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))...

The paper deals with several criteria for the transcendence of infinite products of the form ${\prod }_{n=1}^{\infty }\left[b_n{\alpha }^{a_n}\right]/b_n{\alpha }^{a_n}$ where $\alpha >1$ is a positive algebraic number having a conjugate ${\alpha }^{*}$ such that $\alpha \ne |{\alpha }^{*}|>1$, ${\left\{a_n\right\}}_{n=1}^{\infty }$ and ${\left\{b_n\right\}}_{n=1}^{\infty }$ are two sequences of positive integers with some specific conditions. The proofs are based on the recent theorem of Corvaja and Zannier which relies on the Subspace Theorem (P. Corvaja, U. Zannier: On the rational approximation to the powers of an algebraic number: solution of two problems of Mahler and Mendès France, Acta...

Let $W(m,n;\underline{\psi })$ denote the set of ${\psi }_1,...,{\psi }_n$–approximable points in ${ℝ}^{mn}$. The classical Khintchine–Groshev theorem assumes a monotonicity condition on the approximating functions $\underline{\psi }$. Removing monotonicity from the Khintchine–Groshev theorem is attributed to different authors for different cases of $m$ and $n$. It can not be removed for $m=n=1$ as Duffin–Schaeffer provided the counter example. We deal with the only remaining case $m=2$ and thereby remove all unnecessary conditions from the Khintchine–Groshev theorem.

In this note, we prove that there is no transcendental entire function $f\left(z\right)\in \mathbb{Q}\left[\right[z\left]\right]$ such that $f\left(\mathbb{Q}\right)\subseteq \mathbb{Q}$ and $\mathrm{den}f(p/q)=F\left(q\right)$, for all sufficiently large $q$, where $F\left(z\right)\in \mathbb{Z}\left[z\right]$.

We extend Prasad’s results on the existence of trilinear forms on representations of $G{L}_2$ of a local field, by permitting one or more of the representations to be reducible principal series, with infinite-dimensional irreducible quotient. We apply this in a global setting to compute (unconditionally) the dimensions of the subspaces of motivic cohomology of the product of two modular curves constructed by Beilinson.

We prove a result on approximations to a real number θ by algebraic numbers of degree ≤ 2 in the case when we have certain information about the uniform Diophantine exponent ω̂ for the linear form x₀ + θx₁ + θ²x₂.

In this note we present and comment three equivalent definitions of the so called uniform or Banach density of a set of positive integers.

We compare two sets of (infinite) binary sequences whose suffixes satisfy extremal conditions: one occurs when studying iterations of unimodal continuous maps from the unit interval into itself, but it also characterizes univoque real numbers; the other is a disguised version of the set of characteristic sturmian sequences. As a corollary to our study we obtain that a real number $\beta$ in $(1,2)$ is univoque and self-sturmian if and only if the $\beta$-expansion of $1$ is of the form $1v$, where $v$ is a characteristic...

We compare two sets of (infinite) binary sequences whose suffixes satisfy extremal conditions: one occurs when studying iterations of unimodal continuous maps from the unit interval into itself, but it also characterizes univoque real numbers; the other is a disguised version of the set of characteristic Sturmian sequences. As a corollary to our study we obtain that a real number β in (1,2) is univoque and self-Sturmian if and only if the β-expansion of 1 is of the form 1v, where v is a characteristic...