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Suppose we are given some polynomial with integer coefficients, which we regard as carving out an affine variety $E$, for example:

$$ 3x^2y - 12 x^3y^5 + 27y^9 - 2 = 0 \tag{$*$} $$

(We might consider a bunch of equations, we might work over projective space, but let's keep it simple for now).

We are interested in the number of points on $E$ when we reduce modulo $p$, i.e. over the finite field $\mathbb{F}_p$ as the prime $p$ varies. For our single equation in two variables, as a rough approximation, we would expect $p$ points in general. So for each prime $p$, we define numbers $a_p$ which measure the deviation from this,

$$a_p := p - \text{number of solutions to $(*)$ over $\mathbb{F}_p$}$$

Question: When is it true that the numbers $\{a_p\}$ are "modular", in the sense that there exists a modular form

$$f = \sum_{n=1}^\infty b_n q^n$$

such that $b_p = a_p$ for almost all primes $p$? (The "almost all" is to avoid problems with bad primes. Note that $f$ is still uniquely determined by the above requirement.)

The Modularity Theorem of Breuil, Conrad, Taylor and Diamond says that this is true when $E$ is an elliptic curve, i.e. takes the form $y^2 = 4x^3 - g_2 x - g_3$ for some integers $g_2, g_3$. In that case, $f$ is a weight 2 modular form of level $N$ where $N$ is the "conductor" of $E$.

But is it true for more general varieties?

(Note: I am aware of a generalized "Modularity Theorem" for certain Abelian varieties, but it's not clear to me that what people mean by "Modular" in that context is the same as the simple-minded notion I'm using --- that an adjusted count of points mod $p$ gives the Fourier coefficients of a modular form.)

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    $\begingroup$ First in your rather naive version, modularity holds if the curve is of genus 1 and has exactly one point on the line at infinity. You should better formulate it projectively. For curves of higher genus, the form that you are after must be more complicated as even the zeta-functions of the curve over $\mathbb{F}_p$ needs more information than just the $a_p$. $\endgroup$ – Chris Wuthrich Jan 21 '17 at 11:33
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    $\begingroup$ It would help with your understanding if you read up on the Weil conjectures (this theory works best for smooth projective varieties). These relate the number of $\mathbb{F}_p$-points to the Galois representation on the etale cohomology groups. Modular forms roughly correspond to $2$-dimensional Galois representations, so one would only expect modular forms to occur if one of the pieces of the etale cohomology is $2$-dimensional, or has a $2$-dimensional subspace. This is very uncommon; as pointed out by Chris, most curves of higher genus will not have this property. $\endgroup$ – Daniel Loughran Jan 21 '17 at 14:49
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    $\begingroup$ No. Higher dimensional Galois representations (conjecturally) correspond to automorphic forms on higher dimensional algebraic groups. $\endgroup$ – Daniel Loughran Jan 21 '17 at 15:59
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    $\begingroup$ @BruceBartlett You seem very confident about your viewpoint, but I am afraid it is incorrect, as Daniel has pointed out; you are muddling together the dimension of the Galois representation and the dimension of the algebraic variety, which are totally unrelated. Take e.g. a nonsingular curve of genus 2. Then the $b_p$'s are (up to a small discrepancy coming from how you handle points at infinity) the trace of Frobenius at p on a 4-dimensional Galois representation, occuring as the etale H^1 of the curve, which is dual to the Tate module of its Jacobian. ... $\endgroup$ – David Loeffler Jan 21 '17 at 17:02
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    $\begingroup$ ... [cont'd] This has dimension 4, but motivic weight 1, while the Galois representation associated to a weight k modular form with $k > 2$ has dimension 2 and motivic weight k-1, so there is no way to make these match up. $\endgroup$ – David Loeffler Jan 21 '17 at 17:02
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The correct setting for this construction turns out to be projective varieties, so let me suppose we have a smooth variety $X$ inside $\mathbf{P}^N$, for some $N \ge 1$, defined by the vanishing of some homogenous polynomials $F_1, \dots, F_r$ in variables $x_0, \dots, x_N$, with the $F_i$ having coefficients in $\mathbf{Q}$. Actually, let me assume the $F_i$ have coefficients in $\mathbf{Z}$, which is no loss since we can just multiply up. Then we can make sense of the reduction of $X$ modulo $p$; and we want to study the point counts $\#X(\mathbf{F}_p)$ as a function of $p$, possibly neglecting some finite set $\Sigma$ containing all primes such that the reduction of $X$ mod $p$ is singular.

