arXiv subjects

Elie Studnia

Publications and source records attributed to Elie Studnia.

8 recordsLinked to original sources

arXiv2026

Compactified moduli spaces and Hecke correspondences for elliptic curves with a prescribed $N$-torsion scheme

Given an integer $N \geq 3$, we prove that for any ring $R$ and any finite locally free commutative $R$-group scheme $G$ whose geometric fibres are isomorphic to the $N$-torsion subscheme of some elliptic curve $E$, there is a smooth affine curve $Y_G(N)$ parametrizing elliptic curves over $R$-schemes whose $N$-torsion subscheme is isomorphic to $G$. We also describe compactifications $X_G(N)$ of these curves when $R$ is a regular excellent Noetherian ring in which $N$ is invertible, as well as construct the Hecke correspondences they are endowed with. As an application, we show that the equations for $X_G(N)$ found over base fields for $N=7,8,9,11,13$ (by Halberstadt--Kraus, Poonen--Schaefer--Stoll, Chen and Fisher) are in fact valid over regular excellent Noetherian bases that are $\mathbb{Q}$-algebras. Finally, we describe in detail the equivalence of this construction with the point of view of Galois twists that these authors use.

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arXiv2026

Arithmetic intersections on non-split Cartan modular curves

Let $p$ be a prime number, and let $Δ_1,Δ_2 < 0$ be two coprime fundamental discriminants. When $p$ splits in $\mathbb{Q}(\sqrt{Δ_1})$ and $\mathbb{Q}(\sqrt{Δ_2})$ the height pairings of the corresponding CM divisors on $X_{\mathrm{spl}}^+(p)$ were determined by Gross--Kohnen--Zagier [GKZ87]. When $p$ is inert, we determine the arithmetic intersection numbers of the corresponding divisors on $X_{\mathrm{ns}}^+(p)$ at all finite primes. The key point of our analysis is at the prime of bad reduction $p$: to determine the intersection numbers at $p$, we provide a moduli interpretation for the smooth locus in the regular model of $X_{\mathrm{ns}}^+(p)$ over $\mathrm{Spec}(\mathbb{Z})$ constructed by Edixhoven--Parent [EP24].

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arXiv2026

On obstructions to the Euler system method for Rankin-Selberg convolutions

To apply the Euler system method to a $p$-adic Galois representation $T$, one needs the existence of a $σ\in G_{\mathbb{Q}(μ_{p^{\infty}})}$ such that $V/(σ-1)V$ is free of rank one over the coefficient ring: we say that such a $σ$ is an Euler-suitable element for $V$. Given a non-CM classical newform $f$ of weight $k \geq 2$ and character $χ$, a classical newform $g$ of weight $1$ and character $ψ$, and a prime ideal $\mathfrak{p}$ of residue characteristic $p$ of a sufficiently large number field, we consider the situation where $V=V_{f,g,\mathfrak{p}}$ is the tensor product of the $\mathfrak{p}$-adic representations attached to $f$ and $g$. D. Loeffler asked the following question: is is true that if $χψ\neq 1$, then there is an Euler-suitable element for $V_{f,g,\mathfrak{p}}$ for all but finitely many $\mathfrak{p}$? He gave a positive answer when $f,g$ had coprime conductors. We give several weaker sufficient conditions to answer this question in the affirmative. As an application, we remove some of the technical assumptions in the version of the Bloch-Kato Conjecture proved in arXiv:1503.02888. We also show that the general answer to the question is negative, by constructing a family of counter-examples, and giving additional counter-examples that do not fit in this family.

