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- [x] depends on: #7013 Co-authored-by: Moritz Firsching <firsching@google.com> Co-authored-by: laughinggas <58670661+laughinggas@users.noreply.github.com>
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/- | ||
Copyright (c) 2023 Ashvni Narayanan. All rights reserved. | ||
Released under Apache 2.0 license as described in the file LICENSE. | ||
Authors: Ashvni Narayanan, Moritz Firsching | ||
-/ | ||
import Mathlib.Algebra.Periodic | ||
import Mathlib.NumberTheory.LegendreSymbol.MulCharacter | ||
import Mathlib.Data.ZMod.Algebra | ||
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/-! | ||
# Dirichlet Characters | ||
Let `R` be a commutative monoid with zero. A Dirichlet character `χ` of level `n` over `R` is a | ||
multiplicative character from `ZMod n` to `R` sending non-units to 0. We then obtain some properties | ||
of `toUnitHom χ`, the restriction of `χ` to a group homomorphism `(ZMod n)ˣ →* Rˣ`. | ||
Main definitions: | ||
- `DirichletCharacter`: The type representing a Dirichlet character. | ||
## TODO | ||
- `change_level`: Extend the Dirichlet character χ of level `n` to level `m`, where `n` divides `m`. | ||
- definition of conductor | ||
## Tags | ||
dirichlet character, multiplicative character | ||
-/ | ||
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/-- The type of Dirichlet characters of level `n`. -/ | ||
abbrev DirichletCharacter (R : Type) [CommMonoidWithZero R] (n : ℕ) := MulChar (ZMod n) R | ||
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open MulChar | ||
variable {R : Type} [CommMonoidWithZero R] {n : ℕ} (χ : DirichletCharacter R n) | ||
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namespace DirichletCharacter | ||
lemma toUnitHom_eq_char' {a : ZMod n} (ha : IsUnit a) : | ||
χ a = χ.toUnitHom ha.unit := by simp | ||
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lemma toUnitHom_eq_iff (ψ : DirichletCharacter R n) : | ||
toUnitHom χ = toUnitHom ψ ↔ χ = ψ := by simp | ||
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lemma eval_modulus_sub (x : ZMod n) : | ||
χ (n - x) = χ (-x) := by simp | ||
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lemma periodic {m : ℕ} (hm : n ∣ m) : Function.Periodic χ m := by | ||
intro a | ||
rw [← ZMod.nat_cast_zmod_eq_zero_iff_dvd] at hm | ||
simp only [hm, add_zero] | ||
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end DirichletCharacter |
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