Models Classification Enumeration Zeta function Serre’s obstruction Genus 3 curves: a world to explore Enric Nart Universitat Aut` onoma de Barcelona XVIII Latin American Algebra Colloquium August 2009
Models Classification Enumeration Zeta function Serre’s obstruction Aim It is possible to write endlessly on elliptic curves (This is not a threat) Serge Lang
Models Classification Enumeration Zeta function Serre’s obstruction Aim It is possible to write endlessly on elliptic curves (This is not a threat) Serge Lang k = F q finite field of characteristic p
Models Classification Enumeration Zeta function Serre’s obstruction Models of hyperelliptic genus 3 curves y 2 = f ( x ) Weierstrass models (p > 2) f ( x ) ∈ k [ x ] separable polynomial of degree 7 or 8
Models Classification Enumeration Zeta function Serre’s obstruction Models of hyperelliptic genus 3 curves y 2 = f ( x ) Weierstrass models (p > 2) f ( x ) ∈ k [ x ] separable polynomial of degree 7 or 8 y 2 + y = u ( x ) Artin-Schreier models (p = 2) u ( x ) ∈ k ( x ) has divisor of poles on P 1 : [ x 1 ] + [ x 2 ] + [ x 3 ] + [ x 4 ] [ x 1 ] + [ x 2 ] + 3[ x 3 ] div ∞ ( u ) = 3[ x 1 ] + 3[ x 2 ] [ x 1 ] + 5[ x 2 ] 7[ x 1 ] with x i ∈ P 1 ( k ) The moduli space of hyperelliptic curves has dimension 5
Models Classification Enumeration Zeta function Serre’s obstruction Models of non-hyperelliptic genus 3 curves If C is a non-hyperelliptic curve of genus 3 then the canonical → P 2 is an embedding and the image is a morphism C − non-singular plane quartic: F ( x , y , z ) = 0 Examples x 4 + y 4 + z 4 = 0 x 3 y + y 3 z + z 3 x = 0 (Fermat); (Klein)
Models Classification Enumeration Zeta function Serre’s obstruction Models of non-hyperelliptic genus 3 curves If C is a non-hyperelliptic curve of genus 3 then the canonical → P 2 is an embedding and the image is a morphism C − non-singular plane quartic: F ( x , y , z ) = 0 Examples x 4 + y 4 + z 4 = 0 x 3 y + y 3 z + z 3 x = 0 (Fermat); (Klein) The extrinsic geometry of the embedding in the plane is actually intrinsic. This gives them a lot of structure; for instance, they have (for p > 3) 28 bitangents and 24 flexes The moduli space of non-hyperelliptic curves has dimension 6
Models Classification Enumeration Zeta function Serre’s obstruction Classification of genus 3 curves PROBLEM Classify genus 3 curves up to k -isomorphism
Models Classification Enumeration Zeta function Serre’s obstruction Classification of genus 3 curves PROBLEM Classify genus 3 curves up to k -isomorphism Good models � Shioda Invariants: Dixmier + Ohno Field of moduli vs field of definition Twists. Structure of the automorphism groups Enumeration Stratification of the moduli space: by the automorphism group, by the p -rank, by the number of hyperflexes, ... � Guti´ errez-Shaska dihedral invariants Curves + involutions: ???
