arXiv subjects
Publications and source records attributed to Pedro-José Cazorla García.
In this paper, we develop the modular method for the generalized Fermat equation appearing in the title, within the framework of Darmon's program and using Frey hyperelliptic curves. As an application, we study a conjecture of Laradji, Mignotte, and Tzanakis concerning the equation $5x^2+q^{2n}=y^5$.
In his breakthrough article, Darmon presented a program to study Generalized Fermat Equations (GFE) via abelian varieties of $\text{GL}_2$-type over totally real fields. So far, only Jacobians of some Frey hyperelliptic curves have been used with that purpose. In the present article, we show how most of the known Frey hyperelliptic curves are particular instances of a more general biparametric family of hyperelliptic curves $C(z,s)$. Then, we apply the cluster picture methodology to compute the conductor of $C(z,s)$ at all odd places. As a Diophantine application, we specialize $C(z,s)$ in some particular values $z_0$ and $s_0$, and we find the conductor exponent at odd places of the natural Frey hyperelliptic curves attached to $Ax^p+By^p=Cz^r$ and $Ax^r+By^r=Cz^p$, generalizing the results due to Azon, Curcó-Iranzo, Khawaja, Maistret and Mocanu, and opening the door for future research in GFE with coefficients. Moreover, we show how a new Frey hyperelliptic curve for $Ax^2+By^r=Cz^p$ can be constructed in this way, giving new results on the conductor exponents for this equation. Finally, following the recent approach by Golfieri and Pacetti, we consider the Frey representations attached to a general signature $(q,r,p)$ via hypergeometric motives and, using $C(z,s)$, we compute the wild part of the conductor exponent at primes above $q$ and $r$ of the residual representation modulo a prime above $p$.
Generalised Pythagorean triples are integer tuples $(x,y,z)$ satisfying the equation $E_{a,b,c}: ax^2+by^2+cz^2=0$. A significant amount of research has been devoted towards understanding generalised Pythagorean triples and, in particular, we can now determine whether $E_{a,b,c}$ has solutions and find them in a computationally effective manner. In this paper, we consider an extension of generalised Pythagorean triples to number fields $K$. In particular, we survey and extend the existing results over $\mathbb{Q}$ for determining if $E_{a,b,c}$ has solutions over number fields and if so, to find and parameterise them, as well as to find a minimal solution. Throughout the text, we incorporate numerous examples to make our results accessible to all researchers interested in the topic of generalised Pythagorean triples.
Perfect error correcting codes allow for an optimal transmission of information while guaranteeing error correction. For this reason, proving their existence has been a classical problem in both pure mathematics and information theory. Indeed, the classification of the parameters of $e-$error correcting perfect codes over $q-$ary alphabets was a very active topic of research in the late 20th century. Consequently, all parameters of perfect $e-$error correcting codes were found if $e \ge 3$, and it was conjectured that no perfect $2-$error correcting codes exist over any $q-$ary alphabet, where $q > 3$. In the 1970s, this was proved for $q$ a prime power, for $q = 2^r3^s$ and for only $7$ other values of $q$. Almost $50$ years later, it is surprising to note that there have been no new results in this regard and the classification of $2-$error correcting codes over non-prime power alphabets remains an open problem. In this paper, we use techniques from the resolution of generalised Ramanujan--Nagell equation and from modern computational number theory to show that perfect $2-$error correcting codes do not exist for $172$ new values of $q$ which are not prime powers, substantially increasing the values of $q$ which are now classified. In addition, we prove that, for any fixed value of $q$, there can be at most finitely many perfect $2-$error correcting codes over an alphabet of size $q$.
In this paper, we study the integer solutions of a family of Fermat-type equations of signature $(2, 2n, n)$, $Cx^2 + q^ky^{2n} = z^n$. We provide an algorithmically testable set of conditions which, if satisfied, imply the existence of a constant $B_{C, q}$ such that if $n > B_{C,q}$, there are no solutions $(x, y, z, n)$ of the equation. Our methods use the modular method for Diophantine equations, along with level lowering and Galois theory.
Given a prime number $q$ and a squarefree integer $C_1$, we develop a method to explicitly determine the tuples $(y, n, α)$ for which the difference $y^n-q^α$ has squarefree part equal to $C_1$. Our techniques include the combination of the local information provided by Galois representations of Frey-Hellegouarch curves with the effective resolution of Thue-Mahler equations, as well as the use of improved lower bounds for $q-$adic and complex logarithms. As an application of this methodology, we will completely resolve the case when $1 \le C_1 \le 20$ and $ 2 \le q < 25$.