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A Class of Incomplete Character Sums

Lei Fu, Daqing Wan

TL;DR

This paper addresses the problem of estimating incomplete character sums over finite fields by developing an ℓ-adic cohomology framework based on tensor induction. The authors show that incomplete sums can be converted into complete sums for a suitably induced lisse sheaf $\otimes\text{-Ind}(\mathcal{L}_1)$ on the base, enabling the use of Frobenius eigenvalue bounds and the Grothendieck–Deligne formalism. They derive explicit tensor-induction formulas for Kummer-type and Artin–Schreier–type sheaves, providing concrete decompositions that reduce incomplete sums to complete ones. In the curve case, they obtain sharp bounds via Grothendieck–Ogg–Shafarevich and Swan conductor analyses, yielding practical estimates for sums of the form $\sum\chi(f)$ and $\sum\chi(f)\psi(\mathrm{Tr}(g))$, with the tensor-induction machinery tying together the one-variable estimates and higher-rank cohomological methods.

Abstract

Using $\ell$-adic cohomology of tensor inductions of lisse $\overline{\mathbb Q}_\ell$-sheaves, we study a class of incomplete character sums.

A Class of Incomplete Character Sums

TL;DR

This paper addresses the problem of estimating incomplete character sums over finite fields by developing an ℓ-adic cohomology framework based on tensor induction. The authors show that incomplete sums can be converted into complete sums for a suitably induced lisse sheaf on the base, enabling the use of Frobenius eigenvalue bounds and the Grothendieck–Deligne formalism. They derive explicit tensor-induction formulas for Kummer-type and Artin–Schreier–type sheaves, providing concrete decompositions that reduce incomplete sums to complete ones. In the curve case, they obtain sharp bounds via Grothendieck–Ogg–Shafarevich and Swan conductor analyses, yielding practical estimates for sums of the form and , with the tensor-induction machinery tying together the one-variable estimates and higher-rank cohomological methods.

Abstract

Using -adic cohomology of tensor inductions of lisse -sheaves, we study a class of incomplete character sums.

Paper Structure

This paper contains 4 sections, 9 theorems, 98 equations.

Table of Contents

  1. Tensor Induction
  2. Tensor Induction of Kummer type sheaves
  3. Tensor Induction of Artin-Schreier type sheaves
  4. Character Sums on Curves

Key Result

Proposition 1.1

Let $G$ be a pro-finite group, let $H$ be an open subgroup of $G$ of finite index $d$, and let $\rho:H\to\mathrm{GL}(V)$ be a representation of $H$. (i) Suppose that $H$ is normal in $G$. Let $\{Hg_1,\ldots, Hg_d\}$ be a family of representatives for $H\backslash G$, and let $\rho^{(i)}:H\to \mathrm (ii) Suppose that $H$ is normal in $G$ and $G/H$ is a cyclic group. For any $\sigma\in G$ such that

Theorems & Definitions (18)

  • Proposition 1.1
  • proof
  • Proposition 1.2
  • proof
  • Proposition 2.1
  • Proposition 3.1
  • Remark 3.2
  • Proposition 4.1
  • Theorem 4.2
  • proof
  • ...and 8 more