Counting Frobenius extensions over local function fields

Abstract: We determine the asymptotic growth of extensions of local function fields of characteristic p counted by discriminant, where the Galois group is a subgroup of the affine group AGL_1(p). More general, we solve the corresponding counting problems for all groups which arise in a tower of a cyclic extension of order p over a cyclic extension of degree d coprime to p. This in particular give answers for certain non-abelian groups including S_3, dihedral groups of order 2p, and many Frobenius groups.

• The paper introduces new asymptotic formulas for counting degree n extensions of local function fields with wild ramification, exhibiting periodic fluctuations.
• It rigorously combines combinatorial group theory, Artin-Schreier techniques, and Galois module analysis to compute discriminant exponents.
• The findings challenge the classical Malle conjecture by establishing explicit growth rates for non-abelian extensions in the wild setting.

Asymptotic Counting of Frobenius Extensions over Local Function Fields

Introduction and Context

The paper "Counting Frobenius extensions over local function fields" (2604.02152) introduces an explicit asymptotic analysis of the number of degree $n$ extensions of a local function field $F=\mathbb{F}_q((t))$ of characteristic $p$, with prescribed Galois group $G$, and bounded discriminant exponent. The focus is on non-abelian groups $G$ containing $p$-subgroups, specifically subgroups of the affine group $AGL_1(p)$ and more generally, groups arising as extensions of a cyclic group order $p$ over a cyclic group of order $d$ with $(p,d)=1$.

This setting lies outside the "tame ramification" paradigm and hence falls under the territory where the classical Malle conjecture is known to fail in its original form. The main mathematical tools utilized include combinatorial group theory, Artin-Schreier theory, module-theoretic representation of Galois extensions, and explicit analysis of discriminant exponents.

Main Results

The authors rigorously describe the precise growth rates for the counting functions

• $F=\mathbb{F}_q((t))$0: the number of extensions $F=\mathbb{F}_q((t))$1 (degree $F=\mathbb{F}_q((t))$2) with $F=\mathbb{F}_q((t))$3 and discriminant exponent at most $F=\mathbb{F}_q((t))$4,
• $F=\mathbb{F}_q((t))$5: similar, but for Galois extensions with group $F=\mathbb{F}_q((t))$6 acting as a permutation group on $F=\mathbb{F}_q((t))$7 letters.

The main asymptotic statements (see Theorems in the Introduction) are:

• There exists a $F=\mathbb{F}_q((t))$8-periodic function $F=\mathbb{F}_q((t))$9 so that as $p$0,

$p$1

• There is a $p$2-periodic function $p$3 so that

$p$4

These formulas feature explicit atypical exponents—deviating from naive expectations from Malle's conjecture in the wild case—and periodic fluctuations in the constant, both dictated by the underlying ramification theory.

Structural and Technical Contributions

The paper provides a comprehensive classification of the possible Galois groups $p$5 that arise in towers $p$6, with $p$7 a $p$8-extension (provided by Artin-Schreier theory) and $p$9 a cyclic $G$0-extension. The set of possible $G$1 is parameterized by subsets $G$2 indexing the factorization $G$3 in $G$4, and the order of $G$5 is $G$6 where $G$7 is the sum of degrees of $G$8 for $G$9.

A key technical element is an explicit module-theoretic description of the $G$0-elementary abelian extensions of $G$1, using the $G$2-linear Artin-Schreier functor $G$3 and the module $G$4, considered as a $G$5-module.

The discriminant and conductor exponents of these extensions are computed via explicit bases of $G$6. The authors prove tight periodicity properties for the growth function—arising from the structure of the module $G$7 and the action of $G$8 for $G$9.

Combinatorial counting is performed using generating functions (Dirichlet series) linked to the possible ways of assembling generators for the $p$0-extensions; this demands a fine stratification over ramification indices and the Galois module structure.

Relationship to Prior Work and Malle's Conjecture

Malle's conjecture predicts the asymptotic distribution of number field (or function field) extensions with prescribed Galois group and bounded discriminant. For wild ramification ($p$1 in characteristic $p$2), the conjecture as originally formulated fails: exponents predicted by Malle do not describe the actual asymptotics, which can include higher exponents and extra logarithmic or periodic factors.

The current paper extends previous works (notably, the thesis of Müller [RM] and Potthast [Nicolas]) by covering non-abelian groups in the wild setting and providing precise exponents in the growth formulas for a broad new class of groups, including all dihedral groups of order $p$3, $p$4, affine and some Frobenius groups.

Numerical and Qualitative Characteristics

The asymptotic exponents in the main theorems are explicit functions of $p$5, $p$6 and $p$7, yielding different rates for distinct Galois groups within the same family. In particular, for the groups $p$8 (the obvious non-split extension), the exponent becomes $p$9 and the above formulas specialize, delimiting the asymptotics for these Frobenius and dihedral cases.

The analysis captures, and in some cases generalizes, the phase transition found in previous works for elementary abelian groups: in wild characteristic, discriminant-counts grow much more rapidly than what would be predicted for "tame" groups of the same order.

A robust comparison with the abelian and global (number field) cases is included, showing how the local function field situation exhibits both parallel phenomena—such as the finiteness of extension counts in some cases—and profound differences due to wild ramification.

Implications and Directions for Further Research

The results categorically establish that wild ramification enforces non-classical asymptotic behavior for discriminant counting of local function field extensions with non-abelian group structures, and provides precise asymptotic constants and exponents, resolving open cases for these families.

Potential future directions include:

• Generalization to more complicated towers and iteration of wild extensions;
• Analysis for global function fields with controlled infinite places;
• Explicit determination of the periodic functions $AGL_1(p)$0 (which could yield finer secondary term information);
• Connection of these mass formulas to cohomological or geometric approaches to Hurwitz spaces, especially in positive characteristic;
• Extension of the techniques to counting embedding problems with prescribed local conditions.

Conclusion

This work delivers a comprehensive and technically detailed analysis of the discriminant distribution of non-abelian, wild extensions over local function fields, culminating in sharp asymptotic formulas and explicit construction of the relevant Galois module structures. The results clarify the asymptotic landscape for Galois group distributions in local characteristic $AGL_1(p)$1 and delineate the limitations of conjectural heuristics derived from the tame paradigm. The paper thus provides a rigorous foundation for both arithmetic statistics in the wild case and future theoretical advancements in the study of local and global function fields.