Enumeration of certain varieties over a finite field

Friedlander, John; Shparlinski, Igor

doi:10.1090/S0002-9939-2014-11999-X

Enumeration of certain varieties over a finite field

Authors: John B. Friedlander and Igor E. Shparlinski
Journal: Proc. Amer. Math. Soc. 142 (2014), 2615-2623
MSC (2000): Primary 11E20, 11G10; Secondary 11N36
DOI: https://doi.org/10.1090/S0002-9939-2014-11999-X
Published electronically: April 21, 2014
MathSciNet review: 3209317
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Abstract: Let $\mathbb{F}_q$ be a finite field of elements. E. Howe has shown that there is a natural correspondence between the isogeny classes of two-dimensional ordinary abelian varieties over $\mathbb{F}_q$ which do not contain a principally polarized variety and pairs of positive integers satisfying , where $\gcd (q,b)=1$ and all prime divisors $\ell$ of are in the arithmetic progression $\ell \equiv 1 \pmod 3$ . This arithmetic criterion allows us to give good upper bounds, and for many finite fields good lower bounds, for the frequency of occurrence of isogeny classes of varieties having this property.

[1] Valentin Blomer, Uniform bounds for Fourier coefficients of theta-series with arithmetic applications, Acta Arith. 114 (2004), no. 1, 1-21. MR 2067869 (2005e:11052), https://doi.org/10.4064/aa114-1-1
[2] Valentin Blomer, Ternary quadratic forms, and sums of three squares with restricted variables, Anatomy of integers, CRM Proc. Lecture Notes, vol. 46, Amer. Math. Soc., Providence, RI, 2008, pp. 1-17. MR 2437962 (2010i:11046)
[3] Valentin Blomer and Jörg Brüdern, A three squares theorem with almost primes, Bull. London Math. Soc. 37 (2005), no. 4, 507-513. MR 2143730 (2005m:11186), https://doi.org/10.1112/S0024609305004480
[4] Jörg Brüdern and Étienne Fouvry, Lagrange's four squares theorem with almost prime variables, J. Reine Angew. Math. 454 (1994), 59-96. MR 1288679 (96e:11125), https://doi.org/10.1515/crll.1994.454.59
[5] L. E. Dickson, Quaternary Quadratic Forms Representing all Integers, Amer. J. Math. 49 (1927), no. 1, 39-56. MR 1506600, https://doi.org/10.2307/2370770
[6] Stephen A. DiPippo and Everett W. Howe, Real polynomials with all roots on the unit circle and abelian varieties over finite fields, J. Number Theory 73 (1998), no. 2, 426-450. MR 1657992 (2000c:11101), https://doi.org/10.1006/jnth.1998.2302
[7] W. Duke, On ternary quadratic forms, J. Number Theory 110 (2005), no. 1, 37-43. MR 2114672 (2005h:11078), https://doi.org/10.1016/j.jnt.2004.06.013
[8] W. Duke, J. B. Friedlander, and H. Iwaniec, Weyl sums for quadratic roots, Int. Math. Res. Not. IMRN 11 (2012), 2493-2549. MR 2926988, https://doi.org/10.1093/imrn/rnr112
[9] John Friedlander and Henryk Iwaniec, Quadratic polynomials and quadratic forms, Acta Math. 141 (1978), no. 1-2, 1-15. MR 0476673 (57 #16232)
[10] John Friedlander and Henryk Iwaniec, Opera de cribro, American Mathematical Society Colloquium Publications, vol. 57, American Mathematical Society, Providence, RI, 2010. MR 2647984 (2011d:11227)
[11] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., The Clarendon Press Oxford University Press, New York, 1979. MR 568909 (81i:10002)
[12] Everett W. Howe, Principally polarized ordinary abelian varieties over finite fields, Trans. Amer. Math. Soc. 347 (1995), no. 7, 2361-2401. MR 1297531 (96i:11065), https://doi.org/10.2307/2154828
[13] Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR 2061214 (2005h:11005)
[14] O. T. O'Meara, Introduction to quadratic forms, Springer, Berlin, 1973. MR 1754311
[15] Carl Ludwig Siegel, Über die analytische Theorie der quadratischen Formen, Ann. of Math. (2) 36 (1935), no. 3, 527-606 (German). MR 1503238, https://doi.org/10.2307/1968644