Thanks to Grothendieck, Deligne, and others, we have a very powerful bunch of tools for analysing this problem. The setup is as follows. Choose your favourite prime $\ell$. Then the theory of etale cohomology attaches to $X$ a bunch of finite-dimensional $\mathbf{Q}_\ell$-vector spaces $H^i_{\mathrm{et}}(X_{\overline{\mathbf{Q}}}, \mathbf{Q}_\ell)$ (let me abbreviate this by $H^i_\ell(X)$ to save typing). The dimension of $H^i_\ell$ is the same as the $i$-th Betti number of the manifold $X(\mathbb{C})$; but they encode much more data, because each $H^i_\ell(X)$ is a representation of the Galois group $\operatorname{Gal}(\overline{\mathbf{Q}} / \mathbf{Q})$, unramified outside $\Sigma \cup \{\ell\}$; so for every prime not in this set, and every $i$, we have a number $t_i(p)$, the trace of Frobenius at $p$ on $H^i_\ell$, which turns out to be independent of $\ell$.

Theorem: $\#X(\mathbf{F}_p)$ is the alternating sum $\sum_{i=0}^{2 dim(X)} (-1)^i t_i(p)$.

Now let's analyse $H^i_\ell$ as a Galois representation. Representations of Galois groups needn't be direct sums of irreducibles, but we can replace $H^i_\ell$ by its semisimplification, which does have this property and has the same trace as the original $H^i_\ell$. This semisimplification will look like $V_{i, 1} + \dots + V_{i, r_i}$ where $V_{i, j}$ are irreducible; and the $V_{i, j}$ all have motivic weight $i$, so the same $V$ can't appear for two different $i$'s. So we get a slightly finer decomposition

$\#X(\mathbf{F}_p) = \sum_{i=0}^{2 \mathrm{dim} X} (-1)^i \sum_{j=1}^{k_i} t_{i, j}(p)$

where $t_{i,j}(p)$ is the trace of $Frob_p$ on $V_{i,j}$.

Let me distinguish now between several different types of irreducible pieces:

  • $V_{i, j}$ is a one-dimensional representation. Then $i$ must be even, and the trace of Frobenius on $V_{i, j}$ is $p^{i/2} \chi(p)$ where $\chi$ is a Dirichlet character.
  • $V_{i, j}$ is two-dimensional and comes from a modular form. Then $t_{i,j}(p) = a_p(f)$, and $f$ must have weight $i+1$.
  • $V_{i,j}$ is two-dimensional and doesn't come from a modular form. This can happen, but it's rare, and it's expected that all examples come from another kind of analytic object called a Maass wave form; in particular this forces $i$ to be even.
  • $V_{i, j}$ has dimension $> 2$. Then $V_{i, j}$ cannot be the Galois representation coming from a modular form, because these always have dimension 2.

You seem to want your varieties to have $X(\mathbf{F}_p)$ = (polynomial in $p$) + (coefficient of a modular form). From the above formulae, it's clear that this can only happen if all the $V_{i,j}$ have dimension 1 or 2; there is exactly one with dimension 2 and it comes from a modular form; and the one-dimensional pieces all come from the trivial Dirichlet character. This always happens for genus 1 curves, because the $H^0$ and $H^2$ are always 1-dimensional for a curve, and the genus condition forces the $H^1$ to be two-dimensional.

However, once you step away from genus 1 curves, this is totally not the generic behaviour, and it will only occur for unusual and highly symmetric examples, such as the rigid Calabi-Yaus and extremal $K_3$ surfaces in the links you've posted.

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It seems that this holds as long as one counts with respect to the appropriate "baseline". In this paper they give a class of examples. Let $X$ be a nice enough rigid Calabi-Yau threefold over $\mathbb{Q}$, with Hodge numbers $h^{i,j}$. Then, if I understand them correctly (pg 7), they say that if we set

$a_p := 1 + p^3 + (1+p)p h^{1,1} - \#X(\mathbb{F}_p)$

then these are the Fourier coefficients (for almost all primes $p$) of a modular form of weight 4 and a certain level $N$.

In general, it seems one needs to set the $a_p$ to be the trace of the Frobenius acting on etale cohomology in some degree $i$,

$a_p = \mbox{Tr}(\mbox{Fr}_p : H^i(X) \rightarrow H^i(X)$).

In nice enough cases (such as the one above), one can interpret this trace concretely as a count of the points on $X$ mod p, with respect to an appropriate baseline.

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    $\begingroup$ 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. $\endgroup$ – David Loeffler Jan 22 '17 at 9:29
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    $\begingroup$ 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! $\endgroup$ – David Loeffler Jan 22 '17 at 9:55

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