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Pseudofinite fields with additive and multiplicative character

Stefan Marian Ludwig

TL;DR

This work extends the model theory of pseudofinite fields to include both additive and multiplicative characters by forming and analysing the theory $PF^{+, imes}$, with characters encoded as continuous predicates. It shows that $PF^{+, imes}$ is the characteristic $0$ asymptotic theory of finite fields with sufficiently generic characters, via a careful axiomatisation and a two-pronged QE argument in a symmetric, predicate-rich language. The paper further proves that the Chatzidakis–van den Dries–Macintyre counting measure is uniformly definable in parameters, enabling uniform definable integration of definable predicates, and proves that $PF^{+, imes}$ is simple. A key methodological contribution is the blend of Weil bound character-sum estimates with the Erdős–Turán–Koksma inequality to handle equidistribution in the multiplicative setting, yielding a robust framework for limit theories and definable measures in this enriched pseudofinite context.

Abstract

We introduce the theory $\mathrm{PF}^{+,\times}$ of pseudofinite fields with generic additive and multiplicative character added as continuous logic predicates. Using the Weil bounds on character sums over finite fields as well as the Erdős-Turàn-Koksma inequality we show that it is the asymptotic theory (in characteristic $0$) of finite fields with (sufficiently generic) additive and multiplicative character. Moreover, we establish quantifier elimination in a natural definitional expansion of the language and deduce that integration by the Chatzidakis-van den Dries-Macintyre counting measure is uniformly definable in the parameters. Finally, we show that $\mathrm{PF}^{+,\times}$ is a simple theory.

Pseudofinite fields with additive and multiplicative character

TL;DR

This work extends the model theory of pseudofinite fields to include both additive and multiplicative characters by forming and analysing the theory , with characters encoded as continuous predicates. It shows that is the characteristic asymptotic theory of finite fields with sufficiently generic characters, via a careful axiomatisation and a two-pronged QE argument in a symmetric, predicate-rich language. The paper further proves that the Chatzidakis–van den Dries–Macintyre counting measure is uniformly definable in parameters, enabling uniform definable integration of definable predicates, and proves that is simple. A key methodological contribution is the blend of Weil bound character-sum estimates with the Erdős–Turán–Koksma inequality to handle equidistribution in the multiplicative setting, yielding a robust framework for limit theories and definable measures in this enriched pseudofinite context.

Abstract

We introduce the theory of pseudofinite fields with generic additive and multiplicative character added as continuous logic predicates. Using the Weil bounds on character sums over finite fields as well as the Erdős-Turàn-Koksma inequality we show that it is the asymptotic theory (in characteristic ) of finite fields with (sufficiently generic) additive and multiplicative character. Moreover, we establish quantifier elimination in a natural definitional expansion of the language and deduce that integration by the Chatzidakis-van den Dries-Macintyre counting measure is uniformly definable in the parameters. Finally, we show that is a simple theory.

Paper Structure

This paper contains 13 sections, 31 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Character sum estimates over affine curves
  4. Pseudofinite fields
  5. Continuous logic
  6. Pseudofinite fields with additive character
  7. Axiomatisation
  8. Quantifier elimination
  9. Limit theory
  10. Definable integration
  11. Facts around the Chatzidakis-van den Dries-Macintyre counting measure
  12. Uniform definable integration of definable predicates
  13. Further remarks

Key Result

Theorem A

Let $\Psi_{q},\chi_{q}$ denote a nontrivial additive and a non-trivial multiplicative character on the finite field $\mathbb{F}_{q}$. Let $(F,\Psi,\chi)=\prod_{\mathcal{U}}(\mathbb{F}_{q},\Psi_{q},\chi_{q})$ be any characteristic $0$ ultraproduct of the structures $(\mathbb{F}_{q},\Psi_{q},\chi_{q})

Theorems & Definitions (79)

  • Theorem A: See Theorem \ref{['theoremUPoffinitefieldsmodelofpfplustimes']}
  • Theorem B: See Theorem \ref{['qepfplustimes']}
  • Theorem C: See Theorem \ref{['theorempfplustimesislimittheory']}
  • Theorem D: See Theorem \ref{['corollaryremovinglowerbounds']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.6
  • Definition 2.10
  • Corollary 2.11
  • Theorem 2.12
  • ...and 69 more