On a number of isogeny classes of simple abelian varieties over finite fields
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Mathematische Zeitschrift
On a number of isogeny classes of simple abelian varieties over finite fields Jungin Lee1 Received: 31 July 2019 / Accepted: 4 November 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we investigate the asymptotic behavior of the number sq (g) of isogeny classes of simple abelian varieties of dimension g over a finite field Fq . We prove that the logarithmic asymptotic of sq (g) is the same as the logarithmic asymptotic of the number m q (g) of isogeny classes of all abelian varieties of dimension g over Fq . We also prove that lim sup g→∞
sq (g) = 1. m q (g)
This suggests that there are much more simple isogeny classes of abelian varieties over Fq of dimension g than non-simple ones for sufficiently large g, which can be understood as the opposite situation to a main result of Lipnowski and Tsimerman (Duke Math. 167:3403–3453, 2018). Keywords Abelian variety · Arithmetic statistics · Isogeny class · Finite field Mathematics Subject Classification 11G10 · 11G25 · 14K02
1 Introduction An abelian variety over an arbitrary field k is isogenous to a product of simple abelian varieties. One can naturally ask about the distribution of the dimensions of simple isogeny factors in the set of isogeny classes of abelian varieties over k of given dimension. When k is a prime field F p and a variety is equipped with a principal polarization, there is a nice answer given by Lipnowski and Tsimerman [3]. Note that we may replace 0.99 in the proposition below by any constant c < 1. Proposition 1.1 ([3], Corollary 5.14) Under the assumption of [3, Conjecture 5.2], for a subset of primes p of density at least 1 − 2−9 , the proportion of principally polarized abelian
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Jungin Lee [email protected] Department of Mathematics, Pohang University of Science and Technology, Pohang, Gyeongbuk 37673, Republic of Korea
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varieties over F p which admits an isogeny factor E h for some elliptic curve E and h ≥ 0.99 g approaches 1 as g → ∞. The purpose of this article is to answer the question for abelian varieties (without polarization) over a finite field Fq . An interesting point is that if we do not consider polarizations, most of the isogeny classes have a large simple isogeny factor, which is opposite to the case with principal polarizations. In particular, the number of isogeny classes of simple abelian varieties over Fq is large. We review some background material and summarize the results of this paper in the rest of this section. Let A be an abelian variety over a finite field Fq of dimension g. For a prime q, there is a bijection Q ⊗Z EndFq (A) ∼ = EndQ (V (A)) due to Tate [4] so the q-Frobenius endomorphism on A corresponds to an endomorphism on a Q -vector space V (A). Denote its characteristic polynomial by p A . (It is called the Weil q-polynomial.) Then p A is independent of the choice of , monic, has integer coefficients, of degree 2g and all of its roots are Weil q-numbers (i.e. algebraic integers all of whose Gal(Q/Q)-conj
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