## Duke Mathematical Journal

### How large is $A_{g}(\mathbb{F}_{q})$?

#### Abstract

Let $B(g,p)$ denote the number of isomorphism classes of $g$-dimensional Abelian varieties over the finite field of size $p$. Let $A(g,p)$ denote the number of isomorphism classes of principally polarized $g$-dimensional Abelian varieties over the finite field of size $p$. We derive upper bounds for $B(g,p)$ and lower bounds for $A(g,p)$ for $p$ fixed and $g$ increasing. The extremely large gap between the lower bound for $A(g,p)$ and the upper bound for $B(g,p)$ implies some statistically counterintuitive behavior for Abelian varieties of large dimension over a fixed finite field.

#### Article information

Source
Duke Math. J., Volume 167, Number 18 (2018), 3403-3453.

Dates
Revised: 7 March 2018
First available in Project Euclid: 15 November 2018

https://projecteuclid.org/euclid.dmj/1542250833

Digital Object Identifier
doi:10.1215/00127094-2018-0029

Mathematical Reviews number (MathSciNet)
MR3881200

Zentralblatt MATH identifier
07009769

#### Citation

Lipnowski, Michael; Tsimerman, Jacob. How large is $A_{g}(\mathbb{F}_{q})$ ?. Duke Math. J. 167 (2018), no. 18, 3403--3453. doi:10.1215/00127094-2018-0029. https://projecteuclid.org/euclid.dmj/1542250833

#### References

• [1] J. D. Achter, D. Erman, K. S. Kedlaya, M. M. Wood, and D. Zureick-Brown, A heuristic for the distribution of point counts for random curves over finite field, Philos. Trans. Roy. Soc. A 373 (2015), art. ID 20140310.
• [2] J. De Jong and N. Katz, Counting the number of curves over a finite field, 2000, available at https://www.math.columbia.edu/~dejong/.
• [3] P. Deligne, La conjecture de Weil, I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307.
• [4] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109.
• [5] S. A. DiPippo and E. W. Howe, Real polynomials with all roots on the unit circle and abelian varieties over finite fields, J. Number Theory 73 (1998), 426–450.
• [6] J. S. Ellenberg, A. Venkatesh, and C. Westerland, Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields, Ann. of Math. (2) 183 (2016), 729–786.
• [7] G. Faltings and C.-L. Chai, Degeneration of Abelian Varieties, with an appendix by D. Mumford, Ergeb. Math. Grenzgeb. (3) 22, Springer, Berlin, 1990.
• [8] E. Friedman and L. C. Washington, “On the distribution of divisor class groups of curves over a finite field” in Théorie des nombres (Quebec, PQ, 1987), de Gruyter, Berlin, 1989, 227–239.
• [9] W.-T. Gan and J.-K. Yu, Group schemes and local densities, Duke Math. J. 105 (2000), 497–524.
• [10] G. Harder, A Gauss-Bonnet formula for discrete arithmetically defined groups, Ann. Sci. Éc. Norm. Supér. (4) 4 (1971), 409–455.
• [11] J. Harer and D. Zagier, The Euler characteristic of the moduli space of curves, Invent. Math. 85 (1986), 457–485.
• [12] T. Honda, Isogeny classes of abelian varieties over finite fields, J. Math. Soc. Japan 20 (1968), 83–95.
• [13] R. Jacobowitz, Hermitian forms over local fields, Amer. J. Math. 84 (1962), 441–465.
• [14] N. M. Katz and P. Sarnak, Random Matrices, Frobenius Eigenvalues, and Monodromy, Amer. Math. Soc. Colloq. Publ. 45, Amer. Math. Soc., Providence, 1999.
• [15] R. E. Kottwitz, Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (1992), 373–444.
• [16] R. E. Kottwitz, “Harmonic analysis on reductive $p$-adic groups and Lie algebras” in Harmonic Analysis, the Trace Formula, and Shimura Varieties, Clay Math. Proc. 4, Amer. Math. Soc., Providence, 2005, 393–522.
• [17] R. P. Langlands, “Modular forms and $\ell$-adic representations” in Modular Functions of One Variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Math. 349, Springer, Berlin, 1973, 361–500.
• [18] M. Lipnowski and J. Tsimerman, Cohen-Lenstra heuristics for étale group schemes and symplectic pairings, preprint, arXiv:1610.09304v1 [math.NT].
• [19] S. Louboutin, Explicit bounds for residues of Dedekind zeta functions, values of $L$-functions at $s=1$, and relative class numbers, J. Number Theory 85 (2000), 263–282.
• [20] D. Mumford, Abelian Varieties, Tata Inst. Fund. Res. Stud. Math. 5, Oxford Univ. Press, London, 1970.
• [21] J. Neukirch, Algebraic Number Theory, Grundlehren Math. Wiss. 322, Springer, Berlin, 1999.
• [22] W. Scharlau, Quadratic and Hermitian Forms, Grundlehren Math. Wiss. 270, Springer, Berlin, 1985.
• [23] P. Scholze, The Langlands-Kottwitz approach for the modular curve, Int. Math. Res. Not. IMRN 2011, no. 15, 3368–3425.
• [24] J.-P. Serre, “Bounds for the orders of finite subgroups of $G(k)$” in Group Representation Theory, EPFL Press, Lausanne, 2007, 405–450.
• [25] N.-P. Skoruppa, Quick lower bounds for regulators of number fields, Enseign. Math. (2) 39 (1993), 137–141.
• [26] J. Tate, Endomorphisms of abelian varieties over finite fields, Invent. Math. 2 (1966), 134–144.
• [27] J. Tate, “Classes d’isogénie des variétés abéliennes sur un corps fini (d’après T. Honda)” in Séminaire Bourbaki 1968/1969: Exposés 347–363, no. 352, Lecture Notes in Math. 175, Springer, Berlin, 1971, 95–110.
• [28] W. C. Waterhouse, Abelian varieties over finite fields, Ann. Sci. Éc. Norm. Supér. (4) 2 (1969), 521–560.
• [29] W. C. Waterhouse and J. Milne, “Abelian varieties over finite fields” in 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969), Amer. Math. Soc., Providence, 1971, 53–64.
• [30] Z. Yun, “Orbital integrals and Dedekind zeta functions” in The Legacy of Srinivasa Ramanujan, Ramanujan Math. Soc. Lect. Notes Ser. 20, Ramanujan Math. Soc., Mysore, 2013, 399–420.