Searcharxiv⌕ Search

arXiv subjects

Luis Dieulefait

Publications and source records attributed to Luis Dieulefait.

At least 19 recordsLinked to original sources
arXiv2026

On the computation of base-change lifts and lifts of Hida families

We derive an explicit formula for the Hecke eigenvalues of a Hilbert modular form which is a base-change lift of a classical newform to a totally real number field. We show that for a totally real Galois number field $F$ the $L$-function of a base-change lifted form can be factorized as a product of twists of the $L$-function of the underlying classical form over irreducible representations of $\mathrm{Gal}(F / \mathbb{Q})$. Moreover, we use the formula for the Hecke eigenvalues of a base-change lift to prove the existence of a base-change lift of a Hida family. In particular, we show that a Hida family of classical Hecke eigenforms can be lifted to a formal power series that specializes to the base-change lifts of the Hida family of classical cusp forms.

math.NT↗
arXiv2025

Revisiting the Fermat-type equation $x^{13} + y^{13} = 3z^7$

We solve the Fermat-type equation \[ x^{13} + y^{13} = 3 z^7, \qquad \gcd(x,y,z) = 1 \] combining a unit sieve, the multi-Frey modular method, level raising, computations of systems of eigenvalues modulo 7 over a totally real field, and results for reducibility of certain Galois representations.

math.NT↗
arXiv2025

On Fermat's Last Theorem over the $\mathbb{Z}_3$-extension of $\mathbb{Q}$ and other fields

The main result of the present article is a proof of Fermat's Last Theorem for sufficiently large prime exponents $p$ with $p \equiv 2 \pmod{3}$ over certain number fields. A particular case of these fields are the maximal real subfields of the cyclotomic extensions $\mathbb{Q}(\zeta_{3^n})$ for every $n$. Our strategy consists in combining the modular method with a generalization of an arithmetic result of Pomey to these fields.

math.NT↗
arXiv2023

Modular supercuspidal lifts of weight $2$

Let $F/\mathbb{Q}$ be any totally real number field and $\frak{N}$ an ideal of its ring of integers of norm $N$ and define, for every even $n$, the $[F:\mathbb{Q}]$-dimensional multiweight $\textbf{n}=(n,...,n)$. We prove that for a non CM Hilbert cuspidal Hecke eigenform for $F$, say $f\in S_{\textbf{k}}(\Gamma_0(\frak{N}))$ with $k>2$ even, and a prime $p>\max\{k+1,6\}$ totally split in $F$ such that $p\nmid N$ and such that the residual mod $p$ representation $\overline{\rho}_f$ satisfies that $\mathrm{SL}_2(\mathbb{F}_p)\subseteq \mathrm{Im}(\overline{\rho}_f)$, there exists a lift $\rho_g$ associated to a Hilbert modular cuspform for $F$, say $g\in S_{\textbf{2}}(\frak{N}p^2,\epsilon)$ for some Nebentypus character $\epsilon$ which is supercuspidal at each prime of $F$ over $p$. We also observe that our techniques provide an alternative proof to the corresponding statement for classical Hecke cuspforms already proved by Khare \cite{khare} with classical techniques. Finally, we take the opportunity to include a corrigenda for \cite{dieulefait} which follows from our main result, which provides a congruence that puts the micro good dihedral prime in the level.

math.NT↗
arXiv2022

On Darmon's program for the generalized Fermat equation, I

In 2000, Darmon described a program to study the generalized Fermat equation using modularity of abelian varieties of $\mathrm{GL}_2$-type over totally real fields. The original approach was based on hard open conjectures, which have made it difficult to apply in practice. In this paper, building on the progress surrounding the modular method from the last two decades, we analyze and expand the current limits of this program by developing all the necessary ingredients to use Frey abelian varieties for new Diophantine applications. In particular, we deal with all but the fifth and last step in the modular method for Fermat equations of signature $(r,r,p)$ in almost full generality. As an application, for all integers $n \geq 2$, we give a resolution of the generalized Fermat equation $x^{11} + y^{11} = z^n$ for solutions $(a,b,c)$ such that $a + b$ satisfies certain $2$- or $11$-adic conditions. Moreover, the tools developed can be viewed as an advance in addressing a difficulty not treated in Darmon's original program: even assuming `big image' conjectures about residual Galois representations, one still needs to find a method to eliminate Hilbert newforms at the Serre level which do not have complex multiplication. In fact, we are able to reduce the problem of solving $x^5 + y^5 = z^p$ to Darmon's `big image conjecture', thus completing a line of ideas suggested in his original program, and notably only needing the Cartan case of his conjecture.

