Counting isogenous principally-polarized abelian varieties over finite fields Everett W. Howe Center for Communications Research, La Jolla Arithmetic of Low-Dimensional Abelian Varieties ICERM, 3–7 June 2019 (Corrected and edited slides) email: however@alumni.caltech.edu Web site: ewhowe.com Twitter: @howe Everett W. Howe Counting abelian varieties over finite fields 1 of 29
Motivations Everett W. Howe Counting abelian varieties over finite fields 2 of 29
Formative mathematical experiences Undergraduate: Single-variable complex analysis Everett W. Howe Counting abelian varieties over finite fields 3 of 29
Formative mathematical experiences Undergraduate: Single-variable complex analysis Does a continuous f : C → C satisfy the Cauchy–Riemann relations? Everett W. Howe Counting abelian varieties over finite fields 3 of 29
Formative mathematical experiences Undergraduate: Single-variable complex analysis Does a continuous f : C → C satisfy the Cauchy–Riemann relations? Then it’s infinitely differentiable! Everett W. Howe Counting abelian varieties over finite fields 3 of 29
Formative mathematical experiences Undergraduate: Single-variable complex analysis Does a continuous f : C → C satisfy the Cauchy–Riemann relations? Then it’s infinitely differentiable! Got an open simply-connected proper subset of C ? Everett W. Howe Counting abelian varieties over finite fields 3 of 29
Formative mathematical experiences Undergraduate: Single-variable complex analysis Does a continuous f : C → C satisfy the Cauchy–Riemann relations? Then it’s infinitely differentiable! Got an open simply-connected proper subset of C ? It’s conformally equivalent to the interior of the unit disk! Everett W. Howe Counting abelian varieties over finite fields 3 of 29
Formative mathematical experiences Undergraduate: Single-variable complex analysis Does a continuous f : C → C satisfy the Cauchy–Riemann relations? Then it’s infinitely differentiable! Got an open simply-connected proper subset of C ? It’s conformally equivalent to the interior of the unit disk! Got an intractable integral on the real line to compute? Everett W. Howe Counting abelian varieties over finite fields 3 of 29
Formative mathematical experiences Undergraduate: Single-variable complex analysis Does a continuous f : C → C satisfy the Cauchy–Riemann relations? Then it’s infinitely differentiable! Got an open simply-connected proper subset of C ? It’s conformally equivalent to the interior of the unit disk! Got an intractable integral on the real line to compute? Use Cauchy’s theorem to find its value! Everett W. Howe Counting abelian varieties over finite fields 3 of 29
Formative mathematical experiences Undergraduate: Single-variable complex analysis Does a continuous f : C → C satisfy the Cauchy–Riemann relations? Then it’s infinitely differentiable! Got an open simply-connected proper subset of C ? It’s conformally equivalent to the interior of the unit disk! Got an intractable integral on the real line to compute? Use Cauchy’s theorem to find its value! Life is beautiful and every nice thing is true. Everett W. Howe Counting abelian varieties over finite fields 3 of 29
Formative mathematical experiences Undergraduate: Single-variable complex analysis Does a continuous f : C → C satisfy the Cauchy–Riemann relations? Then it’s infinitely differentiable! Got an open simply-connected proper subset of C ? It’s conformally equivalent to the interior of the unit disk! Got an intractable integral on the real line to compute? Use Cauchy’s theorem to find its value! Life is beautiful and every nice thing is true. Graduate school: Several complex variables Everett W. Howe Counting abelian varieties over finite fields 3 of 29
Formative mathematical experiences Undergraduate: Single-variable complex analysis Does a continuous f : C → C satisfy the Cauchy–Riemann relations? Then it’s infinitely differentiable! Got an open simply-connected proper subset of C ? It’s conformally equivalent to the interior of the unit disk! Got an intractable integral on the real line to compute? Use Cauchy’s theorem to find its value! Life is beautiful and every nice thing is true. Graduate school: Several complex variables Life is brutal and short. Give up now. Everett W. Howe Counting abelian varieties over finite fields 3 of 29
Formative mathematical experiences Undergraduate: Single-variable complex analysis Does a continuous f : C → C satisfy the Cauchy–Riemann relations? Then it’s infinitely differentiable! Got an open simply-connected proper subset of C ? It’s conformally equivalent to the interior of the unit disk! Got an intractable integral on the real line to compute? Use Cauchy’s theorem to find its value! Life is beautiful and every nice thing is true. Graduate school: Several complex variables Life is brutal and short. Give up now. That’s literally all I remember from my several complex variables class. Everett W. Howe Counting abelian varieties over finite fields 3 of 29
