Forms of Differing Degrees Over Number Fields

Frei, Christopher; Madritsch, Manfred G.

doi:10.1112/s0025579316000206

Cited by 6 publications

(6 citation statements)

References 12 publications

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“…Birch's work [3] is generalised to systems of forms with differing degrees by Browning and Heath-Brown [7] over Q and by Frei and Madritsch [16] over number fields. It is extended to linear spaces of solutions by Brandes [5,6].…”

Section: Related Workmentioning

confidence: 99%

Quadratic forms and systems of forms in many variables

Myerson

2018

Invent. math.

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Let F 1 , . . . , F R be quadratic forms with integer coefficients in n variables. When n ≥ 9R and the variety V (F 1 , . . . , F R ) is a smooth complete intersection, we prove an asymptotic formula for the number of integer points in an expanding box at which these forms simultaneously vanish, which in particular implies the Hasse principle for V (F 1 , . . . , F R ). Previous work in this direction required n to grow at least quadratically with R. We give a similar result for R forms of degree d, conditional on an upper bound for the number of solutions to an auxiliary inequality. In principle this result may apply as soon as n > d2 d R. In the case that d ≥ 3, several strategies are available to prove the necessary upper bound for the auxiliary inequality. In a forthcoming paper we use these ideas to apply the circle method to nonsingular systems of forms with real coefficients. Mathematics Subject Classification

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Section: Related Workmentioning

confidence: 99%

Quadratic forms and systems of forms in many variables

Myerson

2018

Invent. math.

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“…Over Q, the following theorem was proved by Browning and Heath-Brown [BHB14], generalising earlier work of Birch [Bir62] which covered, in particular, the case of hypersurfaces. The extension to number fields is due to Skinner [Ski97] and to Frei and Madritsch [FM16].…”

Section: Methods For Rational and Rationally Connected Varietiesmentioning

confidence: 99%

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

Wittenberg¹

2018

Proceedings of Symposia in Pure Mathematics

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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

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“…One feature of the proof that is worth highlighting is our treatment of the singular integral. In recent work, Frei and Madritsch [12] identified an inaccuracy in the work of Skinner [27], and proposed a corrected treatment. Unfortunately, their argument is rather involved, but we are able to give a much simplified proof of the same statement that parallels the treatment over Q.…”

Section: And Letmentioning

confidence: 99%

“…The former paper falls somewhat short of what had been known in the rational case, but in recent work Browning and Vishe [8] found an improved treatment so that now the number field case is almost as well understood as the rational case. Similarly, the recent paper of Browning and Heath-Brown generalising Birch's theorem to systems involving differing degrees [7] has immediately been translated to the number field setting by Frei and Madritsch [12], as has Dietmann's work on small solutions of quadratic forms [11] by Helfrich [16]. In this memoir we aim to continue in this direction by providing a number field version of the author's recent work on linear spaces on hypersurfaces [2,4].…”

Section: Introductionmentioning

confidence: 99%

Linear Spaces on Hypersurfaces over Number Fields

Brandes¹

2017

Michigan Math. J.

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We establish an analytic Hasse principle for linear spaces of affine dimension m on a complete intersection over an algebraic field extension K of Q. The number of variables required to do this is no larger than what is known for the analogous problem over Q. As an application we show that any smooth hypersurface over K whose dimension is large enough in terms of the degree is K-unirational, provided that either the degree is odd or K is totally imaginary.

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Forms of Differing Degrees Over Number Fields

Cited by 6 publications

References 12 publications

Quadratic forms and systems of forms in many variables

Quadratic forms and systems of forms in many variables

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

Linear Spaces on Hypersurfaces over Number Fields

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