O-minimality on twisted universal torsors and Manin's conjecture over number fields

Frei, Christopher; Pieropan, Marta

doi:10.24033/asens.2295

Cited by 16 publications

(27 citation statements)

References 46 publications

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“…For

K = Q

this follows from [, Proposition 2.1]. The proof in can be adapted to arbitrary number fields using similar techniques to ; we do not include the details in the interests of brevity. Since Manin's conjecture is already known to hold for

P^{1}

, all the novelty of Theorem lies in the explicit error term.…”

Section: Rational Points On Conicsmentioning

confidence: 99%

Rational points of bounded height on general conic bundle surfaces

Frei

Loughran

Sofos

2018

Proc. London Math. Soc.

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A conjecture of Manin predicts the asymptotic distribution of rational points of bounded height on Fano varieties. In this paper we use conic bundles to obtain correct lower bounds for a wide class of surfaces over number fields for which the conjecture is still far from being proved. For example, we obtain the conjectured lower bound of Manin's conjecture for any del Pezzo surface whose Picard rank is sufficiently large, or for arbitrary del Pezzo surfaces after possibly an extension of the ground field of small degree.

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“…For

K = Q

P^{1}

, all the novelty of Theorem lies in the explicit error term.…”

Section: Rational Points On Conicsmentioning

confidence: 99%

Rational points of bounded height on general conic bundle surfaces

Frei

Loughran

Sofos

2018

Proc. London Math. Soc.

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“…Let

frakturb

be any principal ideal among these divisors. The number of generators of

frakturb

with all conjugates bounded by

≪ H

≪_{ε} H^{ε / 2}

, which one can see by counting units with bounded conjugates (for example, as in the proof of [5, Lemma 7.2]).◻…”

Section: Minor Arcsmentioning

confidence: 99%

Forms of Differing Degrees Over Number Fields

Frei

Madritsch

2016

Mathematika

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Abstract. Consider a system of polynomials in many variables over the ring of integers of a number field K. We prove an asymptotic formula for the number of integral zeros of this system in homogeneously expanding boxes. As a consequence, any smooth and geometrically integral variety X ⊆ P m K satisfies the Hasse principle, weak approximation and the Manin-Peyre conjecture, if only its dimension is large enough compared to its degree.This generalizes work of Skinner, who considered the case where all polynomials have the same degree, and recent work of Browning and Heath-Brown, who considered the case where K = Q. Our main tool is Skinner's number field version of the Hardy-Littlewood circle method. As a by-product, we point out and correct an error in Skinner's treatment of the singular integral.

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“…Only recently, a generalization of this method to other number fields was started [18][19][20][21]24,25].…”

Section: Introductionmentioning

confidence: 98%

“…This leads sometimes to count lattice points in unbounded regions that require refined techniques [24]. This last paper offers a systematic and explicit way to produce parameterizations in terms of lattice points on twisted universal torsors for varieties over number fields with arbitrary class number and applies it to a specific singular del Pezzo surface of degree 4.…”

Section: Introductionmentioning

confidence: 98%