math.NT↗
arXiv2022

Some extensions of the modular method and Fermat equations of signature $(13,13,n)$

We provide several extensions of the modular method which were motivated by the problem of completing previous work to prove that, for any integer $n \geq 2$, the equation \[ x^{13} + y^{13} = 3 z^n \] has no non-trivial solutions. In particular, we present four elimination techniques which are based on: (1) establishing reducibility of certain residual Galois representations over a totally real field; (2) generalizing image of inertia arguments to the setting of abelian surfaces; (3) establishing congruences of Hilbert modular forms without the use of often impractical Sturm bounds; and (4) a unit sieve argument which combines information from classical descent and the modular method. The extensions are of broader applicability and provide further evidence that it is possible to obtain a complete resolution of a family of generalized Fermat equations by remaining within the framework of the modular method. As a further illustration of this, we complete a theorem of Anni-Siksek to show that, for $\ell, m\ge 5$, the only solutions to the equation $x^{2\ell} + y^{2m} = z^{13}$ are the trivial ones.

math.NT↗
arXiv2021

Potentially diagonalizable modular lifts of large weight

We prove that for a Hecke cuspform $f\in S_k(Γ_0(N),χ)$ and a prime $l>\max\{k,6\}$ such that $l\nmid N$, there exists an infinite family $\{k_r\}_{r\geq 1}\subseteq\mathbb{Z}$ such that for each $k_r$, there is a cusp form $f_{k_r}\in S_{k_r}(Γ_0(N),χ)$ such that the Deligne representation $ρ_{f_{k_r,l}}$ is a crystaline and potentially diagonalizable lift of $\overlineρ_{f,l}$. When $f$ is $l$-ordinary, we base our proof on the theory of Hida families, while in the non-ordinary case, we adapt a local-to-global argument due to Khare and Wintenberger in the setting of their proof of Serre's modularity conjecture, together with a result on existence of lifts with prescribed local conditions over CM fields, a flatness result due to Böckle and a local dimension result by Kisin. We discuss the motivation and tentative future applications of our result in ongoing research on the automorphy of $\mathrm{GL}_{2n}$-representations in the higher level case.

math.NT↗
arXiv2019

Factorization and malleability of RSA modules, and counting points on elliptic curves modulo N

In this paper we address two different problems related with the factorization of an RSA module N. First we can show that factoring is equivalent in deterministic polynomial time to counting points on a pair of twisted Elliptic curves modulo N. Also we settle the malleability of factoring an RSA module, as described in [9], using the number of points of a single elliptic curve modulo N, and Coppersmith's algorithm.

math.NT↗
arXiv2018

On the images of the Galois representations attached to generic automorphic representations of GSp(4)

By making use of Langlands functoriality between GSp(4) and GL(4), we show that the images of the Galois representations attached to "genuine" globally generic automorphic representations of GSp(4) are "large" for almost every prime. Moreover, by using the notion of (n,p)-groups (introduced by Khare, Larsen and Savin) and generic Langlands functoriality from SO(5) to GL(4) we construct automorphic representations of GSp(4) such that the compatible system attached to them has large image for all primes.

math.NT↗
arXiv2018

Raising the level at your favorite prime

In this paper we prove a level raising theorem for some weight $2$ trivial character newforms at almost every prime $p$. This is done by ignoring the residue characteristic at which the level raising appears.

math.NT↗
arXiv2018

On the number of Galois orbits of newforms

Counting the number of Galois orbits of newforms in $S_k(Γ_0(N))$ and giving some arithmetic sense to this number is an interesting open problem. The case $N=1$ corresponds to Maeda's conjecture (still an open problem) and the expected number of orbits in this case is 1, for any $k \ge 16$. In this article we give local invariants of Galois orbits of newforms for general $N$ and count their number. Using an existence result of newforms with prescribed local invariants we prove a lower bound for the number of non-CM Galois orbits of newforms for $Γ_0(N)$ for large enough weight $k$ (under some technical assumptions on $N$). Numerical evidence suggests that in most cases this lower bound is indeed an equality, thus we leave as a Question the possibility that a generalization of Maeda's conjecture could follow from our work. We finish the paper with some natural generalizations of the problem and show some of the implications that a generalization of Maeda's conjecture has.

