数论

We give a systematic treatment of the arithmetic BF theory, introduced by Carlson and Kim. We observe that the Cassels-Tate pairing can be naturally interpreted as an arithmetic BF functional.

Let $(X, d)$ be a compact metric space, and let $Q \subset X$ be countable. Given functions $R: Q \to \mathbb{R}^+$ and $\phi: \mathbb{R}^+ \to \mathbb{R}^+$, we consider the set $E(Q, R, \phi)$ of points $x \in X$ that ``hit'' the…

We show that for any polynomial $f: \mathbb{Z}\to \mathbb{Z}$ with positive leading coefficient and irreducible over $\mathbb{Q}$, if $N$ is large enough then there are two strings of consecutive positive integers $I_{1}=\{n_1-m,\ldots,…

We determine all groups which occur as torsion subgroups of $\mathbb Q$-curves defined over number fields of degrees $3$, $5$ and $7$. In particular, we prove that every torsion subgroup of a $\mathbb Q$-curve defined over a number field of…

In this short note we report on a coincidence of two mathematical quantities that, at first glance, have little to do with each other. On the one hand, there are the Lebesgue constants of the Walsh function system that play an important…

An integer-valued polynomial $P(x,y,z)$ is said to be universal (over $\mathbb Z$) if each nonnegative integer can be written as $P(x,y,z)$ with $x,y,z\in\mathbb Z$. In this paper, we mainly introduce a new technique to determine the…

In this article, we explore a natural extension of the quadratic parametrization introduced in our previous work. By replacing the integer $n$ by $n^s$ ($ s\in\mathbb{R}, s>1$) and allowing the parameters to be real, we obtain for each…

Let $p$ be a prime and let $L$ be a quadratic $\mathbb{Z}_p$-lattice with quadratic form $Q$. For $t\neq 0$ the local representation density $\alpha_p(t;L)$ is the stable normalised growth of the congruence counts of solutions to…

Let $p$ be a prime number, $n$ an integer $\geq 2$, and $L$ a finite extension of $\mathrm{Q}_p$. Let $\rho_L$ be an $n$-dimensional (non-critical but not necessary generic) potentially crystalline $p$-adic Galois representation of the…

Let $x$ be a real number satisfying $x \geq 2$. For any positive integer $n$, we define $s(n)$ as the smallest non-negative integer such that $n + s(n)$ is a perfect square. In this paper, we derive an asymptotic formula for the sum…

Let $[\,\cdot\,]$ denote the floor function. Assume that $\lambda_1, \lambda_2, \lambda_3, \lambda_4, \lambda_5$ are nonzero real numbers, not all of the same sign, that $\lambda_1/\lambda_2$ is irrational, and that $\eta$ is a real number.…

We provide new local characterizations of Hida families of Siegel modular forms with genus two arising from automorphic inductions (stable Yoshida lifts), analogous to the characterizations of Hida families of CM modular forms provided by…

We explain how to obtain the set of solutions of a multivariate polynomial equation modulo a power of a prime number. These solutions are determined by a tree, called the trunk, which makes it possible to reconstruct all solutions. We apply…

All isometries $\sigma$ in a quadratic space over a non-archimedean local field of characteristic not 2 satisfying that any isometry $\tau$ which is conjugate to $\sigma$ in the general linear group is conjugate to $\sigma$ in the…

A Wagstaff prime is a prime number of the form $(2^{\mathfrak{p}}+1)/3$, where $\mathfrak{p}$ is an odd prime. Let $(L_n^{(k)})_{n\geq 2-k}$ be the $k$-Lucas number sequence defined by the recurrence relation $ L_n^{(k)} = L_{n-1}^{(k)} +…

We prove that the 4-dimensional Galois representations associated with a certain Calabi-Yau threefold are reducible, with 2-dimensional composition factors coming from specific modular forms of weights 2 and 4, both level 14. This was…

We study the Dirichlet series associated with the integers whose radix-$b$ representation misses certain (fixed) digits. The existence of a meromorphic continuation to the entire complex plane, which was already well-known as a general fact…

We prove the quantum modularity of the signature of $ \mathrm{SU}(2) $-TQFT for a genus 2 surface, which was conjectured by March\'{e}--Masbaum in 2025. Our approach is based on a quantum modularity of generalized Dedekind sums associated…

The primary objective of this paper is to employ methods from analytic number theory to investigate the mean value properties of a composite function involving the Dirichlet divisor function and a generalized minimal power function.…

We investigate integers whose base $g$ expansion omits a fixed digit and which can be represented as a sum of two prime squares. In the first part of the paper, we apply the Hardy--Littlewood circle method to obtain asymptotic formulas for…