Deterministic primality tests based on tori and elliptic curves

AbstractWe develop a general framework for producing deterministic primality tests based on commutative group schemes over rings of integers. Our focus is on the cases of algebraic tori and elliptic curves. The proposed general machinery provides several series of tests which include, as special cases, tests discovered by Gross and by Denomme and Savin for Mersenne and Fermat primes, primes of the form 22l+1−2l+1, as well as some new ones

Formulas for the unramified Brauer group of a principal homogeneous space of a linear algebraic group

For a smooth compactification V of a principal homogeneous space E under a connected linear algebraic group G defined over a field k of characteristic zero, we present two formulas expressing Br V/Br k in terms of G

Réduction des groupes algébriques commutatifs

We study a problem of constructing an algebraic torus (an abelian variety) over a p-adic field whose Ne Âron model would have a given connected commutative unipotent group as the identity component of its special fibre

Word maps on perfect algebraic groups

We extend Borel's theorem on the dominance of word maps from semisimple algebraic groups to some perfect groups. In another direction, we generalize Borel's theorem to some words with constants. We also consider the surjectivity problem for particular words and groups, give a brief survey of recent results, present some generalizations and variations and discuss various approaches, with emphasis on new ideas, constructions and connections