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Ramification dans le corps des modules

Stéphane Flon (2004)

Annales de l’institut Fourier

Soit f un revêtement de la droite projective défini sur ℚ ¯ , de groupe de monodromie G . Soit K le compositum des corps de rationalité des points de branchement f , et M le corps des modules correspondants. Partant du lien entre corps des modules et espaces de Hurwitz, on étudie la géométrie et l’arithmétique de ces espaces et des espaces de configuration de points complétés pour évaluer la ramification dans M / K des mauvaises places de f qui ne divisent pas l’ordre de G , mais où les points de branchements...

Random Galois extensions of Hilbertian fields

Lior Bary-Soroker, Arno Fehm (2013)

Journal de Théorie des Nombres de Bordeaux

Let L be a Galois extension of a countable Hilbertian field K . Although L need not be Hilbertian, we prove that an abundance of large Galois subextensions of L / K are.

Realizability and automatic realizability of Galois groups of order 32

Helen Grundman, Tara Smith (2010)

Open Mathematics

This article provides necessary and sufficient conditions for each group of order 32 to be realizable as a Galois group over an arbitrary field. These conditions, given in terms of the number of square classes of the field and the triviality of specific elements in related Brauer groups, are used to derive a variety of automatic realizability results.

Reduction and specialization of polynomials

Pierre Dèbes (2016)

Acta Arithmetica

We show explicit forms of the Bertini-Noether reduction theorem and of the Hilbert irreducibility theorem. Our approach recasts in a polynomial context the geometric Grothendieck good reduction criterion and the congruence approach to HIT for covers of the line. A notion of “bad primes” of a polynomial P ∈ ℚ[T,Y] irreducible over ℚ̅ is introduced, which plays a central and unifying role. For such a polynomial P, we deduce a new bound for the least integer t₀ ≥ 0 such that P(t₀,Y) is irreducible...

Relative Bogomolov extensions

Robert Grizzard (2015)

Acta Arithmetica

A subfield K ⊆ ℚ̅ has the Bogomolov property if there exists a positive ε such that no non-torsion point of K × has absolute logarithmic height below ε. We define a relative extension L/K to be Bogomolov if this holds for points of L × ∖ K × . We construct various examples of extensions which are and are not Bogomolov. We prove a ramification criterion for this property, and use it to show that such extensions can always be constructed if some rational prime has bounded ramification index in K.

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