F p 2 -maximal curves with many automorphisms are Galois-covered by - PowerPoint PPT Presentation

F p 2 -maximal curves with many automorphisms are Galois-covered by the Hermitian curve Daniele Bartoli Universit` a degli Studi di Perugia (Italy) (Joint work with Maria Montanucci and Fernando Torres) Finite Geometries - Fifth Irsee Conference 10-16 September 2017

Outline 1 Maximal curves 2 F p 2 -maximal curves with many automorphisms: main result 3 The Fricke-MacBeath curve over finite fields and the F 11 2 -maximal Wiman’s sextic 4 Related questions

Notation and terminology • X ⊆ P r (¯ F q ) projective, geometrically irreducible, non-singular algebraic curve defined over F q • g genus of X If r = 2 then g = ( d − 1)( d − 2) where d = deg ( X ) 2 • X ( F q ) = X ∩ P r ( F q )

Maximal Curves X defined over F q Hasse-Weil Bound |X ( F q ) | ≤ q + 1 + 2 g √ q Definition X is F q -maximal if |X ( F q ) | = q + 1 + 2 g √ q Example Hermitian curve: H q : X q + X = Y q +1 , q = p h |H q ( F q 2 ) | = q 3 + 1 Aut ( H q ) ∼ g = q ( q − 1) / 2 , = PGU (3 , q ) ,

Rational maps and pull-back X ⊆ P r ( K ) and Y ⊆ P s ( K ) • Function field of X � F + I � � K ( X ) = � G / ∈ I = I ( X ) � G + I • Rational map φ : X → Y : is a map given by rational functions φ = ( α 0 : ... : α s ) , for almost all P ∈ X X φ ∗ ( β ) = β ◦ φ φ Y K β

Coverings and Galois-coverings Y is covered by X if there exists a non-constant rational map φ : X → Y K ( X ) : φ ∗ ( K ( Y )) is a finite field extension K ( X ) : φ ∗ ( K ( Y )) Y is a Galois-covered by X ⇐ ⇒ Galois extension

Basic properties of coverings X F q -maximal φ : X → Y non-constant = ⇒ Y is F q -maximal rational map defined over F q (Serre, Kleiman)

Basic properties of coverings X F q -maximal φ : X → Y non-constant = ⇒ Y is F q -maximal rational map defined over F q (Serre, Kleiman) • C non-singular algebraic curve • G finite automorphism group acting on C • X quotient curve of C by G

Basic properties of coverings X F q -maximal φ : X → Y non-constant = ⇒ Y is F q -maximal rational map defined over F q (Serre, Kleiman) • C non-singular algebraic curve • G finite automorphism group acting on C • X quotient curve of C by G Riemann-Hurwitz Formula 2 g ( C ) − 2 = | G | (2 g ( X ) − 2) + D

Classification of F q 2 -maximal curves • g ( X ) ≤ g 1 = q ( q − 1) ( Ihara , 1981) 2

Classification of F q 2 -maximal curves • g ( X ) ≤ g 1 = q ( q − 1) ( Ihara , 1981) 2 ⇒ X ∼ • g = g 1 = = H q ( R ¨ uck − Stichtenoth , 1994)

Classification of F q 2 -maximal curves • g ( X ) ≤ g 1 = q ( q − 1) ( Ihara , 1981) 2 ⇒ X ∼ • g = g 1 = = H q ( R ¨ uck − Stichtenoth , 1994) • 2006: Curve F q 2 -maximal curve not Galois-covered by H q ( Garcia , Stichtenoth ) → F 27 2 -maximal curve

Classification of F q 2 -maximal curves • g ( X ) ≤ g 1 = q ( q − 1) ( Ihara , 1981) 2 ⇒ X ∼ • g = g 1 = = H q ( R ¨ uck − Stichtenoth , 1994) • 2006: Curve F q 2 -maximal curve not Galois-covered by H q ( Garcia , Stichtenoth ) → F 27 2 -maximal curve • 2009: Family fo F q 2 -maximal curves not covered by H q ( Giulietti , Korchm ´ aros ) → F q 6 -maximal curve

Classification of F q 2 -maximal curves • g ( X ) ≤ g 1 = q ( q − 1) ( Ihara , 1981) 2 ⇒ X ∼ • g = g 1 = = H q ( R ¨ uck − Stichtenoth , 1994) • 2006: Curve F q 2 -maximal curve not Galois-covered by H q ( Garcia , Stichtenoth ) → F 27 2 -maximal curve • 2009: Family fo F q 2 -maximal curves not covered by H q ( Giulietti , Korchm ´ aros ) → F q 6 -maximal curve Question Is there an F p 2 -maximal curve not covered by the Hermitian curve?

