数论

In \cite{NZ25}, the authors resolved the rational function analogue of the finiteness results for greatest common divisors of iterates of polynomials established in \cite{HT17}. These results may be viewed as dynamical generalizations of a…

Studying Correlations with Ramanujan Expansions, we arrive to present the new class of, say, Two-Seasons Correlations, abbr. T-S, as a natural set expressing some of the features of, say, H-L-like Correlations; these are the ones that mimic…

The purpose of this paper is to describe explicitly the modules of (Siegel-)Jacobi forms of degree two of index one of any scalar valued weight with respect to some congruence subgroups of small levels $N\leq 4$. Such a structure for the…

Fix a nonzero level $\mathfrak{n} \in \mathbb{F}_q[T]$. In this paper, we first establish a function field analogue of Ligozat's theorem, which serves as our main result and provides a criterion for Drinfeld modular units on the Drinfeld…

Let $F_n$ be the $n$-th Fibonacci number. In this paper, we study the Diophantine equation $F_n+F_m=p^xq^y$ in nonnegative integers $n\ge m$, $x$ and $y$, where $p$ and $q$ are fixed distinct prime numbers. We determine all pairs of primes…

We study traces of Hecke operators on spaces of elliptic cusp forms and Drinfeld cusp forms and show that, modulo any prime power, these traces are periodic in the weight.

We establish Large Galois orbits conjectures for points of unlikely intersections of curves in $Y(1)^n$, upon assumptions on the intersection of such curves with the boundary $X(1)^n\backslash Y(1)^n$, in the Zilber-Pink setting. As a…

We establish an effective version of Siegel's lower bounds for class numbers of imaginary quadratic fields in certain cures in $Y(1)^n$. Our proof goes through the G-functions method of Yves Andr\'e.

Following our work in \cite{papas2022height}, we extend the height bounds established by Y. Andr\'e in his seminal research monograph \cite{andre1989g} for $1$-parameter families of abelian varieties defined over number fields. In our…

We give practical algorithms for computing the divisor class group and the gonality of a curve over a finite field, achieving several orders of magnitude speedup over existing methods for sufficiently large genus or residue field. The…

This paper investigates the existence of a local-global principle for certain twists of abelian varieties defined over number fields. Our main focus is to determine when, for $m$ a positive integer, locally $m$-atic twists of an abelian…

A curve $C$ defined over $\mathbb Q$ is modular of level $N$ if there exists a non-constant morphism from $X_1(N)$ onto $C$ defined over $\mathbb Q$ for some positive integer $N$. We provide a sufficient and necessary condition for the…

We present a unified integral framework based on the Fourier-Laplace transform evaluated along a vertical line in the complex plane. By identifying the moment-generating function (MGF) of a random variable with the weights of these…

Let $\psi$ be a continuous decreasing function defined on all large positive real numbers. We say that a real $m\times n$ matrix $A$ is $\psi$-Dirichlet if for every sufficiently large real number $t$ one can find $\mathbf{p} \in…

We study the Lagarias inequality, an elementary criterion equivalent to the Riemann Hypothesis. Using a continuous extension of the harmonic numbers, we show that the sequence $B_n=\frac{H_n+e^{H_n}\log(H_n)}{n}$ is strictly increasing for…

We establish infinite families of congruences modulo arbitrary powers of $2$ for the three restricted partition functions $M(n), T^\ast(n)$, and $P^\ast(n)$ introduced by Pushpa and Vasuki by employing elementary $q$-series techniques.…

We present the first algorithm for computing class groups and unit groups of arbitrary number fields that provably runs in probabilistic subexponential time, assuming the Extended Riemann Hypothesis (ERH). Previous subexponential algorithms…

We give asymptotic expressions for the number of commuting matrices over finite fields. For this, we use product expansions for the corresponding generating functions.

A number $N$ is a triangular number if it can be written as $N = t(t + 1)/2$ for some nonnegative integer number $t$. A triangular number $N$ is called square if it is a perfect square, that is, $N = d^2$ for some integer number $d$. Square…

We establish a connection between multiple polylogarithms on a torus and the Steinberg module of $\mathbb{Q}$, and show that multiple polylogarithms of depth $d$ and weight $n$ can be expressed via a single function…