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In this paper, we extend a uniformity result of Dimitrov et al. [Uniformity in Mordell-Lang for curves, Ann. of Math. (2) 194(1) (2021) 237–298] to dimension two and use it to get a uniform bound on the cardinality of the set of all quadratic points for non-hyperelliptic non-bielliptic curves which only depend on the Mordell–Weil rank, the genus of the curve and the degree of the number field. [ABSTRACT FROM AUTHOR]

For K / F a finite Galois extension of number fields, the relative Pólya group Po (K / F) is the subgroup of the ideal class group of K generated by all the strongly ambiguous ideal classes in K / F. The notion of Ostrowski quotient Ost (K / F) , as the cokernel of the capitulation map into Po (K / F) , has been recently introduced in [E. Shahoseini, A. Rajaei and A. Maarefparvar, Ostrowski quotients for finite extensions of number fields, Pacific J. Math. 321 (2022) 415–429, doi: 10.2140/pjm.2022.321.415]. In this paper, using some results of [C. D. González-Avilés, Capitulation, ambiguous classes and the cohomology of the units, J. Reine Angew. Math. 613 (2007) 75–97], we find a new approach to define Po (K / F) and Ost (K / F) which is the main motivation for us to investigate analogous notions in the elliptic curve setting. For E an elliptic curve defined over F , we define the Ostrowski quotient Ost (E , K / F) and the coarse Ostrowski quotient Ost c (E , K / F) of E relative to K / F , for which in the latter group we do not take into account primes of bad reduction. Our main result is a nontrivial structure theorem for the group Ost c (E , K / F) and we analyze this theorem, in some details, for the class of curves E over quadratic extensions K / F. [ABSTRACT FROM AUTHOR]

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 ≤ d ≤ 2 0. The key ingredients are: the approach presented in [A. Pacetti and L. V. Torcomian, 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. [ABSTRACT FROM AUTHOR]