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arXiv2025

Congruences modulo $23$ to $y^2=x^3-23$ are trivial

We say that two elliptic curves $E$ and $F$ over $\mathbb{Q}$ are congruent modulo a prime $p$ if their $p$-torsion Galois modules (over the algebraic closure of $\mathbb{Q}$) are isomorphic. Such a congruence is called trivial if there is a rational isogeny between $E$ and $F$ with degree prime to $p$. A version of the Frey-Mazur conjecture states that any congruence modulo any prime $p \geq 19$ is trivial. Given an elliptic curve $E/\mathbb{Q}$ and a prime $p$, it is well-known that there is a twist of the classical modular curve $X(p)$ whose rational points describe the elliptic curves congruent to $E$ modulo $p$. In this article, we apply Mazur's strategy to determine the rational points of such a twisted modular curve under certain assumptions. This involves, among others, the determination of the previously unknown Tate module of its Jacobian and new instances of the Birch and Swinnerton--Dyer conjecture (for abelian varieties not of $\mathrm{GL}_2$-type). In particular, we determine an explicit bound on the conductor of any elliptic curve congruent modulo $p$ to $y^2=x^3-p$ when $p$ is prime and congruent to $5$ modulo $9$, and deduce that any congruence modulo $23$ to $y^2=x^3-23$ is trivial.

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arXiv2025

L-functions and rational points for Galois twists of modular curves

Let $p$ be an odd prime and $E/\mathbb{Q}$ be a rational elliptic curve. There is a smooth affine curve $Y_E(p)$ whose rational points parametrize elliptic curves $F/\mathbb{Q}$ such that $F[p]$ and $E[p]$ are isomorphic Galois modules. This thesis manuscript is an attempt at applying Mazur's strategy to determine the rational points of $Y_E(p)$. One of the results that we prove, as a consequence of the approach, is the following: if $E,F/\mathbb{Q}$ are elliptic curves with isomorphic $23$-torsion Galois modules and $E$ has Weierstrass equation $y^2=x^3-23$, then $E$ and $F$ are isogenous. In Chapter 1, we prove the existence of the curve $Y_E(p)$ and its compactification $X_E(p)$ over more general bases as moduli schemes in the sense of Katz-Mazur, describe their Hecke correspondences, and construct $X_E(p)$ as a Galois twist of $X(p)$. In Chapters 2 and 3, we determine the Tate module of the Jacobians $J_E^α(p)$ of the connected components of $X_E(p)$, thereby refining a result by Virdol. In Chapter 4, we study the factors of the $L$-function of $J_E^α(p)$ previously determined. In general, little can be said, because proving the automorphy of the Galois representations involved seems currently out of reach. However, under suitable automorphy assumptions, we determine the signs of the functional equations satisfied by these $L$-functions. In Chapter 5, we discuss the special case where the image of the Galois action on $E[p]$ is contained in the normalizer of a nonsplit Cartan subgroup. The factors of $L(J_E^α(p),s)$ then satisfy functional equations whose signs we determine. Using the Beilinson-Flach Euler system and results about large Galois image (from Chapter 6), we show that most factors of $J_E^α(p)$ satisfy the Bloch-Kato in rank zero, and prove a formal immersion theorem for certain factors at all but an easily computable set of primes.

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arXiv2023

Refined Selmer equations for the thrice-punctured line in depth two

In [Kim05], Kim gave a new proof of Siegel's Theorem that there are only finitely many $S$-integral points on $\mathbb P^1_{\mathbb Z}\setminus\{0,1,\infty\}$. One advantage of Kim's method is that it in principle allows one to actually find these points, but the calculations grow vastly more complicated as the size of $S$ increases. In this paper, we implement a refinement of Kim's method to explicitly compute various examples where $S$ has size $2$ which has been introduced in [BD19]. In so doing, we exhibit new examples of a natural generalisation of a conjecture of Kim.

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arXiv2020

Quantum Limits for Harmonic Oscillator

In this note we consider high energy eigenfunctions of the harmonic oscillator in $\mathbb{R}^d$ and prove that any invariant measure on the energy surface can be written as a weak limit of eigenfunctions.

math.AP↗

arXiv2019

Quantum ergodicity for pseudo-Laplacians

We prove quantum ergodicity for the eigenfunctions of the pseudo-Laplacian on Riemannian surfaces with finitely many hyperbolic cusps and ergodic geodesic flow.

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