Models Classification Enumeration Zeta function Serre’s obstruction p -rank of a curve r ( C ) := dim F p Jac ( C )[ p ]. It coincides with the length of the side of slope zero of the q -Newton polygon of the characteristic polynomial f Jac( C ) ( x ) = x 6 + ax 5 + bx 4 + cx 3 + qbx 2 + q 2 ax + q 3 of the Frobenius endomorphism of Jac( C ) 3 3 3 ordinary � � � � � � � � � � � � ✟ ✟ ✟✟✟✟ ✟✟✟✟ � � � � ✟ ✟ ✟✟ ✟✟ � � 0 3 6 0 2 4 6 0 1 5 6 r ( C ) = 3 r ( C ) = 2 r ( C ) = 1 ✟ ✟ ✑ 3 3 ✑ ✟✟✟✟✟✟ ✟✟✟✟✟✟ ✑✑✑ ✏✑✑✑ r ( C ) = 0 ✏ ✏✏✏ ✏✏✏ 0 6 0 3 6 supersingular type 1 / 3
Models Classification Enumeration Zeta function Serre’s obstruction Classification in characteristic 2 � | W | − 1 if C is hyperelliptic p = 2 = ⇒ r ( C ) = ⌊| Bit | / 2 ⌋ if C is non-hyperelliptic
Models Classification Enumeration Zeta function Serre’s obstruction Classification in characteristic 2 � | W | − 1 if C is hyperelliptic p = 2 = ⇒ r ( C ) = ⌊| Bit | / 2 ⌋ if C is non-hyperelliptic This makes it possible to classify genus 3 curves with prescribed 2-rank (N-Sadornil 2004), (N-Ritzenthaler 2006). For instance, Hyperelliptic curves with r ( C ) = 0 (NS04) All hyperelliptic C with r ( C ) = 0 are k -isomorphic to y 2 + y = ax 7 + bx 6 + cx 5 + dx 4 + e , a � = 0, e ∈ k / ker(tr) They are all of type 1 / 3 ⇒ ( a , b , c , d , e ) k ∗ ⋊ k �→ ( a ′ , b ′ , c ′ , d ′ , e ′ ) C abcde ≃ C a ′ b ′ c ′ d ′ e ′ ⇐ Aut k ( C ) = C 2 , except for: if q is a cube, then Aut k ( C ) = C 2 × C 7 for the 14 curves y 2 + y = ax 7 + e , a ∈ k ∗ / ( k ∗ ) 7 , e ∈ k / ker(tr)
Models Classification Enumeration Zeta function Serre’s obstruction Classification in characteristic 2 All non-hyperelliptic C with r ( C ) = 0 have exactly one bitangent Non-hyperelliptic curves with r ( C ) = 0 and type 1 / 3 (NR06) ( ax 2 + by 2 + cz 2 + eyz ) 2 = x ( y 3 + x 2 z ), with a , b ∈ k , c ∈ k ∗ / ( k ∗ ) 9 , e ∈ k ∗ µ 9 ( k ) �→ ( a ′ , b ′ , c ′ , e ′ ) C abce ≃ C a ′ b ′ c ′ e ′ ⇐ ⇒ ( a , b , c , e ) Aut k ( C ) = { 1 }
Models Classification Enumeration Zeta function Serre’s obstruction Classification in characteristic 2 All non-hyperelliptic C with r ( C ) = 0 have exactly one bitangent Non-hyperelliptic curves with r ( C ) = 0 and type 1 / 3 (NR06) ( ax 2 + by 2 + cz 2 + eyz ) 2 = x ( y 3 + x 2 z ), with a , b ∈ k , c ∈ k ∗ / ( k ∗ ) 9 , e ∈ k ∗ µ 9 ( k ) �→ ( a ′ , b ′ , c ′ , e ′ ) C abce ≃ C a ′ b ′ c ′ e ′ ⇐ ⇒ ( a , b , c , e ) Aut k ( C ) = { 1 } Supersingular non-hyperelliptic curves (NR06) ( ax 2 + cz 2 + dxy + fxz ) 2 = x ( y 3 + x 2 z ), with a , d , f ∈ k , c ∈ k ∗ / ( k ∗ ) 9 µ 9 ( k ) ⋊ k ( a ′ , c ′ , d ′ , f ′ ) C acdf ≃ C a ′ c ′ d ′ f ′ ⇐ ⇒ ( a , c , d , f ) �→ Aut k ( C ) ≤ C 9 ⋊ V 4