Formative mathematical experiences Undergraduate: Single-variable complex analysis Does a continuous f : C → C satisfy the Cauchy–Riemann relations? Then it’s infinitely differentiable! Got an open simply-connected proper subset of C ? It’s conformally equivalent to the interior of the unit disk! Got an intractable integral on the real line to compute? Use Cauchy’s theorem to find its value! Life is beautiful and every nice thing is true. Graduate school: Several complex variables Life is brutal and short. Give up now. That’s literally all I remember from my several complex variables class. Life lesson: One-dimensional objects are friendly and fun to work with. Everett W. Howe Counting abelian varieties over finite fields 3 of 29
One-dimensional objects and their friends Number rings Maximal orders; non-maximal orders Ideals; ideal class groups Modules over number rings aren’t necessarily one-dimensional, but they’re still pretty friendly Everett W. Howe Counting abelian varieties over finite fields 4 of 29
One-dimensional objects and their friends Number rings Maximal orders; non-maximal orders Ideals; ideal class groups Modules over number rings aren’t necessarily one-dimensional, but they’re still pretty friendly Curves, and things to study about them Over Q : Number of rational points; finding rational points; . . . Over finite fields: Curves with many points for their genus, or few; distribution of Frobenius eigenvalues; . . . Over any field: Automorphism groups; decomposition of Jacobians; . . . Everett W. Howe Counting abelian varieties over finite fields 4 of 29
One-dimensional objects and their friends Number rings Maximal orders; non-maximal orders Ideals; ideal class groups Modules over number rings aren’t necessarily one-dimensional, but they’re still pretty friendly Curves, and things to study about them Over Q : Number of rational points; finding rational points; . . . Over finite fields: Curves with many points for their genus, or few; distribution of Frobenius eigenvalues; . . . Over any field: Automorphism groups; decomposition of Jacobians; . . . . . . Hold on there bucko, Jacobians are higher-dimensional objects! Everett W. Howe Counting abelian varieties over finite fields 4 of 29
The dream Can we understand Jacobians, general abelian varieties, polarizations, and so forth, using one-dimensional objects ? Deligne (1969) For ordinary abelian varieties over finite fields: yes . Centeleghe and Stix (2015) For (not quite completely general) abelian varieties over finite prime fields: yes . This talk: I will sketch Deligne’s result and some follow-on work, and use it to address the question of determining the number of principally-polarized varieties in a simple ordinary isogeny class. Everett W. Howe Counting abelian varieties over finite fields 5 of 29
Deligne modules Everett W. Howe Counting abelian varieties over finite fields 6 of 29
Ordinary abelian varieties Suppose k is a finite field of characteristic p , A is a g -dimensional abelian variety over k , f is the characteristic polynomial of Frobenius (the Weil polynomial ) for A . ( f ∈ Z [ x ] is monic, degree 2 g , and its complex roots have magnitude √ q .) We say that A is ordinary if one of the following equivalent conditions holds: # A ( k )[ p ] = p g ; The local-local group scheme α p can’t be embedded into A ; Exactly half of the roots of f in Q p are p -adic units; The middle coefficient of f (that is, the coefficient of x g ) is coprime to p . Everett W. Howe Counting abelian varieties over finite fields 7 of 29
The category of Deligne modules We define the category L q of Deligne modules over F q by specifying its objects and morphisms. Objects Pairs ( T , F ) , where: T is a finitely-generated free Z -module of even rank, and F is an endomorphism of T such that The endomorphism F ⊗ Q of the Q -vector space T ⊗ Q is semi-simple, and its complex eigenvalues have magnitude √ q ; Exactly half of the roots in Q p of the characteristic polynomial of F are p -adic units; There is an endomorphism V of T with FV = q . Morphisms from ( T 1 , F 1 ) to ( T 2 , F 2 ) Z -module morphisms ϕ : T 1 → T 2 such that F 2 ϕ ( x ) = ϕ ( F 1 x ) for all x ∈ T 1 . Everett W. Howe Counting abelian varieties over finite fields 8 of 29
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