Lightning Talks June 2, 2020 Session I 2:15 - 2:20 Caleb Springer, - PowerPoint PPT Presentation

Lightning Talks June 2, 2020 Session I 2:15 - 2:20 Caleb Springer, Penn State 2:20 - 2:25 Jacob Mayle, University of Illinois at Chicago 2:25 - 2:30 Pip Goodman, University of Bristol 2:30 - 2:35 Jeroen Hanselman, Universität Ulm 2:35 -2:40 Oana Adascalitei, Boston University 2:40 - 2:45 Vishal Arul, MIT 2:45 - 2:50 Emre Sertöz, Leibniz University Hannover

The Structure of the Group of Rational Points of an Abelian Variety over a Finite Field Caleb Springer The Pennsylvania State University June 2, 2020

B ACKGROUND The Goal Given an abelian variety A defined over F q , recognize the group of rational points A ( F q ) as a module over the endomorphism ring End F q ( A ) . ◮ Lenstra solved this problem completely for elliptic curves .

B ACKGROUND The Goal Given an abelian variety A defined over F q , recognize the group of rational points A ( F q ) as a module over the endomorphism ring End F q ( A ) . ◮ Lenstra solved this problem completely for elliptic curves . ◮ In the same paper, Lenstra showed that his result does not immediately generalize to all principally polarized ordinary abelian varieties. What we still want: A generalization of Lenstra’s theorem that is true, assuming some conditions that are automatic for elliptic curves.

M AIN R ESULT Fix g ≥ 1. Let A / F q be simple of dimension g with Frobenius π . Write R = End F q ( A ) , and let Z be the center of R . (a) If [ Q ( π ) : Q ] = 2 g and R is a Gorenstein ring , then = R / R ( π n − 1 ) . A ( F q n ) ∼

M AIN R ESULT Fix g ≥ 1. Let A / F q be simple of dimension g with Frobenius π . Write R = End F q ( A ) , and let Z be the center of R . (a) If [ Q ( π ) : Q ] = 2 g and R is a Gorenstein ring , then = R / R ( π n − 1 ) . A ( F q n ) ∼ (b) If ( π n − 1 ) Z is the product of invertible prime ideals in Z , then there is an isomorphism of Z -modules = ( Z / Z ( π n − 1 )) d . A ( F q n ) ∼ where d = 2 g / [ K : Q ] . The R -module structure comes from an isomorphism of rings R / R ( π n − 1 ) ∼ = Mat d ( Z / Z ( π n − 1 )) .

Rigidity in Elliptic Curve Local-Global Principles Jacob Mayle June 2, 2020 University of Illinois at Chicago Workshop on Arithmetic Geometry, Number Theory, and Computation

Elliptic curve local-global principles Let K be a number field, E / K be an elliptic curve, and ℓ be an odd prime.

Elliptic curve local-global principles Let K be a number field, E / K be an elliptic curve, and ℓ be an odd prime. Define T ℓ := { primes p ⊆ O K : E p has nontrivial F p -rational ℓ -torsion } ,

Elliptic curve local-global principles Let K be a number field, E / K be an elliptic curve, and ℓ be an odd prime. Define T ℓ := { primes p ⊆ O K : E p has nontrivial F p -rational ℓ -torsion } , I ℓ := { primes p ⊆ O K : E p admits an F p -rational ℓ -isogeny } .

Elliptic curve local-global principles Let K be a number field, E / K be an elliptic curve, and ℓ be an odd prime. Define T ℓ := { primes p ⊆ O K : E p has nontrivial F p -rational ℓ -torsion } , I ℓ := { primes p ⊆ O K : E p admits an F p -rational ℓ -isogeny } . Let δ ( T ℓ ) and δ ( I ℓ ) be the densities of these sets among the prime ideals of O K .

Elliptic curve local-global principles Let K be a number field, E / K be an elliptic curve, and ℓ be an odd prime. Define T ℓ := { primes p ⊆ O K : E p has nontrivial F p -rational ℓ -torsion } , I ℓ := { primes p ⊆ O K : E p admits an F p -rational ℓ -isogeny } . Let δ ( T ℓ ) and δ ( I ℓ ) be the densities of these sets among the prime ideals of O K . Theorem (Katz 1981). If δ ( T ℓ ) = 1, then E is K -isogenous to an elliptic curve with nontrivial K -rational ℓ -torsion.

Elliptic curve local-global principles Let K be a number field, E / K be an elliptic curve, and ℓ be an odd prime. Define T ℓ := { primes p ⊆ O K : E p has nontrivial F p -rational ℓ -torsion } , I ℓ := { primes p ⊆ O K : E p admits an F p -rational ℓ -isogeny } . Let δ ( T ℓ ) and δ ( I ℓ ) be the densities of these sets among the prime ideals of O K . Theorem (Katz 1981). If δ ( T ℓ ) = 1, then E is K -isogenous to an elliptic curve with nontrivial K -rational ℓ -torsion. �� − 1 � Theorem (Sutherland 2012). Suppose ℓ �∈ K . If δ ( I ℓ ) = 1, then E ℓ admits an ℓ -isogeny over a quadratic extension of K .

