Jacobi Sums #
This file defines the Jacobi sum of two multiplicative characters χ
and ψ
on a finite
commutative ring R
with values in another commutative ring R'
:
jacobiSum χ ψ = ∑ x : R, χ x * ψ (1 - x)
(see jacobiSum
) and provides some basic results and API lemmas on Jacobi sums.
References #
We essentially follow
but generalize where appropriate.
This is based on Lean code written as part of the bachelor's thesis of Alexander Spahl.
Jacobi sums: definition and first properties #
Jacobi sums over finite fields #
The Jacobi sum of two multiplicative characters on a nontrivial finite commutative ring F
can be written as a sum over F \ {0,1}
.
The Jacobi sum of twice the trivial multiplicative character on a finite field F
equals #F-2
.
If χ
and φ
are multiplicative characters on a finite field F
such that
χφ
is nontrivial, then g(χφ) * J(χ,φ) = g(χ) * g(φ)
.
If χ
and φ
are multiplicative characters on a finite field F
with values
in another field F'
and such that χφ
is nontrivial, then J(χ,φ) = g(χ) * g(φ) / g(χφ)
.
If χ
and φ
are multiplicative characters on a finite field F
with values in another
field F'
such that χ
, φ
and χφ
are all nontrivial and char F' ≠ char F
, then
J(χ,φ) * J(χ⁻¹,φ⁻¹) = #F
(in F'
).
If χ
and φ
are multiplicative characters on a finite field F
satisfying χ^n = φ^n = 1
and with values in an integral domain R
, and μ
is a primitive n
th root of unity in R
,
then the Jacobi sum J(χ,φ)
is in ℤ[μ] ⊆ R
.
If χ
and ψ
are multiplicative characters of order dividing n
on a finite field F
with values in an integral domain R
and μ
is a primitive n
th root of unity in R
,
then J(χ,ψ) = -1 + z*(μ - 1)^2
for some z ∈ ℤ[μ] ⊆ R
. (We assume that #F ≡ 1 mod n
.)
Note that we do not state this as a divisibility in R
, as this would give a weaker statement.
If χ
is a multiplicative character of order n ≥ 2
on a finite field F
,
then g(χ)^n = χ(-1) * #F * J(χ,χ) * J(χ,χ²) * ... * J(χ,χⁿ⁻²)
.