International Journal of Number Theory

We treat two different equations involving powers of singular moduli. On the one hand, we show that, with two possible (explicitly specified) exceptions, two distinct singular moduli $j(\tau ),j({\tau }^{\prime })$ such that the numbers $1$, $j{(\tau )}^m$ and $j{({\tau }^{\prime })}^n$ are linearly dependent over $\mathbb{Q}$ for some positive integers $m,n$, must be of degree at most $2$. This partially generalizes a result of Allombert, Bilu and Pizarro-Madariaga, who studied CM-points belonging to straight lines in ${ℂ}^2$ defined over $\mathbb{Q}$. On the other hand, we show that, with obvious exceptions, the product of any two powers of singular moduli cannot be a non-zero rational number. This generalizes a result of Bilu, Luca and Pizarro-Madariaga, who studied CM-points belonging to a hyperbola $xy=A$, where $A\in \mathbb{Q}$.

Over the last two decades, there has been a wave of activity establishing the Sato-Tate kind of distribution in various families of elliptic curves over prime fields. Typically the goal here is to prove this for families which are as thin as possible. We consider a function field analogue of this question, that is, for high degree extensions of a finite field where new effects allow us to study families, which are much thinner that those typically investigated over prime fields.

We use results by Chenevier to interpolate the classical Jacquet–Langlands correspondence for Hilbert modular forms, which gives us an extension of Chenevier’s results to totally real fields. From this we obtain an isomorphism between eigenvarieties attached to Hilbert modular forms and those attached to modular forms on a totally definite quaternion algebra over a totally real field of even degree.

Let $E$ be a $\mathbb{Q}$-curve without complex multiplication. We address the problem of deciding whether $E$ is geometrically isomorphic to a strongly modular $\mathbb{Q}$-curve. We show that the question has a positive answer if and only if $E$ has a model that is completely defined over an abelian number field. Next, if $E$ is completely defined over a quadratic or biquadratic number field $L$, we classify all strongly modular twists of $E$ over $L$ in terms of the arithmetic of $L$. Moreover, we show how to determine which of these twists come, up to isogeny, from a subfield of $L$.

Hubert, Mauduit and Sárközy introduced pseudorandom measures for finite pseudorandom binary lattices. Gyarmati, Mauduit, Sárközy and Stewart presented some natural and flexible constructions, which are the two-dimensional extensions and modifications of a few one-dimensional constructions. The upper estimates for the pseudorandom measures of their binary lattices are based on the principle that character sums or exponential sums in two variables can be estimated by fixing one of the variables.

In this paper, we constructed two large families of $n$ dimensional pseudorandom binary lattices by using the multiplicative inverse modulo $p$, and study the properties: pseudorandom measure, collision and avalanche effect.

For a general class of non-negative functions defined on integral ideals of number fields, upper bounds are established for their average over the values of certain principal ideals that are associated to irreducible binary forms with integer coefficients.

We provide explicit bounds on the difference of heights of the $j$-invariants of isogenous elliptic curves defined over $\overline{\mathbb{Q}}$. The first one is reminiscent of a classical estimate for the Faltings height of isogenous abelian varieties, which is indeed used in the proof. We also use an explicit version of Silverman’s inequality and isogeny estimates by Gaudron and Rémond. We give applications in the study of Vélu’s formulas and of modular polynomials.

We consider a family of linear recurrence sequences $T^{(k)}:={\{T_n^{(k)}\}}_n$ of order $k$ whose first $k$ terms are $0,1,\dots ,1$ and each term afterwards is the sum of the preceding $k$ terms. In this paper, we study the zero–multiplicity on $T^{(k)}$ when the indices are extended to all integers. In particular, we give a upper bound (dependent on $k$) for the largest positive integer $n$ such that $T_{-n}^{(k)}=0$ and show that $T^{(5)}$ has zero–multiplicity unitary when the indices are extended to all the integers.

The distribution of the primes of the forms $\lfloor n^{\alpha }\rfloor$ and $\lfloor n\alpha +\beta \rfloor$ are studied extensively, where $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$. In this paper, we will consider several new type problems on the distribution of the primes involving the ceiling (floor) function. For any real number $𝜃$ with $0<𝜃\le 1$, let ${\pi }_{𝜃}^{\prime }(n)$ be the number of integers $k$ with $1\le k\le n^{𝜃}$ such that $\lceil n/k\rceil$ is prime and let ${\pi }_{𝜃}^{\prime \prime }(n)$ be the number of primes $p$ for which there exists an integer $k$ with $1\le k\le n^{𝜃}$ such that $p=\lceil n/k\rceil$, where $\lceil x\rceil$ denotes the least integer not less than $x$. These are closely related to the number of the prime factors of the denominator of the Bernoulli polynomial $B_n(x)-B_n$. In this paper, we study asymptotic properties of ${\pi }_{𝜃}^{\prime }(n)$ and ${\pi }_{𝜃}^{\prime \prime }(n)$. The methods in this paper are also effective for corresponding distribution functions of the primes involving the floor function.

It is well known that sequences such as the Fibonacci numbers and the factorials satisfy Benford’s Law; that is, leading digits in these sequences occur with frequencies given by $P(d)={log}_{10}(1+1/d)$, $d=1,2,\dots ,9$. In this paper, we investigate leading digit distributions of arithmetic sequences from a local point of view. We call a sequence locally Benford distributed of order $k$ if, roughly speaking, $k$-tuples of consecutive leading digits behave like $k$ independent Benford-distributed digits. This notion refines that of a Benford distributed sequence, and it provides a way to quantify the extent to which the Benford distribution persists at the local level. Surprisingly, most sequences known to satisfy Benford’s Law have rather poor local distribution properties. In our main result we establish, for a large class of arithmetic sequences, a “best-possible” local Benford Law; that is, we determine the maximal value $k$ such that the sequence is locally Benford distributed of order $k$. The result applies, in particular, to sequences of the form $\{a^n\}$, $\{a^{n^d}\}$, and $\{n^{\beta }{a}^{n^{\alpha }}\}$, as well as the sequence of factorials $\{n!\}$ and similar iterated product sequences.

Chebyshev presented a conjecture after observing the apparent bias towards primes congruent to $3(mod4)$. His conjecture is equivalent to a version of the Generalized Riemann Hypothesis. Fujii strengthened this conjecture; we strengthen it still further using detailed computations of zeroes of Dirichlet $L$-functions.

Let $G$ be a finite abelian group and $S$ be a subset of $G$. Let $\mathrm{\Sigma }(S)$ denote the set of group elements which can be expressed as a sum of a nonempty subset of $S$. We say that $S$ is zero-sum free if $0\notin \mathrm{\Sigma }(S)$. In this paper, we show that if $S$ is zero-sum free with $\left|S\right|\ge 6$, then $|\mathrm{\Sigma }(S)|\ge 3\left|S\right|$. Furthermore, if $S$ is zero-sum free and $\left|S\right|\ge 7$, with the assumption that the subgroup generated by $S$ is noncyclic and the order $|〈S〉|\ge 146$, we are able to show that $|\mathrm{\Sigma }(S)|\ge 4\left|S\right|-3$.