The Grunwald problem and approximation properties for homogeneous spaces

Demarche, Cyril; Arteche, Giancarlo Lucchini; Neftin, Danny

doi:10.5802/aif.3104

Cited by 20 publications

(18 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Let {G m } m be a family of finite groups such that, for every m, G m satisfies one of the three conditions (1), (2) or (3) of Theorem 1. Then, by the above cited results, the Grunwald problem has a positive answer for G m : more precisely, this follows from the Grunwald-Wang theorem (see [Gru33], [Wan50]) in case (1), from Neukirch's result [Neuk79] in case (2) and from the recent work of Demarche, Lucchini Arteche and Neftin [DLAN17] in case (3).…”

Section: The Grunwald Problem and Proof Of Theoremmentioning

confidence: 95%

A note on Galois groups and local degrees

Checcoli

2018

manuscripta math.

View full text Add to dashboard Cite

In this paper, we consider infinite Galois extensions of number fields and study the relation between their local degrees and the structure of their Galois groups. It is known that, if K is a number field and L/K is an infinite Galois extension of group G, then the local degrees of L are uniformly bounded at all rational primes if and only if G has finite exponent. In this note we show that the non uniform boundedness of the local degrees is not equivalent to any group theoretical property. More precisely, we exhibit several groups that admit two different realisations over a given number field, one with bounded local degrees at a given set of primes and one with infinite local degrees at the same primes.

show abstract

Section: The Grunwald Problem and Proof Of Theoremmentioning

confidence: 95%

A note on Galois groups and local degrees

Checcoli

2018

manuscripta math.

View full text Add to dashboard Cite

show abstract

“…Moreover, if P(T ) ∈ S , then, as in the proof of Lemma 4.6, there is a set S of prime numbers of positive density α such that no prime number p ∈ S is a prime divisor of P(T ). 18 Set P(T ) = a 0 + a 1 T + • • • + a N T N . As condition ( * ) holds, P(T ) has no root in Q.…”

Section: On the Local-global Principle For Specializationsmentioning

confidence: 99%

Density Results for Specialization Sets of Galois covers

König¹,

Legrand²

2019

J. Inst. Math. Jussieu

View full text Add to dashboard Cite

We provide evidence for this conclusion: given a finite Galois cover f : X → P 1 Q of group G, almost all (in a density sense) realizations of G over Q do not occur as specializations of f . We show that this holds if the number of branch points of f is sufficiently large, under the abc-conjecture and, possibly, the lower bound predicted by the Malle conjecture for the number of Galois extensions of Q of given group and bounded discriminant. This widely extends a result of Granville on the lack of Q-rational points on quadratic twists of hyperelliptic curves over Q with large genus, under the abc-conjecture (a diophantine reformulation of the case G = Z/2Z of our result). As a further evidence, we exhibit a few finite groups G for which the above conclusion holds unconditionally for almost all covers of P 1 Q of group G. We also introduce a local-global principle for specializations of Galois covers f : X → P 1 Q and show that it often fails if f has abelian Galois group and sufficiently many branch points, under the abc-conjecture. On the one hand, such a local-global conclusion underscores the "smallness" of the specialization set of a Galois cover of P 1 Q . On the other hand, it allows to generate conditionally "many" curves over Q failing the Hasse principle, thus generalizing a recent result of Clark and Watson devoted to the hyperelliptic case.

show abstract

“…However, they cannot be ignored in the formulation of Conjecture 2.3: there exist smooth and proper rationally connected threefolds X over Q such that [Har96]). Another example of a rationally connected variety X over a number field k such that X(A k ) Br(X) = X(A k ) Br 1 (X) is given in [DLAN17]. Transcendental elements and their influence on the Brauer-Manin set have received a lot of attention for other classes of varieties as well (see [Wit04], [Ier10], [HVAV11], [Pre13], [HVA13], [IS15], [New16], [CV15], [MSTVA16] for examples of K3 and Enriques surfaces for which transcendental elements play a role in the Brauer-Manin set).…”

Section: Over Number Fields: General Contextmentioning

confidence: 99%

Rational points and zero-cycles on rationally connected varieties over number fields

Wittenberg¹

2018

Proceedings of Symposia in Pure Mathematics

View full text Add to dashboard Cite

We report on progress in the qualitative study of rational points on rationally connected varieties over number fields, also examining integral points, zero-cycles, and non-rationally connected varieties. One of the main objectives is to highlight and explain the many recent interactions with analytic number theory. p. 174]). Let X be a smooth, proper, geometrically irreducible variety defined over a number field k. If X is rationally connected, then X(k) is dense in X(A k ) Br(X) . Conjecture 2.3By "X is rationally connected", we mean that the variety X ⊗ k k is rationally connected in the sense of Campana, Kollár, Miyaoka and Mori, where k denotes an algebraic closure of k. Equivalently, for any algebraically closed field K containing k, two general K-points of X can be joined by a rational curve defined over K. We recall that examples of rationally connected varieties include geometrically unirational varieties (trivial), Fano varieties (see [Cam92], [KMM92]), and fibrations into rationally connected varieties over a rationally connected base (see [GHS03]).Conjecture 2.3 is wide open even for surfaces.Remarks 2.4. (i) The group Br 0 (X) = Im(Br(k) → Br(X)) does not contribute to the Brauer-Manin set: the global reciprocity law implies the equality X(A k ) B = X(A k ) B+Br 0 (X) for any subgroup B ⊆ Br(X). When Br(X)/Br 0 (X) is finite, the subset X(A k ) Br(X) ⊆ X(A k ) is therefore cut out by finitely many conditions. In particular, in this case, it is an open subset of X(A k ).(ii) When X is rationally connected, or, more generally, when X is simply connected and H 2 (X, O X ) = 0, an analysis of the Hochschild-Serre spectral sequence and of the Brauer group of X ⊗ k k implies that Br(X)/Br 0 (X) is finite (see [CTS13b, Lemma 1.1]). Presumably, this quotient should be finite under the sole assumption that X is simply connected, but this is out of reach of current knowledge. (As follows from [CTS13a] and [CTS13b, §4], the finiteness of the ℓ-primary torsion subgroup of Br(X)/Br 0 (X) is equivalent, when X is simply connected, to the ℓ-adic Tate conjecture for divisors on X.) (iii) Letting v be any archimedean place of k and noting that the image of the projection map X(A k ) Br(X) → X(k v ) is a union of connected components, we see

show abstract

The Grunwald problem and approximation properties for homogeneous spaces

Cited by 20 publications

References 14 publications

A note on Galois groups and local degrees

A note on Galois groups and local degrees

Density Results for Specialization Sets of Galois covers

Rational points and zero-cycles on rationally connected varieties over number fields

Contact Info

Product

Resources

About