Rigidity of automorphisms of a function field

Let F / k be a function field of genus g with exact constant field k. An automorphism σ ∈ Aut(F / k) that fixes 2g + 3 distinct rational places is the identity. This is the rigidity statement behind the finiteness of Aut(F / k) in genus at least two: an automorphism group acting on a finite set of at least 2g + 3 rational places embeds into the permutations of that set.

Main results

• TauCeti.eq_one_of_two_mul_genus_add_three_le_card: an automorphism of F / k fixing at least 2g + 3 rational places is the identity.
• TauCeti.eq_of_forall_smul_eq_of_two_mul_genus_add_three_le_card: two automorphisms agreeing on at least 2g + 3 rational places are equal.

References

• H. Stichtenoth, Algebraic Function Fields and Codes, 2nd ed., GTM 254, Springer, 2009, Proposition 1.6.6 and Exercise 3.17.
• G. D. Villa Salvador, Topics in the Theory of Algebraic Function Fields, Birkhäuser, 2006, Chapter 9.

theorem TauCeti.eq_one_of_two_mul_genus_add_three_le_card {k : Type u_1} {F : Type u_2} [Field k] [Field F] [Algebra k F] (hF : IsFunctionField k F) (hex : IsIntegrallyClosedIn k F) {σ : Gal(F/k)} {S : Finset (Place k F)} (hS : ∀ P ∈ S, P.degree = 1 ∧ σ • P = P) (hcard : 2 * genus k F + 3 ≤ S.card) : σ = 1

Rigidity of automorphisms of a function field (Stichtenoth, Exercise 3.17): over an exact constant field, an automorphism of F / k fixing at least 2g + 3 distinct rational places is the identity.

theorem TauCeti.eq_of_forall_smul_eq_of_two_mul_genus_add_three_le_card {k : Type u_1} {F : Type u_2} [Field k] [Field F] [Algebra k F] (hF : IsFunctionField k F) (hex : IsIntegrallyClosedIn k F) {σ τ : Gal(F/k)} {S : Finset (Place k F)} (hS : ∀ P ∈ S, P.degree = 1 ∧ σ • P = τ • P) (hcard : 2 * genus k F + 3 ≤ S.card) : σ = τ

Automorphisms are determined by their action on 2g + 3 rational places: over an exact constant field, two automorphisms of F / k that move each place of a set of at least 2g + 3 rational places to the same place are equal. Hence Aut(F / k) acts faithfully on every invariant set of at least 2g + 3 rational places.