Models Classification Enumeration Zeta function Serre’s obstruction Number of rational points in the moduli space M h M nh M 3 3 3 q 5 − q 4 q 6 − q 5 + 1 q 6 − q 4 + 1 ordinary q 4 − 2 q 3 + q 2 q 5 − q 4 q 5 − 2 q 3 + q 2 2-rank two 2( q 3 − q 2 ) q 4 − q 3 q 4 + q 3 − 2 q 2 2-rank one q 3 − q 2 q 2 q 3 type 1 / 3 q 2 q 2 supersingular 0 q 6 + 1 q 6 + q 5 + 1 q 5 total
Models Classification Enumeration Zeta function Serre’s obstruction Number of curves r ( C ) hyperelliptic non-hyperelliptic 2 q 5 − 2 q 4 + 2 q 3 − 4 q 2 + 2 q q 6 − q 5 + q 4 − 3 q 3 + 5 q 2 − 6 q + 7 3 2 q 4 − 4 q 3 + 3 q 2 − q q 5 − q 4 + q 3 − 2 q 2 + 2 q − 1 2 4 q 3 − 2 q 2 − 2 q 4 − 2 q 2 + q 1 2 q 2 + [12] q ≡ 1 (mod 7) q 3 − q 2 0 ( 1 3 ) 2 q 2 − q + [4 q − 2] q ≡ 1 (mod 3) 0 (ss) 0 +[6] q ≡ 1 (mod 9) 2 q 5 + 2 q 3 − q 2 + q − 2 q 6 + q 4 − q 3 + 2 q 2 + 4 − total +[12] q ≡ 1 (mod 7) [4 q − 2] q ≡− 1 (mod 3) + [6] q ≡ 1 (mod 9)
Models Classification Enumeration Zeta function Serre’s obstruction Number of hyperelliptic curves if p > 2 2 q 5 + 2 q 3 − 2 − 2[ q 2 − q ] 4 | q +1 + 2[ q − 1] p > 3 + [4] 8 | q − 1 + +[12] 7 | q − 1 + [2] p =7 + [2] q ≡ 1 , 5 (mod 12) Among them, the number of self-dual curves is � 0 , if q ≡ 1 (mod 4) 2 q 2 − 2 q + [2] p > 3 + [4] 8 | q +1 , if q ≡ 3 (mod 4)
Models Classification Enumeration Zeta function Serre’s obstruction Zeta function If N n := # C ( F q n ), there exist a , b , c ∈ Z such that N n � n x n = Z ( C / F q , x ) = exp n ≥ 1 1 + ax + bx 2 + cx 3 + qbx 4 + q 2 ax 5 + q 3 x 6 (1 − x )(1 − qx )
Models Classification Enumeration Zeta function Serre’s obstruction Zeta function If N n := # C ( F q n ), there exist a , b , c ∈ Z such that N n � n x n = Z ( C / F q , x ) = exp n ≥ 1 1 + ax + bx 2 + cx 3 + qbx 4 + q 2 ax 5 + q 3 x 6 (1 − x )(1 − qx ) PROBLEM What polynomials occur as the numerator of the zeta function of a projective smooth genus 3 curve over F q ? For what values of ( N 1 , N 2 , N 3 ) ∈ Z 3 there exists a projective smooth genus 3 curve C over F q such that # C ( F q ) = N 1 , # C ( F q 2 ) = N 2 , # C ( F q 3 ) = N 3 ?
Models Classification Enumeration Zeta function Serre’s obstruction Jacobians enter into the game The characteristic polynomial of the Frobenius endomorphism of Jac( C ) is f Jac( C ) ( x ) = x 6 + ax 5 + bx 4 + cx 3 + qbx 2 + q 2 ax + q 3 We know all Weil polynomials that occur as f A ( x ) for some abelian threefold A / k . Thus, we need only to identify inside this family, the subfamily of all Weil polynomials of Jacobians: � � f Jac( C ) ( x ) | C / k genus 3 curve ⊆ { f A ( x ) | A / k abelian 3-fold }
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