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Abelian varieties over finite fields with a specified characteristic polynomial modulo \(\ell\). (English) Zbl 1156.11322

From the introduction: Let \(\mathbb F\) be a finite field of characteristic \(p\) and order \(q\), and \(\ell\) a prime not equal to \(p\). Let \[ P(T)= (T^{2g} + q^g) + a_1(T^{2g-1} + q^{g-1}T) + \dots+ a_{g-1}(T^{g+1} + qT^{g-1}) + a_gT^g \]
be a polynomial. The goal of this paper is to estimate the number of isogeny classes of abelian varieties over \(\mathbb F\) of dimension \(g\) for which the characteristic polynomial for the action of Frobenius is congruent to \(P(T)\) modulo \(\ell\).
As an application we find the proportion of isogeny classes of abelian varieties with a rational point of order \(\ell\).

MSC:

11G10 Abelian varieties of dimension \(> 1\)
14G15 Finite ground fields in algebraic geometry
14K10 Algebraic moduli of abelian varieties, classification
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References:

[1] Jeffrey D. Achter and Joshua Holden, Notes on an analogue of the Fontaine-Mazur conjecture. J. Théor. Nombres Bordeaux 15 no.3 (2003), 627-637. · Zbl 1077.11080
[2] 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), 426-450. · Zbl 0931.11023
[3] Gerhard Frey, Ernst Kani and Helmut Völklein, Curves with infinite \({K}\)-rational geometric fundamental group. In Helmut Völklein, David Harbater, Peter Müller and J. G. Thompson, editors, Aspects of Galois theory (Gainesville, FL, 1996), volume 256 of London Mathematical Society Lecture Note Series, 85-118. Cambridge Univ. Press, 1999. · Zbl 0978.14021
[4] Joshua Holden, On the Fontaine-Mazur Conjecture for number fields and an analogue for function fields. J. Number Theory 81 (2000), 16-47. · Zbl 0997.11096
[5] Y. Ihara, On unramified extensions of function fields over finite fields. In Y. Ihara, editor, Galois Groups and Their Representations, volume 2 of Adv. Studies in Pure Math. 89-97. North-Holland, 1983. · Zbl 0542.14011
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