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Newton and Schinzel sequences in quadratic fields

David Adam, Paul-Jean Cahen (2010)

Actes des rencontres du CIRM

We give the maximal length of a Newton or a Schinzel sequence in a quadratic extension of a global field. In the case of a number field, the maximal length of a Schinzel sequence is 1, except in seven particular cases, and the Newton sequences are also finite, except for at most finitely many cases, all real. We give the maximal length of these sequences in the special cases. We have similar results in the case of a quadratic extension of a function field 𝔽 q ( T ) , taking in account that the ring of integers...

Notes on an analogue of the Fontaine-Mazur conjecture

Jeffrey D. Achter, Joshua Holden (2003)

Journal de théorie des nombres de Bordeaux

We estimate the proportion of function fields satisfying 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 .

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