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Nuno Freitas

Publications and source records attributed to Nuno Freitas.

At least 19 recordsLinked to original sources
arXiv2026

Congruences and ramified primes in fields of coefficients of newforms

We investigate the splitting behavior of $\ell$ in the coefficient field of a newform $f$ of level $N$, under the assumption that $f$ is congruent modulo a prime above $\ell$ to another newform $g$ whose level divides $N/p^2$ for some prime $p\mid N$. In particular, we show that the maximal real subfield of the $\ell$-th cyclotomic field, $\mathbb{Q}(\zeta_\ell + \zeta_\ell^{-1})$, is contained in the coefficient field of $f$. We conclude by presenting explicit examples that illustrate these results.

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arXiv2025

Revisiting the equation $x^2+y^3=z^p$

Let $E/\mathbb Q$ be an elliptic curve and $p \geq 3$ a prime. The modular curve $X_E^-(p)$ parameterizes elliptic curves with $p$-torsion modules anti-symplectically isomorphic to $E[p]$. The work of Freitas--Naskr\k{e}cki--Stoll uses the modular method to show that all primitive non-trivial solutions of the Fermat-type equation $x^2 + y^3 = z^p$ give rise to rational points on $X_E^-(p)$ with $E \in \{27a1,54a1,96a1,288a1,864a1,864b1,864c1 \}$. Using a criterion classifying the existence of local points due to the first two authors, we show that, for $E$ any of the curves with conductor 864 and certain primes $p \equiv 19 \pmod{24}$, we have $X_E^-(p)(\mathbb Q_\ell) = \emptyset$. Furthermore, for each $E$ in the list and any $p$, we prove that either $X_E^-(p)$ can be discarded using the same criterion, or it cannot be discarded using purely local information.

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arXiv2025

On inertial types of elliptic curves

We classify the inertial Weil-Deligne types arising from elliptic curves over all finite extensions $F/\mathbb Q_p$. Based on this classification, we give a fully explicit description of the types and implement an algorithm that computes all inertial types of elliptic curves defined over a given $F$. As an application, we determine all inertial types arising from elliptic curves over any extension $F/\mathbb Q_p$ of degree at most 3.

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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.

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arXiv2025

On the generalized Fermat equation $x^{13} + y^{13} = z^n$

Let $n \in \mathbb{Z}_{\geq 2}$. We study the generalized Fermat equation \[x^{13}+y^{13}=z^n, \quad x,y,z \in \mathbb{Z}, \quad \gcd(x,y,z)=1.\] Using a combination of techniques, including the modular method, classical descent, unit sieves, and Chabauty and Mordell--Weil sieve methods over number fields, we show that for $n=5$ all its solutions $(a,b,c)$ are trivial, i.e. satisfy $abc=0$. Under the assumption of GRH, we also show that for $n=7$ there are only trivial solutions. Furthermore, we provide partial results towards solving the equation for general $n \in \mathbb{Z}_{\geq 2}$, in particular that any solution $(a,b,c)$ with $13\mid c$ is trivial.

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arXiv2025

Comparing Galois representations in the residually reducible case

Let $n \geq 2$ and $p$ be a prime. Let $K$ be a number field and consider two Galois representations $\rho_1, \rho_2 : \operatorname{Gal}(\overline{K} / K) \to \operatorname{GL}_n(\mathbb{Z}_p)$ having residual image a $p$-group. We explain and implement an algorithm that makes effective a result of Lo\"ic Greni\'e to decide wether the semisimplifications of $\rho_1$ and $\rho_2$ are isomorphic. As an application, we show that an irreducible representation $\rho : G_{\mathbb{Q}(\sqrt{-3})} \to \operatorname{GL}_2(\mathbb{Z}_3)$ unramified outside 3 is determined by the characteristic polynomials of Frobenius elements at five primes of small norm. As an additional check, we apply it to a 2-adic example studied by Greni\'e, recovering Greni\'e's result in a fully automated way.

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arXiv2025

Symplectic criteria for elliptic curves, revisited

Let $\ell$ and $p \geq 3$ be different primes. Let $E/\mathbb{Q}_\ell$ and $E'/\mathbb{Q}_\ell$ be elliptic curves with isomorphic $p$-torsion. Assume that $E$ has potentially multiplicative reduction. We classify when all $G_{\mathbb{Q}_\ell}$-isomorphisms $\phi : E[p] \to E'[p]$ have the same symplectic type and prove two new criteria to determine the type in that case. In particular, when both curves have multiplicative reduction, our results cover the case of unramified $p$-torsion which is not covered by the original criterion due to Kraus and Oesterl\'e. We also give a variant of a symplectic criterion for the case when both $E$ and~$E'$ have good reduction and provide an algorithm to apply it. As an application, we determine the symplectic type of all the mod $p \geq 5$ congruences between rational elliptic curves with conductor $\leq 500 000$ that satisfy the hypothesis of either of our criteria at some prime~$\ell$.

