Summary: arXiv:math.NT/0105087v227Mar2004 Journal de Th´eorie des Nombres
de Bordeaux 15 (2003), 627­637
Notes on an analogue of the Fontaine-Mazur
conjecture
par Jeffrey D. ACHTER et Joshua HOLDEN
R´esum´e. On estime le proportion des corps de fonctions qui rem-
plissent des conditions qui impliquent un analogue de la conjec-
ture de Fontaine et Mazur. En passant, on calcule le propor-
tion des vari´et´es abeli´ennes (ou Jacobiennes) sur un corps fini qui
poss`edent un point rationnel d'orde .
Abstract. We estimate the proportion of function fields satis-
fying certain conditions which imply a function field analogue of
the Fontaine-Mazur conjecture. As a byproduct, we compute the
fraction of abelian varieties (or even Jacobians) over a finite field
which have a rational point of order .
1. Introduction
The paper [10] discusses the following conjecture, originally stated by
Fontaine and Mazur in [8]:
Conjecture 1.1 (Fontaine-Mazur, as restated in [1]). Let F be a number
field and any prime. There does not exist an infinite everywhere unram-

Source: Achter, Jeff - Department of Mathematics, Colorado State University

Collections: Environmental Sciences and Ecology; Mathematics