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Classically, singular cohomology is an important tool for studying topological spaces, in particular, complex varieties. In the mid-twentieth century it was realized that there are many analogues of singular cohomology for varieties that work in other settings, for instance, $\ell$-adic cohomology of varieties over finite fields. The theory of motives seeks to unify these cohomology theories by constructing objects, termed "motives", which bundle all of them together at once. Grothendieck's original definition formalizes the intuition that a motive should be a "chunk" of a variety, reflecting the abelian nature of cohomology theories.

(The cohomology theories relevant to motives, which work with some flavor of locally-constant sheaf or local system, should not be confused with the cohomology of quasi-coherent sheaves, which is of a different nature.)

My impression of motives is that the subject is deep and difficult. One always hears of Grothendieck's focus on the standard conjectures, which imply that the category of (numerical equivalence) motives is semisimple. The next part of the story is that the standard conjectures are impossibly difficult.

In light of this impression, I was surprised to learn (from Deligne's second Corvallis article) that we can completely understand the category of motives arising from zero-dimensional varieties. This is a very nice exercise in unwrapping the definition. To avoid saying something wrong, let me take $\mathbb Q$ as the base field. Then the category of motives of zero-dimensional varieties is simply the category of finite-dimensional $\mathbb Q$-representations of the absolute Galois group $G=\text{Gal}(\bar{\mathbb Q}/\mathbb Q)$. The subcategory corresponding to the honest zero-dimensional varieties is the category of permutation representations. Forming an abelian category from this subcategory, one is forced to adjoin all other Galois representations. We call these motives "Artin motives" in honor of Artin's L-functions, which are attached to complex finite-dimensional Galois representations. To make the analogy tighter we should consider Artin motives with coefficients in some number field $E$, which becomes the coefficient field of the corresponding Galois representations.

Now let me introduce another theme, modularity. We are told (?) that "motives are supposed to be related to modular forms". I understand this to have the following more precise meaning: we hope that for many motives $M$ one can find a modular form $f$ so that there is an equality of $L$-functions: $$ L(M,s) = L(f,s). $$ Apparently this hope has been realized by the work of Wiles and his intellectual successors when $M$ is an elliptic curve over $\mathbb Q$, though I have not myself traced through the connection. When $M$ is a zero-dimensional variety, its $L$-function encodes information about the number of zeros of $M$ modulo the various primes, and the modular connection should give us some insight into this arithmetic problem. Modularity for such a motive would generalize quadratic reciprocity in some sense.

Serre's article "On a Theorem of Jordan" nicely discusses this circle of ideas for the Artin motives $M_n$ cut out by the polynomial $x^n - x - 1$, as $n\geq 2$ varies. For $n\leq 4$, Serre nimbly uses class field theory and the Langlands–Tunnell theorem, which can be phrased as modularity results, to study $|M_n(\mathbb F_p)|$. But for $n\geq 5$, he says

No explicit connection with modular forms (or modular representations) is known, although some must exist because of the Langlands program.

I find this claim really surprising. How could Serre's void of knowledge possibly coexist with our understanding of modularity for higher-dimensional varieties, à la Wiles et al? One would think that zero-dimensional varieties would be much easier than higher-dimensional ones.

Question: Are there any precise modularity conjectures for Artin motives over $\mathbb Q$?

  • If so, what are they? Which kinds of modular (or even automorphic) forms are expected to arise? (Are they holomorphic? Or Maaß forms? What is the expected level? etc.)
  • If not, then why are modularity conjectures difficult to formulate?
  • Can one hope to formulate such conjectures by experimentation (using a computer, or otherwise)?

Based on Serre's article, one would expect the level of the modular form to equal the discriminant.

The final part of the question is motivated by the ease with which we can count roots of univariate polynomials mod $p$. I am not so familiar with computing Hecke eigenvalues of modular forms, but I expect there to be good computational methods there as well. You could then try to compare the number sequences in the two settings . . .

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  • $\begingroup$ Pardon my naivety, but why does class field theory not essentially answer this in the sense that the $L$-function an Artin-character and the corresponding Hecke-character agree? Also out of curiosity, "motives are supposed to be related to modular forms" wouldn't that be more accurate to expect them to be related to automorphic forms? $\endgroup$ Dec 29, 2022 at 1:37
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    $\begingroup$ Class field theory answers this question only for characters of 1-dimensional representations. Otherwise, we expect there to be an associated automorphic form, and whether it's a Maas form or a modular form depends on properties of the character (even or odd). The first major result in this direction was the Langlands-Tunnel theorem on the automorphicity of galois representations into $S_3$, and Wiles' theorem heavily relies on it. Since then the biggest result is Khare-Winterberger's theorem, aka Serre's modularity conjecture. $\endgroup$ Dec 29, 2022 at 9:37
  • $\begingroup$ I see. So this would tell us about modularity of Galois representations that factor through $\text{GL}_2(\mathbb F_q)$. Are there conjectures in this setting where one replaces $\text{GL}_2$ by some other reductive group, say, $\text{GL}_n$ or a unitary group? $\endgroup$ Dec 29, 2022 at 10:45
  • $\begingroup$ Serre’s conjecture (now a theorem) is that odd irreducible representations into $GL_2(\mathbb{F}_q)$ are indeed modular (in particular, odd irreducible complex $2$-dim representations). The modularity for even representations into $GL_2(\mathbb{C})$ is afaik completely open apart from Tunnell-Langlands. My lay-ish understanding is that for other groups such as $GL_n$, we expect there to be an automorphic form with the same $L$-function (and then some) but it’s regarded as much more difficult for various reasons (such as, allegedly, the equal Hodge-Tate weights). $\endgroup$
    – Aphelli
    Jan 6, 2023 at 23:20

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