[Paper Review] Fox-Neuwirth-Fuks cells, quantum shuffle algebras, and Malle's conjecture for function fields

Jordan S. Ellenberg, TriThang Tran+1 more|arXiv (Cornell University)|Jan 17, 2017

Advanced Algebra and Geometry2 references8 citations

TL;DR

This paper proves the upper bound in the weak Malle conjecture for function fields $\mathbb{F}_q(t)$ by establishing a connection between the cohomology of Hurwitz spaces and quantum shuffle algebras, showing that the number of $G$-extensions of $\mathbb{F}_q(t)$ with discriminant at most $X$ is bounded by $C(G)X^{a(G)}\log(X)^{e(G)}$ for $q$ sufficiently large and coprime to $\#G$, where $e(G)$ is bounded by the Gelfand-Kirillov dimension of an associated graded ring.

ABSTRACT

The purpose of this paper is to prove the upper bound in Malle's conjecture on the distribution of finite extensions of $\mathbb{F}_q(t)$ with specified Galois group. As in previous work of Ellenberg-Venkatesh-Westerland, our result is based upon computations of the homology of braid groups with certain (exponential) coefficients. However, the approach in this paper is new, relying on a connection between the cohomology of Hurwitz spaces and the cohomology of quantum shuffle algebras.

Motivation & Objective

• To establish the upper bound in the weak Malle conjecture for the number of finite extensions of $\mathbb{F}_q(t)$ with a given Galois group $G$.
• To connect the cohomology of Hurwitz spaces over finite fields to quantum shuffle algebras and Nichols algebras, enabling cohomological computations via topological and algebraic methods.
• To show that the growth rate of such extensions is bounded by $X^{a(G)}\log(X)^{e(G)}$ for $q$ large enough and coprime to $\#G$, with $e(G)$ computable from group-theoretic data.
• To provide a general framework applicable to arbitrary transitive groups $G \leq S_m$ beyond previously known cases, particularly for function fields.

Proposed method

• Utilizes the arithmetic geometry of moduli spaces of Hurwitz covers over $\mathbb{F}_q(t)$, relating their $\ell$-adic cohomology to topological singular cohomology via comparison theorems.
• Applies the Lefschetz trace formula to count $\mathbb{F}_q$-rational points on Hurwitz spaces, linking point counts to traces of Frobenius on cohomology groups.
• Relies on the theory of quantum shuffle algebras and braided commutative algebras associated to a braided vector space $V$ determined by $G$, which model the cohomology of the Hurwitz spaces.
• Computes the Gelfand-Kirillov dimension $d$ of a graded ring $R$ associated to $G$, which determines the exponent $e(G) = d - 1$ in the upper bound.
• Uses the action of Frobenius on cohomology and monodromy data to ensure that the cohomological contribution grows as $n^{d-1}$, leading to the desired lower bound on point counts.
• Applies results from [Woo21] on the reduced Schur cover and the structure of $U'(G,c)$ to verify that Frobenius acts trivially under divisibility conditions on $q-1$, such as $3 \mid q-1$ for $A_4$.

Experimental results

Research questions

• RQ1 What is the asymptotic growth rate of the number of $G$-extensions of $\mathbb{F}_q(t)$ with discriminant bounded by $X$?
• RQ2 Can the upper bound in the weak Malle conjecture be established for all finite transitive groups $G \leq S_m$ over function fields?
• RQ3 How does the cohomology of Hurwitz spaces relate to quantum shuffle algebras and Nichols algebras in the context of arithmetic statistics?
• RQ4 What conditions on $q$ ensure that Frobenius acts trivially on the relevant cohomology groups, enabling point-counting arguments?
• RQ5 What is the precise exponent $e(G)$ in the logarithmic factor of the upper bound, and how is it determined from group-theoretic data?

Key findings

• The number of $G$-extensions of $\mathbb{F}_q(t)$ with discriminant at most $X$ is bounded by $C(G)X^{a(G)}\log(X)^{e(G)}$ for all $q > Q(G)$ coprime to $\#G$, proving the upper bound in the weak Malle conjecture.
• The exponent $e(G)$ in the logarithmic factor is bounded above by $|G| - 1$ and can be taken as $d - 1$, where $d$ is the Gelfand-Kirillov dimension of a graded ring $R$ associated to $G$.
• For $G = A_4$, when $q \equiv 1 \pmod{3}$, the number of quartic extensions with Galois group contained in $A_4$ and discriminant $\leq X$ satisfies $C_1 X^{1/2} (\log X)^2 \leq N_{A_4}(\mathbb{F}_q(t); X) \leq C_2 X^{1/2} (\log X)^2$, matching Malle’s prediction up to constants.
• The cohomology of Hurwitz spaces is linked to quantum shuffle algebras, allowing the use of algebraic structures to compute topological and arithmetic invariants.
• The Frobenius action on $U'(G,c)$ is trivial when $q-1$ is divisible by the order of the relevant conjugacy classes, such as $3 \mid q-1$ for $A_4$, which is essential for the point-counting argument.
• The method yields a finite, combinatorially computable bound on $e(G)$, making the upper bound effective for any given $G$.