[Paper Review] A short proof of a Chebotarev density theorem for function fields

Michiel Kosters|arXiv (Cornell University)|Apr 25, 2014

Advanced Topology and Set Theory1 references5 citations

TL;DR

This paper presents a new formulation of the Chebotarev density theorem for function fields over perfect fields with procyclic absolute Galois groups, offering a short, error-free proof that includes ramified primes by parametrizing Frobenius classes via twists of the base function field. The key contribution is a measure-theoretic parametrization of conjugacy classes using rational primes in constant field extensions, avoiding error terms through geometric and Galois-theoretic techniques.

ABSTRACT

In this article we discuss a version of the Chebotarev density for function fields over perfect fields with procyclic absolute Galois groups. Our version of this density theorem differs from other versions in two aspects: we include ramified primes and we do not have an error term.

Motivation & Objective

• To generalize the Chebotarev density theorem to function fields over perfect fields with procyclic absolute Galois groups, extending beyond finite fields.
• To include ramified primes in the density statement, unlike classical versions that restrict to unramified primes.
• To eliminate error terms in the equidistribution statement by using a parametrization via constant field extensions and Galois twists.
• To unify and generalize existing results by allowing non-finite base fields and non-full constant fields in the extension.

Proposed method

• Define a probability measure (P,M)(γ) on the Galois group G that assigns mass to conjugacy classes based on Frobenius data from primes in the extension M/K.
• Use the decomposition and inertia groups to define the Frobenius class (P,M/K) as a union of conjugacy classes, even in the presence of ramification.
• Construct a twist Mγ of the function field M via a constant field extension k_m, such that rational primes in Mγ correspond to primes in K with a given Frobenius class.
• Establish a bijection between rational primes in Mγ and rational primes in K with a specified Frobenius class via the map φ, using Galois descent and fixed field theory.
• Prove equivalence of three conditions: γ in (P',M/K), Δ in (P',k_mM/K), and P'|_{M_γ} rational, to relate the parametrization to the original Frobenius condition.
• Use the Hasse-Weil bound to estimate the number of rational primes in Mγ, leading to the main theorem’s quantitative conclusion.

Experimental results

Research questions

• RQ1 Can the Chebotarev density theorem be reformulated to include ramified primes without introducing error terms?
• RQ2 How can Frobenius conjugacy classes be parametrized using rational primes in twisted function fields?
• RQ3 What is the role of constant field extensions in constructing a parametrization of Frobenius classes?
• RQ4 How does the absence of an error term affect the structure of the density statement in the function field setting?
• RQ5 Can the classical Chebotarev theorem be recovered as a special case of this new parametrization framework?

Key findings

• The paper establishes a measure-theoretic parametrization of Frobenius conjugacy classes via rational primes in constant field twists, with no error term.
• For any γ ∈ G with conjugacy class Γ, the number of rational primes P ∈ K with (P,M/K) ∩ Γ ≠ ∅ is equal to #N × (P,M)(γ), where #N is the size of the geometric Galois group.
• The number of rational primes in Mγ with γ in their Frobenius class is asymptotically bounded by the Hasse-Weil bound, giving |#S - (q+1)| ≤ 2g_{k'}(M)√q.
• The parametrization via twists ensures that every Frobenius class is captured exactly through the fiber structure of the map φ, with fiber size proportional to deg_k(Q)/h.
• The proof shows that the Frobenius class (P,M/K) is well-defined and independent of the choice of prime above P, even in the ramified case.
• The construction works for any perfect field k with procyclic absolute Galois group, generalizing results from finite fields to broader classes like quasi-finite fields and maximal p-power extensions.