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Geometric Gauss Sums and Gross-Koblitz Formula over Function Fields

Ting-Wei Chang

TL;DR

This work develops a comprehensive theory of geometric Gauss sums for function fields in positive characteristic. By establishing a Gross-Koblitz-Thakur–type formula that connects geometric Gauss sums to $v$-adic gamma values, the authors derive foundational properties including a reflection formula, Stickelberger-type factorization, and Hasse-Davenport relations, and they determine absolute values and signs at infinity. The approach hinges on interpreting geometric Gauss sums as reductions of twisted Coleman functions and leveraging duality between the Carlitz and adjoint-Carlitz structures, together with residue, trace, and Poonen pairings. The results extend Thakur’s arithmetic Gauss sums and unify multiple strands of function-field arithmetic, yielding explicit algebraicity statements and a robust framework for two-variable gamma functions and their applications. Overall, the paper provides a deep, technically rich bridge between Gauss-sum phenomena and $v$-adic gamma theories in the geometric setting, with potential implications for Stickelberger-type phenomena and Brumer–Stark-type unit constructions in function fields.

Abstract

In this paper, we introduce an analog of Gauss sums over function fields in positive characteristic. We establish several fundamental properties, including reflection formula, Stickelberger's theorem, and Hasse-Davenport relations. In addition, we determine their absolute values and signs at infinity. While these results parallel the classical theory of Gauss sums as well as Thakur's "arithmetic" analogs over function fields, our approach differs completely from both of the preceding cases. Specifically, we first prove a Gross-Koblitz-type formula relating geometric Gauss sums to special $v$-adic gamma values. The properties of geometric Gauss sums then follow from the specializations of this formula together with the functional equations of $v$-adic gamma functions.

Geometric Gauss Sums and Gross-Koblitz Formula over Function Fields

TL;DR

This work develops a comprehensive theory of geometric Gauss sums for function fields in positive characteristic. By establishing a Gross-Koblitz-Thakur–type formula that connects geometric Gauss sums to -adic gamma values, the authors derive foundational properties including a reflection formula, Stickelberger-type factorization, and Hasse-Davenport relations, and they determine absolute values and signs at infinity. The approach hinges on interpreting geometric Gauss sums as reductions of twisted Coleman functions and leveraging duality between the Carlitz and adjoint-Carlitz structures, together with residue, trace, and Poonen pairings. The results extend Thakur’s arithmetic Gauss sums and unify multiple strands of function-field arithmetic, yielding explicit algebraicity statements and a robust framework for two-variable gamma functions and their applications. Overall, the paper provides a deep, technically rich bridge between Gauss-sum phenomena and -adic gamma theories in the geometric setting, with potential implications for Stickelberger-type phenomena and Brumer–Stark-type unit constructions in function fields.

Abstract

In this paper, we introduce an analog of Gauss sums over function fields in positive characteristic. We establish several fundamental properties, including reflection formula, Stickelberger's theorem, and Hasse-Davenport relations. In addition, we determine their absolute values and signs at infinity. While these results parallel the classical theory of Gauss sums as well as Thakur's "arithmetic" analogs over function fields, our approach differs completely from both of the preceding cases. Specifically, we first prove a Gross-Koblitz-type formula relating geometric Gauss sums to special -adic gamma values. The properties of geometric Gauss sums then follow from the specializations of this formula together with the functional equations of -adic gamma functions.

Paper Structure

This paper contains 69 sections, 41 theorems, 223 equations.

Table of Contents

  1. Introduction
  2. Classical Gauss sums
  3. Gauss sums in positive characteristic
  4. Thakur's arithmetic Gauss sums
  5. Geometric Gauss sums
  6. The Galois group and Gauss sum monomials
  7. The properties of geometric Gauss sums
  8. Reflection formula
  9. Analog of Stickelberger's theorem
  10. Analog of Hasse-Davenport product relation
  11. Absolute values and signs at infinity
  12. A comparison of Gauss sums
  13. Gross-Koblitz-Thakur formulas
  14. v-adic gamma functions
  15. Gross-Koblitz-Thakur formulas
  16. ...and 54 more sections

Key Result

Theorem 1.3.1

For any $x \in \mathfrak{n}^{-1}A \setminus A$, we have In particular, the prime factorization of $\mathcal{G}^{\textnormal{geo}}_x$ only involves primes above $v$.

Theorems & Definitions (83)

  • Theorem 1.3.1: Reflection formula, Theorem \ref{['thm-gauss-sum-reflection']}
  • Theorem 1.3.2: Analog of Stickelberger's theorem, Theorem \ref{['thm-stickelberger']}
  • Theorem 1.3.3: Analog of Hasse-Davenport product relation, Theorem \ref{['thm-hd-geo-product']}
  • Proposition 1.3.4: $\infty$-adic absolute values, Proposition \ref{['prop-absolute-values']}
  • Remark 1.3.5: Signs at infinity
  • Theorem 1.4.1: Analogs of Gross-Koblitz-Thakur formula, Theorem \ref{['thm-gkt-formula-for-geo']}
  • Remark 1.4.2
  • Remark 1.4.3
  • Proposition 2.2.1
  • proof
  • ...and 73 more