Notes on an analogue of the Fontaine-Mazur conjecture. (English) Zbl 1077.11080

Let \(\ell\) be a prime number. According to a function field analogue of the Fontaine-Mazur conjecture [J.-M. Fontaine, B. Mazur, “Geometric Galois representations”. International Press Ser. Number Theory 1, 41–78 (1995; Zbl 0839.14011)], it should be rare to find a function field over a finite field with \(\ell\not=0\) that has an infinite unramified pro-\(\ell\) extension virtually with uniformly powerful geometric Galois group. Exceptional counter examples are found in [Y. Ihara, Adv. Stud. Pure Math. 2, 89–97 (1983; Zbl 0542.14011)] and [G. Frey, E. Kani, H. Völklein, Lond. Math. Soc. Lect. Note Ser. 256, 85–118 (1999; Zbl 0978.14021)]. In [J. Number Theory 81, No. 1, 16–47 (2000; Zbl 0997.11096)], the second author gave two sorts of criteria for the non-existence of such an infinite extension in terms of the Frobenius property of the Jacobian \(\ell\)-torsion. Applying Katz’ equidistribution theorem to these criteria, the authors show that most of the (under some mild assumption, approximately \(1-\frac{1}{\ell^2}\) of all) function fields do not have such infinite extensions. By a similar argument, it is shown that approximately \(\frac{\ell}{\ell^2-1}\) of abelian varieties over a finite field have non-trivial rational \(\ell\)-torsion points.


11R58 Arithmetic theory of algebraic function fields
11R32 Galois theory
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