Classification of F q 2 -maximal curves • g ( X ) ≤ g 1 = q ( q − 1) ( Ihara , 1981) 2 ⇒ X ∼ • g = g 1 = = H q ( R ¨ uck − Stichtenoth , 1994) • 2006: Curve F q 2 -maximal curve not Galois-covered by H q ( Garcia , Stichtenoth ) → F 27 2 -maximal curve • 2009: Family fo F q 2 -maximal curves not covered by H q ( Giulietti , Korchm ´ aros ) → F q 6 -maximal curve Question Is there an F p 2 -maximal curve not covered by the Hermitian curve? Is there an F p 2 -maximal curve not Galois-covered by the Hermitian curve?

Situation up to p = 5 • p = 2 , 3: trivial (Every F p 2 -maximal curve is Galois-covered by H p )

Situation up to p = 5 • p = 2 , 3: trivial (Every F p 2 -maximal curve is Galois-covered by H p ) • p = 5: Every F 25 -maximal curve is Galois-covered by H 5

Situation up to p = 5 • p = 2 , 3: trivial (Every F p 2 -maximal curve is Galois-covered by H p ) • p = 5: Every F 25 -maximal curve is Galois-covered by H 5 M (5) := { g ( X ) | X is F 25 -maximal } = { 0 , 1 , 2 , 3 , 4 , 10 } = H 5 : y 6 = x 5 + x ⇒ X ∼ 1 g ( X ) = 10 ⇐ = Y 4 : y 3 = x 5 + x ⇒ X ∼ 2 g ( X ) = 4 ⇐ = Y 3 : y 6 = x 5 + 2 x 4 + 3 x 3 + 4 x 2 + 3 xy 3 ⇒ X ∼ 3 g ( X ) = 3 ⇐ = Y 2 : y 2 = x 5 + x ⇒ X ∼ 4 g ( X ) = 2 ⇐ = Y 1 : x 3 + y 3 + 1 = 0 ⇒ X ∼ 5 g ( X ) = 1 ⇐

Main Result: a partial answer Almost all the known examples of maximal curves not Galois-covered by H q have large automorphism group ⇓ We investigated curves with | Aut ( X ) | > 84( g ( X ) − 1)

Main Result: a partial answer Almost all the known examples of maximal curves not Galois-covered by H q have large automorphism group ⇓ We investigated curves with | Aut ( X ) | > 84( g ( X ) − 1) Theorem (B.-Montanucci-Torres, 2017) X F p 2 -maximal curve, p ≥ 7 , g ( X ) ≥ 2 , | Aut ( X ) | > 84( g ( X ) − 1) = ⇒ X Galois-covered by H p

Main Result: a partial answer Almost all the known examples of maximal curves not Galois-covered by H q have large automorphism group ⇓ We investigated curves with | Aut ( X ) | > 84( g ( X ) − 1) Theorem (B.-Montanucci-Torres, 2017) X F p 2 -maximal curve, p ≥ 7 , g ( X ) ≥ 2 , | Aut ( X ) | > 84( g ( X ) − 1) = ⇒ X Galois-covered by H p • (B. Gunby, A. Smith, A. Yuan, 2015): X defined over F p 2 , p ≥ 7, g ( X ) ≥ 2, | Aut ( X ) | ≥ max { 84( g − 1) , g 2 } = X m : y m = x p − x ⇒ X ∼ = ( X m is not F p 2 -maximal for each m . . . )

Main Result: a partial answer Theorem (B.-Montanucci-Torres, 2017) X F p 2 -maximal curve, p ≥ 7 , g ( X ) ≥ 2 , | Aut ( X ) | > 84( g ( X ) − 1) = ⇒ X Galois-covered by H p