Elliptic curve local-global principles Let K be a number field, E / K be an elliptic curve, and ℓ be an odd prime. Define T ℓ := { primes p ⊆ O K : E p has nontrivial F p -rational ℓ -torsion } , I ℓ := { primes p ⊆ O K : E p admits an F p -rational ℓ -isogeny } . Let δ ( T ℓ ) and δ ( I ℓ ) be the densities of these sets among the prime ideals of O K . Theorem (Katz 1981). If δ ( T ℓ ) = 1, then E is K -isogenous to an elliptic curve with nontrivial K -rational ℓ -torsion. �� − 1 � Theorem (Sutherland 2012). Suppose ℓ �∈ K . If δ ( I ℓ ) = 1, then E ℓ admits an ℓ -isogeny over a quadratic extension of K . Qestion. If δ ( T ℓ ) � = 1, then how large may δ ( T ℓ ) be? Similarly for δ ( I ℓ ) .

Rigidity of the locally everywhere conditions Let G ( ℓ ) ⊆ GL 2 ( F ℓ ) denote the image of the mod ℓ Galois representation of E .

Rigidity of the locally everywhere conditions Let G ( ℓ ) ⊆ GL 2 ( F ℓ ) denote the image of the mod ℓ Galois representation of E . It follows from properties of G ( ℓ ) and the Chebotarev density theorem that

Rigidity of the locally everywhere conditions Let G ( ℓ ) ⊆ GL 2 ( F ℓ ) denote the image of the mod ℓ Galois representation of E . It follows from properties of G ( ℓ ) and the Chebotarev density theorem that 1. δ ( T ℓ ) is the proportion of matrices in G ( ℓ ) with 1 as an eigenvalue,

Rigidity of the locally everywhere conditions Let G ( ℓ ) ⊆ GL 2 ( F ℓ ) denote the image of the mod ℓ Galois representation of E . It follows from properties of G ( ℓ ) and the Chebotarev density theorem that 1. δ ( T ℓ ) is the proportion of matrices in G ( ℓ ) with 1 as an eigenvalue, 2. δ ( I ℓ ) is the proportion of matrices in G ( ℓ ) with some eigenvalue in F ℓ .

Rigidity of the locally everywhere conditions Let G ( ℓ ) ⊆ GL 2 ( F ℓ ) denote the image of the mod ℓ Galois representation of E . It follows from properties of G ( ℓ ) and the Chebotarev density theorem that 1. δ ( T ℓ ) is the proportion of matrices in G ( ℓ ) with 1 as an eigenvalue, 2. δ ( I ℓ ) is the proportion of matrices in G ( ℓ ) with some eigenvalue in F ℓ . Considering subgroups of GL 2 ( ℓ ) case-by-case along Dickson’s theorem, we prove:

Rigidity of the locally everywhere conditions Let G ( ℓ ) ⊆ GL 2 ( F ℓ ) denote the image of the mod ℓ Galois representation of E . It follows from properties of G ( ℓ ) and the Chebotarev density theorem that 1. δ ( T ℓ ) is the proportion of matrices in G ( ℓ ) with 1 as an eigenvalue, 2. δ ( I ℓ ) is the proportion of matrices in G ( ℓ ) with some eigenvalue in F ℓ . Considering subgroups of GL 2 ( ℓ ) case-by-case along Dickson’s theorem, we prove: Theorem (M. 2020). δ ( T ℓ ) , δ ( I ℓ ) �∈ ( 3 4 , 1 ) .

Rigidity of the locally everywhere conditions Let G ( ℓ ) ⊆ GL 2 ( F ℓ ) denote the image of the mod ℓ Galois representation of E . It follows from properties of G ( ℓ ) and the Chebotarev density theorem that 1. δ ( T ℓ ) is the proportion of matrices in G ( ℓ ) with 1 as an eigenvalue, 2. δ ( I ℓ ) is the proportion of matrices in G ( ℓ ) with some eigenvalue in F ℓ . Considering subgroups of GL 2 ( ℓ ) case-by-case along Dickson’s theorem, we prove: Theorem (M. 2020). δ ( T ℓ ) , δ ( I ℓ ) �∈ ( 3 4 , 1 ) . This rigidity differentiates the local-global principles of Katz & Sutherland with, for instance, the Hasse-Minkowski theorem where failures are quite limited.

References S. Anni, A local–global principle for isogenies of prime degree over number fields , J. Lond. Math. Soc. (2) 89 (2014), no. 3, 745–761. N.M. Katz, Galois properties of torsion points on abelian varieties , Invent. Math. 62 (1981), no. 3, 481–502. J. Mayle, Rigidity in elliptic curve local-global principles , arXiv:2005.05881 (2020). A.V. Sutherland, A local-global principle for rational isogenies of prime degree , J. Théor. Nombres Bordeaux 24 (2012), no. 2, 475–485. I. Vogt, A local-global principle for isogenies of composite degree , arXiv:1801.05355 (2018). Thank you!

Superelliptic curves with large Galois images Pip Goodman University of Bristol 2nd June 2020 Pip Goodman University of Bristol Superelliptic curves with large Galois images 2nd June 2020 1 / 4

Notation r a prime f ∈ Q [ x ] a polynomial without repeated roots C superelliptic curve associated to the smooth affine model y r = f ( x ) J the jacobian of C Theorem (G.’20) Suppose 2 r | d can be written as the sum of two primes q 1 < q 2 and there exists a prime q 2 + 2 < q 3 < d . Then we may construct an explicit polynomial f ∈ Q [ x ] of degree d such that for all primes l outside of a finite explicit set the image of the representation ρ l : G Q → Aut( J [ l ]) is as large as possible. Pip Goodman University of Bristol Superelliptic curves with large Galois images 2nd June 2020 2 / 4