The modulus of a shifted Gauss sum

Proved

Weil.norm_gaussSum_mulShift

by tabbott · Sep 2, 2026 · Mathlib c5ea003 (Lean v4.30.0)

exponential-sumsfinite-fieldsgauss-sumsnumber-theoryweil-bound

Let $F$ be a finite field, $\chi$ a non-trivial multiplicative character of $F$ valued in $\mathbb{C}$ , $\psi$ a non-trivial additive character of $F$ , and $a \in F$ with $a \ne 0$ . Then

\lVert g(\chi, \psi_a)\rVert = \sqrt{\#F},

where $\psi_a(x) = \psi(ax)$ . Shifting the additive character multiplies the Gauss sum by a unimodular factor $\chi(a)^{-1}$ , so the modulus is unchanged. (Over a field, a non-trivial additive character is automatically primitive.)

Preamble

import Mathlib.NumberTheory.GaussSum
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum
import Mathlib.NumberTheory.MulChar.Lemmas
import Mathlib.GroupTheory.SpecificGroups.Cyclic
import Mathlib.GroupTheory.Index
import Mathlib.Analysis.SpecialFunctions.Complex.CircleAddChar
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.Complex.Basic
import Mathlib.Algebra.Field.GeomSum
import Mathlib.Algebra.Order.BigOperators.Group.Finset

set_option autoImplicit false
set_option linter.unusedSectionVars false
set_option linter.unusedVariables false

universe u_1 u_2

open AddChar MulChar Finset

Formal statement

namespace Weil
theorem norm_gaussSum_mulShift : ∀ {F : Type u_1} [inst : Field F] [inst_1 : Fintype F] [DecidableEq F] (χ : MulChar F ℂ), χ ≠ 1 → ∀ (ψ : AddChar F ℂ), ψ ≠ 1 → ∀ {a : F}, a ≠ 0 → ‖gaussSum χ (AddChar.mulShift ψ a)‖ = √(↑(Fintype.card F) : ℝ) := by sorry
end Weil

Source

Ireland & Rosen, A Classical Introduction to Modern Number Theory, 2nd ed., Springer GTM 84, Ch. 8 (Gauss and Jacobi Sums)

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