数论

We prove inequalities relating the Taguchi heights, respectively the graded heights, of four Drinfeld modules arranged in a ``parallelogram of isogenies''. This inequality is the analogue for Drinfeld modules of the parallelogram inequality…

A natural number $n$ is called semi-prime if it is a product of two primes or a square of a prime. We denote $\mathbb{P}_2$ the set of all semi-primes. Our goal is to prove that for fixed integer number $a$ and sufficiently large $x$ the…

We establish an asymptotic formula with a power-saving error term for the twisted mixed moment of Dirichlet $L$-functions and automorphic $L$-functions twisted by all primitive characters modulo $q$, valid for all admissible moduli. As a…

We prove that the holomorphic solutions of Picard-Fuchs differential equations associated with one-parameter families of abelian varieties with real multiplication admit power series expansions with $S$-integral coefficients at a maximal…

The technique of P\'{o}lya-Hurwitz of partial fractions is implemented to investigate the zeros of finite $q$-Hankel transforms, which are defined in terms of the third $q$-Bessel function of Jackson. The new approach, which is a…

In this paper, we give error bounds on the number of monic irreducible polynomials $a_0+a_1x+\dots+a_{n-1}x^{n-1}+x^n$ over a finite field $\mathbb{F}_q$ of degree $n$ with $(a_0, a_1, \dots, a_{n-1}, 1)$ lying in a fixed affine algebraic…

Inspired by a problem proposed by Mahler, we will address the following related question, 'How well can irrationals in a missing digit set be approximated by rationals with polynomial denominators?' and prove some related results. To…

Periodic lattices are natural generalizations of lattices, which arise naturally in diophantine approximations with rationals of bounded denominators. In this paper, we prove analogues of classical theorems in geometry of numbers for…

Let $p\geq 3$ be a prime number and $K$ be a quadratic imaginary field in which $p$ splits as $\mathfrak{p}\overline{\mathfrak{p}}$. Let $\mathcal{F}$ be a cuspidal Bianchi eigenform over $K$ of weight $(k,k)$, where $k\geq 0$ is an…

Motivated by the works of L. Carlitz, R. Chapman and Z.-W. Sun on cyclotomic matrices, in this paper, we investigate certain cyclotomic matrices concerning the Jacobi sums over finite fields, which can be viewed as finite field analogues of…

In 1985, Schoof devised an algorithm to compute zeta functions of elliptic curves over finite fields by directly computing the numerators of these rational functions modulo sufficiently many primes (see \cite{schoof_1985}). If $E/K$ is an…

We present a strengthening of the proof of the St\'ephanois theorem. We follow the modular version by Waldschmidt, which is based in a suggestion by Daniel Bertrand, but it also applies to the original proof. The improvement is not in the…

In this paper we prove the canonical period of a Hilbert modular form with respect to the base change of a real quadratic extension differs from the square of its own canonical period only by a $p$-adic unit under some conditions. We prove…

Let $p$ be a prime number. If a number field $k$ has at least one complex place, there are infinitely many $\mathbb{Z}_p$-extensions over $k$, and some authors studied the behavior of Iwasawa invariants of these $\mathbb{Z}_p$-extensions.…

Let $\ell$ be an odd prime, $q$ an odd prime power such that $q \not\equiv 0 \pmod \ell$, and $m$ the order of $q$ in $\F_\ell^\times$. We propose an explicit $L$-polynomial of hyperelliptic function field $K:=\F_q(T,…

The multiplier systems of $\eta^{2k}(\tau)$ and $\theta^{2k}(\tau)$ $(k\in\mathbb{Z})$ are characters. In this paper, we determine their kernels, Ker$\,\nu_{\eta^{2k}}$ and Ker$\,\nu_{ \theta^{2k} } $.

We study properties of recently introduced Wieferich primes for Drinfeld modules, as their relation with Fermat equations and finitess or non-finiteness of their number. We also introduce Mersenne numbers for Drinfeld modules, and study the…

In $2020$, Bhargava, Shankar, Taniguchi, Thorne, Tsimerman, and Zhao proved that for a finite extension $K/\mathbb{Q}$ of degree $n\geq 5$, the size of the $2$-torsion class group is bounded by $\#…

Conjectures of Braverman and Kazhdan, Ng\^o and Sakellaridis have motivated the development of Schwartz spaces for certain spherical varieties. We prove that under suitable assumptions these Schwartz spaces are naturally a representation of…

Let $K$ be a field of characteristic $0$ and $E/K$ an elliptic curve over $K$. For a finite extension $L/K$ and a prime~$\ell$, we provide Galois-theoretic sufficient conditions on $L/K$ under which…