Research interests

Papers

[7] Netan Dogra & Samuel Le Fourn, Quadratic Chabauty for modular curves and modular forms of rank one, [ PDF ].
In this paper, we provide refined sufficient conditions for the quadratic Chabauty method to produce a finite set of points, with the conditions on the rank of the Jacobian replaced by conditions on the rank of a quotient of the Jacobian plus an associated space of Chow-Heegner points. We then apply this condition to prove the finiteness of this set for any modular curves $X_0^+(N)$ and $X_{\rm{ns}}^+(N)$ of genus at least 2 with $N$ prime. The proof relies on the existence of a quotient of their Jacobians whose Mordell-Weil rank is equal to its dimension (and at least 2), which is proven via analytic estimates for orders of vanishing of L-functions of modular forms, thanks to a Kolyvagin-Logachev type result.

[6] Samuel Le Fourn, Tubular approaches to Baker's method on curves and varieties, submitted [arXiv].

[+] Abstract
Baker's method, relying on estimates on linear forms in logarithms of algebraic numbers, allows one to prove in several situations the effective finiteness of integral points on varieties. In this article, we give a generalisation of results of Levin regarding Baker's method for varieties, and explain how, quite surprisingly, it mixes (under additional hypotheses) with Runge's method to improve some known estimates in the case of curves by bypassing (or more generally reducing) the need for linear forms in $p$-adic logarithms.

[5] Samuel Le Fourn & Filip Najman, Torsion of $\Q$-curves over quadratic fields, Mathematical Research Letters, to appear [ arXiv | bib].

[+] Abstract
We find all the possible torsion groups of $\Q$-curves over quadratic fields and determine which groups appear finitely and which appear infinitely often.

[4] Samuel Le Fourn, A tubular variant of Runge's method in all dimensions, with applications to integral points on Siegel modular varieties, Algebra & Number Theory , 2019 [ arXiv | bib| DOI | Sage worksheet ]

[+] Abstract
Runge's method is a tool to figure out integral points on some curves, effectively in terms of height. This method has been generalised to varieties of any dimension, unfortunately its conditions of application are often too restrictive. In this paper, we provide a further generalisation intended to be more flexible while still effective, and exemplify its applicability by giving finiteness results for integral points on some Siegel modular varieties. As a special case, we obtain a totally explicit finiteness result for integral points on the Siegel modular variety $A_2(2)$.

[3] Samuel Le Fourn, Sur la méthode de Runge et les points entiers de certaines variétés modulaires de Siegel, Comptes Rendus de l'Académie des Sciences, 2017 [ arXiv | bib| DOI ].

[+] Abstract
We present results on the integral points of some modular varieties. These results are based on a generalisation of the so-called Runge's method to higher dimensions, which will be explained first. In particular, we obtain an explicit result for the Siegel modular variety $A_2(2)$.

[2] Samuel Le Fourn, Nonvanishing of central values of ${L}$-functions of newforms in ${S}_2 ({\Gamma}_0 (dp^2))$ twisted by quadratic characters, Canadian Mathematical Bulletin , 2017. [ arXiv | bib| DOI]

[+] Abstract
We prove that for $d \in \{ 2,3,5,7,13 \}$ and $K$ a quadratic (or rational) field of discriminant $D$ and Dirichlet character $\chi$, if a prime $p$ is large enough compared to $D$, there is a newform $f \in S_2(\Gamma_0(dp^2))$ with sign $(+1)$ with respect to the Atkin-Lehner involution $w_{p^2}$ such that $L(f \otimes \chi,1) \neq 0$. This result is obtained through an estimate of a weighted sum of twists of $L$-functions which generalises a result of Ellenberg. It relies on the approximate functional equation for the $L$-functions $L(f \otimes \chi, \cdot)$ and a Petersson trace formula restricted to Atkin-Lehner eigenspaces. An application of this nonvanishing theorem will be given in terms of existence of rank zero quotients of some twisted jacobians, which generalises a result of Darmon and Merel.

[1] Samuel Le Fourn, Surjectivity of Galois representations associated with quadratic $\Q$-curves, Mathematische Annalen, 2016. [ arXiv | bib| DOI]

[+] Abstract
We prove in this paper an uniform surjectivity result for Galois representations associated with non-CM $\Q$-curves over imaginary quadratic fields, using various tools for the proof, such as Mazur's method, isogeny theorems, Runge's method and analytic estimates of sums of $L$-functions.

PhD

I did my PhD under the supervision of Pierre Parent (Université de Bordeaux), from 2011 to 2015. Here is the manuscript (in French), titled « Points entiers et rationnels sur des courbes et variétés modulaires de dimension supérieure » (in French) and an abstract.

[+] Abstract

This thesis concerns the study of integral and rational points on some modular curves and varieties. After a brief introduction which describes the motivation and the setting of this topic as well as the main results of this thesis, the manuscript follows a threefold development.

The first chapter focuses on $\Q$-curves, and on the morphisms $\operatorname{Gal}(\Qb / \Q) \rightarrow \operatorname{PGL}_2(\Fp)$ that we can build with a $\Q$-curve for every prime $p$. We prove that, under good hypotheses, for $p$ large enough with respect to the discriminant of the definition field of the $\Q$-curve, such a morphism is surjective, which solves a particular case of Serre's uniformity problem (still open in general). The main tools of the chapter are Mazur's method (based here on results of Ellenberg), Runge's method, and isogeny theorems, following the strategy of Bilu and Parent.

The second chapter covers analytic estimates of weighted sums of $L$-function values of modular forms, in the fashion of techniques designed by Duke and Ellenberg. The initial goal of such a result is the application of Mazur's method in the first chapter.

The third chapter is devoted to the search for generalisations of Runge's method for higher-dimensional varieties. Here we prove anew a result of Levin inspired by this method, before proving an enhanced version called “tubular Runge”, more generally applicable. In the perspective of studying integral points of modular varieties, we finally give an example of application of this theorem to the reduction of an abelian surface in a product of elliptic curves.

Keywords

Elliptic curves, modular curves, Galois representations, Serre's uniformity problem, Mazur's method, isogeny theorems, Petersson trace formula, Runge's method, Siegel modular varieties, theta functions.