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arXiv2025

Local points on twists of $X(p)$ with applications

Let $E/\mathbb Q$ be an elliptic curve and $p \geq 3$ a prime. The modular curve $X_E^-(p)$ parametrizes elliptic curves with $p$-torsion modules anti-symplectically isomorphic to $E[p]$. We give a complete classification of when $X_E^-(p)(\mathbb Q_\ell)$ is non-empty, for all primes $\ell\neq p$; our result also includes $\ell=p$ in most cases when $E$ is semistable at $p$. We give two different applications. First, we classify CM curves $E/\mathbb Q$ where the modular curve $X_E^-(p)$ is a counterexample to the Hasse principle for infinitely many $p$. Assuming the Frey--Mazur conjecture, we prove that for at least $60\%$ of rational elliptic curves $E$, the modular curve $X_E^-(p)$ is a counterexample to the Hasse principle for at least $50\%$ of primes $p$. Secondly, we introduce a new technique to the elimination stage of the modular method and apply it to show that $x^3+y^3=5^\alpha z^p$ has no non-trivial primitive solutions for various primes $p$ satisfying $(\alpha/p)=-1$. Moreover, as a by-product of our work, we simplify the assumptions of several local symplectic criteria due to the first author and Alain Kraus.

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arXiv2025

The Generalized Fermat Equation $x^2 + y^3 = z^{25}$

We consider the generalized Fermat equation (*) $x^2 + y^3 = z^{25}$. Using the known parameterization of the primitive integral solutions to $x^2 + y^3 = z^5$ (due to Edwards), we reduce the solution of (*) to the solution of five specific equations of the form $H(u,v) = w^5$, where $H$ is homogeneous of degree $10$ with coefficients in a sextic number field $K$, $u$ and $v$ are coprime (rational) integers, and $w \in K$.

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arXiv2024

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 $p$.

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arXiv2023

On Darmon's program for the Generalized Fermat equation, II

We obtain additional Diophantine applications of the methods surrounding Darmon's program for the generalized Fermat equation developed in the first part of this series of papers. As a first application, we use a multi-Frey approach combining two Frey elliptic curves over totally real fields, a Frey hyperelliptic curve over $\mathbb{Q}$ due to Kraus, and ideas from the Darmon program to give a complete resolution of the generalized Fermat equation $$x^7 + y^7 = 3 z^n$$ for all integers $n \ge 2$. Moreover, we explain how the use of higher dimensional Frey abelian varieties allows a more efficient proof of this result due to additional structures that they afford, compared to using only Frey elliptic curves. As a second application, we use some of these additional structures that Frey abelian varieties possess to show that a full resolution of the generalized Fermat equation $x^7 + y^7 = z^n$ depends only on the Cartan case of Darmon's big image conjecture. In the process, we solve the previous equation for solutions $(a,b,c)$ such that $a$ and $b$ satisfy certain $2$ or $7$-adic conditions and all $n \ge 2$.

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arXiv2023

Two results on $x^r + y^r = dz^p$

This note proves two theorems regarding Fermat-type equation $x^r + y^r = dz^p$ where $r \geq 5$ is a prime. Our main result shows that, for infinitely many integers~$d$, the previous equation has no non-trivial primitive solutions such that $2 \mid x+y$ or $r \mid x+y$, for a set of exponents $p$ of positive density. We use the modular method with a symplectic argument to prove this result.

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arXiv2022

On the symplectic type of isomorphims of the p-torsion of elliptic curves

Let $p \geq 3$ be a prime. Let $E/\mathbb{Q}$ and $E'/\mathbb{Q}$ be elliptic curves with isomorphic $p$-torsion modules $E[p]$ and $E'[p]$. Assume further that either (i) every $G_\mathbb{Q}$-modules isomorphism $ϕ: E[p] \to E'[p]$ admits a multiple $λ\cdot ϕ$ with $λ\in \mathbb{F}_p^\times$ preserving the Weil pairing; or (ii) no $G_\mathbb{Q}$-isomorphism $ϕ: E[p] \to E'[p]$ preserves the Weil pairing. This paper considers the problem of deciding if we are in case (i) or (ii). Our approach is to consider the problem locally at a prime $\ell \neq p$. Firstly, we determine the primes $\ell$ for which the local curves $E/\mathbb{Q}_\ell$ and $E'/\mathbb{Q}_\ell$ contain enough information to decide between (i) or (ii). Secondly, we establish a collection of criteria, in terms of the standard invariants associated to minimal Weierstrass models of $E/\mathbb{Q}_\ell$ and $E'/\mathbb{Q}_\ell$, to decide between (i) and (ii). We show that our results give a complete solution to the problem by local methods away from $p$. We apply our methods to show the non-existence of rational points on certain hyperelliptic curves of the form $y^2 = x^p - \ell$ and $y^2 = x^p - 2\ell$ where $\ell$ is a prime; we also give incremental results on the Fermat equation $x^2 + y^3 = z^p$. As a different application, we discuss variants of a question raised by Mazur concerning the existence of symplectic isomorphisms between the $p$-torsion of two non-isogenous elliptic curves defined over $\mathbb{Q}$.

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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.

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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.

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arXiv2021

Local criteria for the unit equation and the asymptotic Fermat's Last Theorem

Let F be a totally real number field of odd degree. We prove several purely local criteria for the asymptotic Fermat's Last Theorem to hold over F, and also for the non-existence of solutions to the unit equation over F. For example, if 2 totally ramifies and 3 splits completely in F, then the asymptotic Fermat's Last Theorem holds over F.

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arXiv2020

On the unit equation over cyclic number fields of prime degree

Let $\ell \ne 3$ be a prime. We show that there are only finitely many cyclic number fields $F$ of degree $\ell$ for which the unit equation $$λ+ μ= 1, \qquad λ,~μ\in \mathcal{O}_F^\times$$ has solutions. Our result is effective. For example, we deduce that the only cyclic quintic number field for which the unit equation has solutions is $\mathbb{Q}(ζ_{11})^+$.

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