Torsion points on curves and common divisors of $a^k-1$ and $b^k-1$

Nir Ailon; Zéev Rudnick

Displaying similar documents to “Torsion points on curves and common divisors of $a^{k} - 1$ and $b^{k} - 1$ ”

On the torsion of the Jacobians of the hyperelliptic curves y² = xⁿ + a and y² = x(xⁿ+a)

Tomasz Jędrzejak (2016)

Acta Arithmetica

Similarity:

Consider two families of hyperelliptic curves (over ℚ), $C^{n, a} : y ² = x ⁿ + a$ and $C_{n, a} : y ² = x (x ⁿ + a)$ , and their respective Jacobians $J^{n, a}$ , $J_{n, a}$ . We give a partial characterization of the torsion part of $J^{n, a} (ℚ)$ and $J_{n, a} (ℚ)$ . 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 $J_{8, a} (ℚ)$ . Namely, we show that $J_{8, a} {(ℚ)}_{t o r s} = J_{8, a} (ℚ) [2]$ . In addition, we characterize the torsion parts of $J_{p, a} (ℚ)$ , where p is an odd...

Characterization of the torsion of the Jacobians of two families of hyperelliptic curves

Tomasz Jędrzejak (2013)

Acta Arithmetica

Similarity:

Consider the families of curves $C^{n, A} : y ² = x ⁿ + A x$ and $C_{n, A} : y ² = x ⁿ + A$ where A is a nonzero rational. Let $J^{n, A}$ and $J_{n, A}$ denote their respective Jacobian varieties. The torsion points of $C^{3, A} (ℚ)$ and $C_{3, A} (ℚ)$ are well known. We show that for any nonzero rational A the torsion subgroup of $J^{7, A} (ℚ)$ is a 2-group, and for A ≠ 4a⁴,-1728,-1259712 this subgroup is equal to $J^{7, A} (ℚ) [2]$ (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...

The space $S_{α, β}$ and σ-core

Bruno de Malafosse (2006)

Studia Mathematica

Similarity:

We give some new properties of the space $S_{α, β}$ and we apply them to the σ-core theory. These results generalize those by Choudhary and Yardimci.

G-matrices, $J$ -orthogonal matrices, and their sign patterns

Frank J. Hall, Miroslav Rozložník (2016)

Czechoslovak Mathematical Journal

Similarity:

A real matrix $A$ is a G-matrix if $A$ is nonsingular and there exist nonsingular diagonal matrices $D_{1}$ and $D_{2}$ such that $A^{- T} = D_{1} A D_{2}$ , where $A^{- T}$ denotes the transpose of the inverse of $A$ . Denote by $J = diag (\pm 1)$ a diagonal (signature) matrix, each of whose diagonal entries is $+ 1$ or $- 1$ . A nonsingular real matrix $Q$ is called $J$ -orthogonal if $Q^{T} J Q = J$ . Many connections are established between these matrices. In particular, a matrix $A$ is a G-matrix if and only if $A$ is diagonally (with positive diagonals) equivalent to a column permutation...

A note on the torsion of the Jacobians of superelliptic curves $y^{q} = x^{p} + a$

Tomasz Jędrzejak (2016)

Banach Center Publications

Similarity:

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 $J_{3, 5, a} (ℚ)$ (resp. $J_{q, p, a} (ℚ)$ ). 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...

A computation of positive one-peak posets that are Tits-sincere

Marcin Gąsiorek, Daniel Simson (2012)

Colloquium Mathematicae

Similarity:

A complete list of positive Tits-sincere one-peak posets is provided by applying combinatorial algorithms and computer calculations using Maple and Python. The problem whether any square integer matrix $A \in ₙ (ℤ)$ is ℤ-congruent to its transpose $A^{t r}$ is also discussed. An affirmative answer is given for the incidence matrices $C_{I}$ and the Tits matrices $C ̂_{I}$ of positive one-peak posets I.