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Samuel Le Fourn

Publications and source records attributed to Samuel Le Fourn.

13 recordsLinked to original sources
arXiv2025

Percolative properties of the random coprime colouring

Given $u$ and $v$ in $\mathbb{Z}^d$, say that $u$ is visible from $v$ if the segment from $u$ to $v$ contains exactly two elements, which are $u$ and $v$. Take $X$ "uniformly at random in $\mathbb{Z}^d$" and colour each vertex $u$ of $\mathbb{Z}^d$ in white if $u$ is visible from $X$ and in black otherwise. Previous independent works of Pleasants-Huck and of the third author give a precise meaning to this definition. This paper is dedicated to the study of this random colouring from the point of view of percolation theory: given a reasonable graph structure on $\mathbb{Z}^d$, how many infinite black (resp. white) connected components are there?

math.PR↗
arXiv2025

Rational points on $X_0(N)^*$ when $N$ is non-squarefree

Let $N$ be a non-squarefree integer such that the quotient $X_0(N)^*$ of the modular curve $X_0(N)$ by the full group of Atkin-Lehner involutions has positive genus. Elkies conjectures that the rational points on $X_0(N)^*$ are only cusps or CM points when $N$ is large enough. We establish an integrality result for the $j$-invariants of non-cuspidal rational points on $X_0(N)^*$, representing a significant step toward resolving a key subcase of Elkies' conjecture. To this end, we prove the existence of rank-zero quotients of certain modular Jacobians $J_0(pq)$. Furthermore, we provide a complete classification of the rational points on $X_0(N)^*$ of genus $1 \leq g \leq 5$, when they are finite. In the process we identify exceptional rational points on $X_0(147)^*$ and $X_0(75)^*$ which were not known before.

math.NT↗
arXiv2025

Variation of height in an isogeny class over a function field

We give optimal estimates on the variation of the differential and modular heights within an isogeny class of abelian varieties defined over the function field of a curve (in any characteristic). We also prove a parallelogram inequality for abelian varieties in this context, and deduce corollaries of these results.

math.NT↗
arXiv2023

Explicit height estimates for CM curves of genus 2

In this paper we make explicit the constants of Habegger and Pazuki's work from 2017 on bounding the discriminant of cyclic Galois CM fields corresponding to genus 2 curves with CM by them and potentially good reduction outside a predefined set of primes. We also simplify some of the arguments.

math.NT↗
arXiv2023

Torsion bounds for a fixed abelian variety and varying number field

Let $A$ be an abelian variety defined over a number field $K$. For a finite extension $L/K$, the cardinality of the group $A(L)_{\operatorname{tors}}$ of torsion points in $A(L)$ can be bounded in terms of the degree $[L:K]$. We study the smallest real number $β_A$ such that for any finite extension $L/K$ and $\varepsilon>0$, we have $|A(L)_{\operatorname{tors}}| \leq C \cdot [L:K]^{β_A+\varepsilon}$, where the constant $C$ depends only on $A$ and $\varepsilon$ (and not $L$). Assuming the Mumford--Tate conjecture for $A$, we will show that $β_A$ agrees with the conjectured value of Hindry and Ratazzi. We also give a similar bound for the maximal order of a torsion point in $A(L)$.

math.NT↗
arXiv2020

Residual Galois representations of elliptic curves with image contained in the normaliser of a non-split Cartan

It is known that if $p>37$ is a prime number and $E/\mathbb{Q}$ is an elliptic curve without complex multiplication, then the image of the mod $p$ Galois representation $$ \barρ_{E,p}:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\rightarrow \operatorname{GL}(E[p]) $$ of $E$ is either the whole of $\operatorname{GL}(E[p])$, or is \emph{contained} in the normaliser of a non-split Cartan subgroup of $\operatorname{GL}(E[p])$. In this paper, we show that when $p>1.4\times 10^7$, the image of $\barρ_{E,p}$ is either $\operatorname{GL}(E[p])$, or the \emph{full} normaliser of a non-split Cartan subgroup. We use this to show the following result, partially settling a question of Najman. For $d\geq 1$, let $I(d)$ denote the set of primes $p$ for which there exists an elliptic curve defined over $\mathbb{Q}$ and without complex multiplication admitting a degree $p$ isogeny defined over a number field of degree $\leq d$. We show that, for $d\geq 1.4\times 10^7$, we have $$ I(d)=\{p\text{ prime}:p\leq d-1\}. $$

math.NT↗
arXiv2019

Quadratic Chabauty for modular curves and modular forms of rank one

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 $X_{\mathrm{ns} }^+ (N)$ and $X_0 ^+ (N)$ 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.

math.NT↗
arXiv2019

Tubular approaches to Baker's method for curves and varieties

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.

math.NT↗
arXiv2017

A "tubular" variant of Runge's method in all dimensions, with applications to integral points on Siegel modular varieties

Runge's method is a tool to figure out integral points on 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 $A_2(2)$.

math.NT↗
arXiv2016

Nonvanishing of central values of $L$-functions of newforms in $S_2 (Γ_0 (dp^2))$ twisted by quadratic characters

We prove that for $d \in \{ 2,3,5,7,13 \}$ and $K$ a quadratic (or rational) field of discriminant $D$ and Dirichlet character $χ$, if a prime $p$ is large enough compared to $D$, there is a newform $f \in S_2(Γ_0(dp^2))$ with sign $(+1)$ with respect to the Atkin-Lehner involution $w_{p^2}$ such that $L(f \otimes χ,1) \neq 0$. 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 $L(f \otimes χ, \cdot)$ 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.

math.NT↗
arXiv2013

Surjectivity of Galois representations associated with quadratic Q-curves

We prove in this paper an uniform surjectivity result for Galois representations associated with non-CM $\mathbb{Q}$-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.

math.NT↗