math.NT↗
arXiv2017

Constructing Hilbert modular forms whithout exceptional primes

In this paper we construct families of Hilbert modular newforms without exceptional primes. This is achieved by generalizing the notion of good-dihedral primes, introduced by Khare and Wintenberger in their proof of Serre's modularity conjecture, to totally real fields.

math.NT↗
arXiv2017

On congruences between normalized eigenforms with different sign at a Steinberg prime

Let $f$ be a newform of weight $2$ on $Γ_0(N)$ with Fourier $q$-expansion $f(q)=q+\sum_{n\geq 2} a_n q^n$, where $Γ_0(N)$ denotes the group of invertible matrices with integer coefficients, upper triangular mod $N$. Let $p$ be a prime dividing $N$ once, $p\parallel N$, a Steinberg prime. Then, it is well known that $a_p\in\{1,-1\}$. We denote by $K_f$ the field of coefficients of $f$. Let $λ$ be a finite place in $K_f$ not dividing $2p$ and assume that the mod $λ$ Galois representation attached to $f$ is irreducible. In this paper we will give necessary and sufficient conditions for the existence of another Hecke eigenform $f'(q)=q+\sum_{n\geq 2} a'_n q^n$ $p$-new of weight $2$ on $Γ_0(N)$ and a finite place $λ'$ of $K_{f'}$ such that $a_p=-a'_p$ and the Galois representations $\barρ_{f,λ}$ and $\barρ_{f',λ'}$ are isomorphic.

math.NT↗
arXiv2017

A multi-Frey approach to Fermat equations of signature $(r,r,p)$

In this paper, we give a resolution of the generalized Fermat equations $$x^5 + y^5 = 3 z^n \text{ and } x^{13} + y^{13} = 3 z^n,$$ for all integers $n \ge 2$, and all integers $n \ge 2$ which are not a multiple of $7$, respectively, using the modular method with Frey elliptic curves over totally real fields. The results require a refined application of the multi-Frey technique, which we show to be effective in new ways to reduce the bounds on the exponents $n$. We also give a number of results for the equations $x^5 + y^5 = d z^n$, where $d = 1, 2$, under additional local conditions on the solutions. This includes a result which is reminiscent of the second case of Fermat's Last Theorem, and which uses a new application of level raising at $p$ modulo $p$.

math.NT↗
arXiv2016

Automorphy of $\mathrm{GL}_2\otimes \mathrm{GL}_n$ in the self-dual case

In this paper we establish a new case of Langlands functoriality. More precisely, we prove that the tensor product of the compatible system of Galois representations attached to a level-1 classical modular form and the compatible system attached to an n-dimensional RACP automorphic representation of GL_n of the adeles of Q is automorphic, for any positive integer n, under some natural hypotheses (namely regularity and irreducibility), and a mild restriction on the level of the n-dimensional representation.

math.NT↗
arXiv2016

Possible connection between a generalized Maeda's conjecture and local types

Here we follow on the proposed generalization of Maeda's conjecture made in [2]. We report on computations that suggest a relation between the number of local types and the number of non-CM newform Galois orbits. We extend the conjecture into spaces with non-trivial Nebentypus and provide a formula for the number of non-CM orbits for all levels and trivial Nebentypus. We also provide some numerical evidence towards further generalizations of this conjecture to totally real fields as well as further strengthening of it by proposing a structure for the corresponding Galois groups.

math.NT↗
arXiv2016

A result on the equation $x^p + y^p = z^r$ using Frey abelian varieties

We prove a diophantine result on generalized Fermat equations of the form $x^p + y^p = z^r$ which for the first time requires the use of Frey abelian varieties of dimension $\geq 2$ in Darmon's program. For that, we provide an irreducibility criterion for the mod $\mathfrak{p}$ representations attached to certain abelian varieties of $\text{GL}_2$-type over totally real fields.

math.NT↗