arXiv subjects

Kevin Destagnol

Publications and source records attributed to Kevin Destagnol.

9 recordsLinked to original sources

arXiv2025

Counting quadratic points on Fano varieties

This paper initiates the systematic study of the number of points of bounded height on symmetric squares of weak Fano varieties. We provide a general framework for establishing the point count on $\text{Sym}^2 X$. In the specific case of surfaces, we relate this to the Manin--Peyre conjecture for $\text{Hilb}^2 X$, and prove the conjecture for an infinite family of non-split quadric surfaces. In order to achieve the predicted asymptotic, we show that a type II thin set of a new flavour must be removed. To establish our counting result for the specific family of surfaces, we generalise existing lattice point counting techniques to lattices defined over rings of integers. This reduces the dimension of the problem and yields improved error terms. Another key tool we develop is a collection of results for summing Euler products over quadratic extensions. We use this to show moments of $L$-functions at $s=1$ are constant on average in quadratic twist families.

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arXiv2025

Local solubility in generalised Ch\^atelet varieties

We obtain asymptotic formulas for averages of general multivariate arithmetic functions evaluated at polynomial arguments using recent work of Rydin Myerson and Rome-Yamagishi. We give several applications of our results in analytic number theory and arithmetic geometry. For example, we improve on the number of variables needed to prove the Hasse principle for certain polynomial systems, and we count the number of fibers with a rational point in families of high-dimensional Ch\^atelet varieties, allowing for arbitrarily large subordinate Brauer groups.

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arXiv2024

Averages of arithmetic functions over polynomials in many variables

We estimate the average of any arithmetic function $k$ over the values of any smooth polynomial in many variables provided only that $k$ has a distribution in arithmetic progressions of fixed modulus. We give several applications of this result including the analytic Hasse principle for an intersection of two cubics in 21 variables and asymptotics for the number of integer solutions of a non-algebraic variety.

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arXiv2019

Rational points and prime values of polynomials in moderately many variables

We derive the Hasse principle and weak approximation for pencils of certain varieties in the spirit of work by Colliot-Thélène,Sansuc and Harpaz-Skorobogatov-Wittenberg. Our varieties are defined through polynomials in many variables and part of our work is devoted to establishing Schinzel's hypothesis for polynomials of this kind. This last part is achieved by using arguments behind Birch's well-known result regarding the Hasse principle for complete intersections with the notable difference that we prove our result in 50% fewer variables than in the classical Birch setting. We also study the problem of square-free values of an integer polynomial with 66.6% fewer variables than in the Birch setting.

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arXiv2018

On a certain non-split cubic surface

In this note, we establish an asymptotic formula for the number of rational points of bounded height on the singular cubic surface $$ x_0(x_1^2 + x_2^2)=x_3^3 $$ with a power-saving error term, which verifies the Manin-Peyre conjectures for this surface.

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arXiv2018

The power-saving Manin-Peyre's conjectures for a senary cubic

Using recent work of the first author~\cite{Bet}, we prove a strong version of the Manin-Peyre's conjectures with a full asymptotic and a power-saving error term for the two varieties respectively in $\mathbb{P}^2 \times \mathbb{P}^2$ with bihomogeneous coordinates $[x_1:x_2:x_3],[y_1:y_2,y_3]$ and in $\mathbb{P}^1\times \mathbb{P}^1 \times \mathbb{P}^1$ with multihomogeneous coordinates $[x_1:y_1],[x_2:y_2],[x_3:y_3]$ defined by the same equation $x_1y_2y_3+x_2y_1y_3+x_3y_1y_2=0$. We thus improve on recent work of Blomer, Brüdern and Salberger \cite{BBS} and provide a different proof based on a descent on the universal torsor of the conjectures in the case of a del Pezzo surface of degree 6 with singularity type $\mathbf{A}_1$ and three lines (the other existing proof relying on harmonic analysis \cite{CLT}). Together with~\cite{Blomer2014} or with recent work of the second author \cite{Dest2}, this settles the study of the Manin-Peyre's conjectures for this equation.

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arXiv2018

Description de torseurs quasi-versels pour une famille de surfaces fibrées en coniques

Generalising work of La Bretèche, Browning and Peyre and of the author, we describe the geometry required by Manin's principle and Peyre's conjecture for a family of conic bundle surfaces containing Châtelet surfaces. These surfaces $S_{a,F}$ are the conic bundle surfaces obtained as smooth minimal proper model of $$ Y^2-aZ^2=F(X,1) $$ with $a \in \mathbf{Z}$ squarefree and $F \in \mathbf{Z}[x_1,x_2]$ a binary form of \textit{even} degree $n$ without repeated roots and whose irreducible factors over $\mathbf{Q}$ remain irreducible over $\mathbf{Q}\left( \sqrt{a} \right)$.

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arXiv2017

La conjecture de Manin pour une famille de variétés en dimension supérieure

Inspired by a method of La Bretèche relying on some unique factorisation, we generalize work of Blomer, Brüdern, and Salberger to prove Manin's conjecture in its strong form conjectured by Peyre for some infinite family of varieties of higher dimension. The varieties under consideration in this paper correspond to the projective varieties defined by the following equation $$ x_1 y_2y_3\cdots y_n+x_2y_1y_3 \cdots y_n+ \cdots+x_n y_1 y_2 \cdots y_{n-1}=0. $$ in $\mathbb{P}^{2n-1}_{\mathbb{Q}}$ for all $n \geqslant 3$. This paper comes with an Appendix by Per Salberger.

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arXiv2016

La conjecture de Manin pour certaines surfaces de Châtelet

Following the line of attack from La Bretèche, Browning and Peyre, we prove Manin's conjecture in its strong form conjectured by Peyre for a family of Châtelet surfaces which are defined as minimal proper smooth models of affine surfaces of the form $$ Y^2-aZ^2=F(X,1), $$ where $a=-1$, $F \in \mathbb{Z}[x_1,x_2]$ is a polynomial of degree 4 whose factorisation into irreducibles contains two non proportional linear factors and a quadratic factor which is irreducible over $\mathbb{Q}[i]$. This result deals with the last remaining case of Manin's conjecture for Châtelet surfaces with $a=-1$ and essentially settles Manin's conjecture for Châtelet surfaces with $a<0$.

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