Timeline for Modular forms from counting points on algebraic varieties over a finite field

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Jan 22, 2017 at 9:55	comment	added	David Loeffler		To address your last paragraph: if you define $a_p$ in this way, it has no hope of being the coefficient of a modular form unless $H^i(X)$ is two-dimensional; and the "baseline" is just the alternating sum of Frobenius traces on $H^j$ for $j \ne i$. In the examples you've given, this is a polynomial in $p$, but that is only because the varieties you've written down are extremely special. I suggest you contemplate the situation where $X$ is the product of two elliptic curves, in which case the point counts are clearly built up from modular-forms somehow, but not in the way you describe!
Jan 22, 2017 at 9:29	comment	added	David Loeffler		The key word here is "nice enough". These rigid Calabi--Yaus are much more "structured" than a generic 3-fold would be, so you can express their point-counts purely in terms of polynomials in p and modular forms. But most varieties of dimension > 1, and most curves of genus > 1, will not have this property -- e.g. for a genus 2 curve $C$ which is "generic" in the sense that the Jacobian $J(C)$ is not isogenous over $\overline{\mathbb{Q}}$ to a product of elliptic curves, there is no way to express $\#C(\mathbf{F}_p)$ in terms of polynomials in $p$ and $a_p$'s of modular forms.
Jan 21, 2017 at 21:01	history	answered	Bruce Bartlett	CC BY-SA 3.0