Narrow Results

In a previous paper, the second author proved that the equation

This paper is devoted to the generalized Fermat equation x^p + y^q = z^r, where p, q and r are integers, and x, y and z are nonzero coprime integers. We begin by surveying the exponent triples (p, q, r), including a number of infinite families, for which the equation has been solved to date, detailing the techniques involved. In the remainder of the paper, we attempt to solve the remaining infinite families of generalized Fermat equations that appear amenable to current techniques. While the main tools we employ are based upon the modularity of Galois representations (as is indeed true with all previously solved infinite families), in a number of cases we are led via descent to appeal to a rather intricate combination of multi-Frey techniques.

Let $E$ be a $\mathbb{Q}$-curve without complex multiplication. We address the problem of deciding whether $E$ is geometrically isomorphic to a strongly modular $\mathbb{Q}$-curve. We show that the question has a positive answer if and only if $E$ has a model that is completely defined over an abelian number field. Next, if $E$ is completely defined over a quadratic or biquadratic number field $L$, we classify all strongly modular twists of $E$ over $L$ in terms of the arithmetic of $L$. Moreover, we show how to determine which of these twists come, up to isogeny, from a subfield of $L$.

The purpose of this paper is to show how the modular method together with different techniques can be used to prove non-existence of primitive non-trivial solutions of the equation $x^2+d{y}^6=z^p$ for square-free values $1\le d\le 20$. The key ingredients are: the approach presented in [A. Pacetti and L. V. Torcomian, $\mathbb{Q}$-curves, Hecke characters and some Diophantine equations, Math. Comp. 91(338) (2022) 2817–2865] (in particular its recipe for the space of modular forms to be computed) together with the use of the symplectic method (as developed in [E. Halberstadt and A. Kraus, Courbes de Fermat: Résultats et problèmes, J. Reine Angew. Math. 548 (2002) 167–234], although we give a variant over ramified extensions needed in our applications) to discard solutions and the use of a second Frey curve, aiming to prove large image of residual Galois representations.

Let N ≡ 1(mod 4) be a square-free positive integer, let ε be the primitive quadratic character of conductor N, and let f ∈ S₂(Γ₀(N), ε) be a newform with fourier coefficients in a quadratic field. Shimura associates to f an elliptic curve E defined over the real quadratic field which is isogenous over F to its Galois conjugate. Let M/F be a quadratic extension of F which is neither CM nor totally real, and denote by E_M the twist of E with respect to M/F. The first main result of this article is that, if L′(E_M/F, 1) ≠ 0, then E_M(F) has rank one and III(E/F) is finite. The proof rests on a fundamental result⁴² of Yuan, Zhang and Zhang and on an explicit Heegner point in E_M(F) arising from the modular parametrization X₁(N) → E. We formulate a conjecture relating this Heegner point to Stark-Heegner points⁸ arising from ATR cycles of real dimension one on Hilbert modular surfaces, and present some numerical evidence for this conjecture.