Rational points of bounded height and the Weil restriction

Loughran, Daniel

doi:10.1007/s11856-015-1245-x

Cited by 6 publications

(7 citation statements)

References 34 publications

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“…Theorem 1.1 yields for the first time the correct lower bound for Manin's conjecture for the Fermat cubic surface over any number field, and in particular over Q. As an application, we are able to extend their counterexample to arbitrary number fields, and moreover to improve upon the lower bounds obtained in [2, Theorem 3.1] (other counterexamples over arbitrary number fields have also been constructed in [26,27]…”

Section: Counterexamples To Manin's Conjecturementioning

confidence: 76%

“…Theorem yields for the first time the correct lower bound for Manin's conjecture for the Fermat cubic surface over any number field, and in particular over

double-struckQ

. As an application, we are able to extend their counterexample to arbitrary number fields, and moreover to improve upon the lower bounds obtained in [, Theorem 3.1] (other counterexamples over arbitrary number fields have also been constructed in ). Theorem Let

K

be a number field and

Y : a_{0} x_{0}^{3} + a_{1} x_{1}^{3} + a_{2} x_{2}^{3} + a_{3} x_{3}^{3} = 0 \subset {double-struckP}_{K}^{3} \times {double-struckP}_{K}^{3} .

For any dense open subset

U \subset Y

and any anticanonical height function

H

Y

we have

N_{U, H} false(B false) ≫_{U, H} \{\begin{matrix} left B {false(prefixlog B false)}^{6}, & left if double-struckQ false(\sqrt{- 3} false) \subseteq K, \\ left B {false(prefixlog B false)}^{3}, & left if double-struckQ false(\sqrt{- 3} false) ⊄ K . \end{matrix}

…”

Section: Introductionmentioning

confidence: 73%

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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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Section: Counterexamples To Manin's Conjecturementioning

confidence: 76%

“…Theorem yields for the first time the correct lower bound for Manin's conjecture for the Fermat cubic surface over any number field, and in particular over

double-struckQ

K

be a number field and

Y : a_{0} x_{0}^{3} + a_{1} x_{1}^{3} + a_{2} x_{2}^{3} + a_{3} x_{3}^{3} = 0 \subset {double-struckP}_{K}^{3} \times {double-struckP}_{K}^{3} .

For any dense open subset

U \subset Y

and any anticanonical height function

H

Y

we have

N_{U, H} false(B false) ≫_{U, H} \{\begin{matrix} left B {false(prefixlog B false)}^{6}, & left if double-struckQ false(\sqrt{- 3} false) \subseteq K, \\ left B {false(prefixlog B false)}^{3}, & left if double-struckQ false(\sqrt{- 3} false) ⊄ K . \end{matrix}

…”

Section: Introductionmentioning

confidence: 73%

Rational points of bounded height on general conic bundle surfaces

Frei

Loughran

Sofos

2018

Proc. London Math. Soc.

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“…toric varieties [5]). Nevertheless this conjecture is false as stated and there are now counter-examples over any number field [4,42]. One of the aims of this paper is to try to generalise Manin's conjecture to the counting functions (1.2).…”

Section: Introductionmentioning

confidence: 99%

The number of varieties in a family which contain a rational point

Loughran

2018

J. Eur. Math. Soc.

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We consider the problem of counting the number of varieties in a family over a number field which contain a rational point. In particular, for products of Brauer-Severi varieties and closely related counting functions associated to Brauer group elements. Using harmonic analysis on toric varieties, we provide a positive answer to a question of Serre on such counting functions in some cases. We also formulate some conjectures on generalisations of Serre's problem.

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“…Then

\begin{matrix} true \\ right H ((x_{1} : \dots : x_{s})) : = \prod_{v \in Ω_{K}} {∥ (x_{1}, \dots, x_{s}) ∥}_{v}^{n_{v} (s - D)} \end{matrix}

defines an anticanonical height function on the rational points

X (K)

. The proof of [6, Theorem 4.8] shows that the conclusion of Theorem implies the Manin–Peyre conjecture for

X

with respect to the height

H

.…”

Section: Introductionmentioning

confidence: 98%

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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Rational points of bounded height and the Weil restriction

Cited by 6 publications

References 34 publications

Rational points of bounded height on general conic bundle surfaces

Rational points of bounded height on general conic bundle surfaces

The number of varieties in a family which contain a rational point

Forms of Differing Degrees Over Number Fields

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