This paper focuses on the Diophantine equation $xⁿ+p^{\alpha }yⁿ=Mz³$, with fixed α, p, and M. We prove that, under certain conditions on M, this equation has no non-trivial integer solutions if $n\ge ℱ(M,p^{\alpha })$, where $ℱ(M,p^{\alpha })$ is an effective constant. This generalizes Theorem 1.4 of the paper by Bennett, Vatsal and Yazdani [Compos. Math. 140 (2004), 1399-1416].

We study integral similitude 3 × 3-matrices and those positive integers which occur as products of their row elements, when matrices are symmetric with the same numbers in each row. It turns out that integers for which nontrivial matrices of this type exist define elliptic curves of nonzero rank and are closely related to generalized cubic Fermat equations.

We prove several results regarding some invariants of elliptic curves on average over the family of all elliptic curves inside a box of sides A and B. As an example, let E be an elliptic curve defined over ℚ and p be a prime of good reduction for E. Let $e_E\left(p\right)$ be the exponent of the group of rational points of the reduction modulo p of E over the finite field ${}_p$. Let be the family of elliptic curves $E_{a,b}:y^2=x^3+ax+b$, where |a| ≤ A and |b| ≤ B. We prove that, for any c > 1 and k∈ ℕ, $1/\left|\right|{\sum }_{E\in }{\sum }_{p\le x}{e}_E^k\left(p\right)=C_{k}li\left(x^{k+1}\right)+O(\left(x^{k+1}\right)/{\left(logx\right)}^c$)$$as x → ∞, as long as $A,B>exp\left(c_1{\left(logx\right)}^{1/2}\right)$ and $AB>x{\left(logx\right)}^{4+2c}$,...

We give a necessary condition for a surjective representation Gal$(\overline{\mathbb{Q}}/\mathbb{Q})\to {\mathrm{PGL}}_2\left({𝔽}_3\right)$ to arise from the $3$-torsion of a $\mathbb{Q}$-curve. We pay a special attention to the case of quadratic $\mathbb{Q}$-curves.

T. Dokchitser [Acta Arith. 126 (2007)] showed that given an elliptic curve E defined over a number field K then there are infinitely many degree 3 extensions L/K for which the rank of E(L) is larger than E(K). In the present paper we show that the same is true if we replace 3 by any prime number. This result follows from a more general result establishing a similar property for the Jacobian varieties associated with curves defined by an equation of the shape f(y) = g(x) where f and g are polynomials...

Let $E$ be a central $\mathbb{Q}$-curve over a polyquadratic field $k$. In this article we give an upper bound for prime divisors of the order of the $k$-rational torsion subgroup $E_{tors}\left(k\right)$ (see Theorems 1.1 and 1.2). The notion of central $\mathbb{Q}$-curves is a generalization of that of elliptic curves over $\mathbb{Q}$. Our result is a generalization of Theorem 2 of Mazur [12], and it is a precision of the upper bounds of Merel [15] and Oesterlé [17].

We use the so-called second 2-descent method to find several series of non-congruent numbers. We consider three different 2-isogenies of the congruent elliptic curves and their duals, and find a necessary condition to estimate the size of the images of the 2-Selmer groups in the Selmer groups of the isogeny.

Given an elliptic curve E over a function field K = ℚ(T₁,...,Tₙ), we study the behavior of the canonical height $h{̂}_{E_{\omega }}$ of the specialized elliptic curve $E_{\omega }$ with respect to the height of ω ∈ ℚⁿ. We prove that there exists a uniform nonzero lower bound for the average of the quotient $\left(h{̂}_{E_{\omega }}\left(P_{\omega }\right)\right)/h\left(\omega \right)$ over all nontorsion P ∈ E(K).