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Consider the families of curves $C^{n,A}:y²=xⁿ+Ax$ and $C_{n,A}:y²=xⁿ+A$ where A is a nonzero rational. Let $J^{n,A}$ and $J_{n,A}$ denote their respective Jacobian varieties. The torsion points of $C^{3,A}\left(\mathbb{Q}\right)$ and $C_{3,A}\left(\mathbb{Q}\right)$ are well known. We show that for any nonzero rational A the torsion subgroup of $J^{7,A}\left(\mathbb{Q}\right)$ is a 2-group, and for A ≠ 4a⁴,-1728,-1259712 this subgroup is equal to $J^{7,A}\left(\mathbb{Q}\right)\left[2\right]$ (for a excluded values of A, with the possible exception of A = -1728, this group has a point of order 4). This is a variant of the corresponding results for $J^{3,A}$ (A ≠ 4) and $J^{5,A}$. We also almost...

It is classical that a natural number n is congruent iff the rank of ℚ -points on Eₙ: y² = x³-n²x is positive. In this paper, following Tada (2001), we consider generalised congruent numbers. We extend the above classical criterion to several infinite families of real number fields.

Consider two families of hyperelliptic curves (over ℚ), $C^{n,a}:y²=xⁿ+a$ and $C_{n,a}:y²=x(xⁿ+a)$, and their respective Jacobians $J^{n,a}$, $J_{n,a}$. We give a partial characterization of the torsion part of $J^{n,a}\left(\mathbb{Q}\right)$ and $J_{n,a}\left(\mathbb{Q}\right)$. More precisely, we show that the only prime factors of the orders of such groups are 2 and prime divisors of n (we also give upper bounds for the exponents). Moreover, we give a complete description of the torsion part of $J_{8,a}\left(\mathbb{Q}\right)$. Namely, we show that $J_{8,a}{\left(\mathbb{Q}\right)}_{tors}=J_{8,a}\left(\mathbb{Q}\right)\left[2\right]$. In addition, we characterize the torsion parts of $J_{p,a}\left(\mathbb{Q}\right)$, where p is an odd prime, and...

This article is a short version of the paper published in J. Number Theory 145 (2014) but we add new results and a brief discussion about the Torsion Conjecture. Consider the family of superelliptic curves (over ℚ) $C_{q,p,a}:y^q=x^p+a$, and its Jacobians $J_{q,p,a}$, where 2 < q < p are primes. We give the full (resp. partial) characterization of the torsion part of $J_{3,5,a}\left(\mathbb{Q}\right)$ (resp. $J_{q,p,a}\left(\mathbb{Q}\right)$). The main tools are computations of the zeta function of $C_{3,5,a}$ (resp. $C_{q,p,a}$) over ${}_l$ for primes l ≡ 1,2,4,8,11 (mod 15) (resp. for primes l ≡ -1 (mod qp))...

We study the family of curves $F_m\left(p\right):x^p+y^p=m$, where p is an odd prime and m is a pth power free integer. We prove some results about the distribution of root numbers of the L-functions of the hyperelliptic curves associated to the curves $F_m\left(p\right)$. As a corollary we conclude that the jacobians of the curves $F_m\left(5\right)$ with even analytic rank and those with odd analytic rank are equally distributed.

We consider the Diophantine equation $(x+y)(x²+Bxy+y²)=D{z}^p$, where B, D are integers (B ≠ ±2, D ≠ 0) and p is a prime >5. We give Kraus type criteria of nonsolvability for this equation (explicitly, for many B and D) in terms of Galois representations and modular forms. We apply these criteria to numerous equations (with B = 0, 1, 3, 4, 5, 6, specific D’s, and p ∈ (10,10⁶)). In the last section we discuss reductions of the above Diophantine equations to those of signature (p,p,2).