Main Result: a partial answer Theorem (B.-Montanucci-Torres, 2017) X F p 2 -maximal curve, p ≥ 7 , g ( X ) ≥ 2 , | Aut ( X ) | > 84( g ( X ) − 1) = ⇒ X Galois-covered by H p • An F p 2 -maximal “GK-curve” cannot exist

Main Result: a partial answer Theorem (B.-Montanucci-Torres, 2017) X F p 2 -maximal curve, p ≥ 7 , g ( X ) ≥ 2 , | Aut ( X ) | > 84( g ( X ) − 1) = ⇒ X Galois-covered by H p • An F p 2 -maximal “GK-curve” cannot exist • Find F p 2 -maximal curves not Galois-covered by the Hermitian curve • Can we extend it to | Aut ( X ) | ≤ 84( g − 1)?

Sketch of the proof Lemma X F p 2 -maximal curve X �∼ p 2 ∤ | G | = F p 2 H p = ⇒ G ≤ Aut ( X ), p | | G |

Sketch of the proof Lemma X F p 2 -maximal curve X �∼ p 2 ∤ | G | = F p 2 H p = ⇒ G ≤ Aut ( X ), p | | G | Theorem (Garcia-Tafazolian, 2008) • q = p h , X F q 2 -maximal curve • ∃ H ≤ Aut ( X ) abelian, | H | = q , X / H rational = F q 2 H m : x q + x = y m X ∼ ∃ m | q + 1 :

Sketch of the proof Lemma X F p 2 -maximal curve X �∼ p 2 ∤ | G | = F p 2 H p = ⇒ G ≤ Aut ( X ), p | | G | Theorem (Garcia-Tafazolian, 2008) • q = p h , X F q 2 -maximal curve • ∃ H ≤ Aut ( X ) abelian, | H | = q , X / H rational = F q 2 H m : x q + x = y m X ∼ ∃ m | q + 1 : H m is covered by H q : x q + x = y q +1 : G = { ϕ λ : ( x , y ) �→ ( x , λ y ) | λ ( q +1) / m = 1 } H m ∼ = H p / G ,

How many automorphisms? • g ( C ) ≥ 2 = ⇒ | Aut ( C ) | < ∞ (Schmid, Iwasawa-Tamagawa, Roquette, Rosentlich, Garcia)

How many automorphisms? • g ( C ) ≥ 2 = ⇒ | Aut ( C ) | < ∞ (Schmid, Iwasawa-Tamagawa, Roquette, Rosentlich, Garcia) • p = 0, g ( C ) ≥ 2 = ⇒ | Aut ( C ) | ≤ 84( g − 1) (Hurwitz)

How many automorphisms? • g ( C ) ≥ 2 = ⇒ | Aut ( C ) | < ∞ (Schmid, Iwasawa-Tamagawa, Roquette, Rosentlich, Garcia) • p = 0, g ( C ) ≥ 2 = ⇒ | Aut ( C ) | ≤ 84( g − 1) (Hurwitz) Example: Klein quartic: K : X 3 + Y + XY 3 = 0, g = 3, Aut ( K ) = PSL (2 , 7), | Aut ( K ) | = 168 = 84(3 − 1)

How many automorphisms? • g ( C ) ≥ 2 = ⇒ | Aut ( C ) | < ∞ (Schmid, Iwasawa-Tamagawa, Roquette, Rosentlich, Garcia) • p = 0, g ( C ) ≥ 2 = ⇒ | Aut ( C ) | ≤ 84( g − 1) (Hurwitz) Example: Klein quartic: K : X 3 + Y + XY 3 = 0, g = 3, Aut ( K ) = PSL (2 , 7), | Aut ( K ) | = 168 = 84(3 − 1) • gcd ( p , | Aut ( C ) | ) = 1 = ⇒ | Aut ( C ) | ≤ 84( g − 1)

What if | Aut ( X ) | ≤ 84( g − 1)? • p > 0: No classifications of Hurwitz groups | G | = 84( g − 1) • Partial classification if X is classical (Schoeneberg’s Lemma)