数论

We study obstructed deformation problems for two-dimensional residual Galois representations arising from weight~$2$ newforms of level~$N$. Using Poitou-Tate duality, we isolate local and global sources of obstructions and give concrete…

Inspired by the work of Bourgain and Garaev (2013), we provide new bounds for certain weighted bilinear Kloosterman sums in polynomial rings over a finite field. As an application, we build upon and extend some results of Sawin and…

Even though Zaremba's conjecture remains open, Bourgain and Kontorovich solved the problem for a full density subset. Nevertheless, there are only a handful of explicit sequences known to satisfy the strong version of the conjecture, all of…

In this paper, we give error bounds for the distance distribution of Reed-Muller codes, extending prior work on the distance distribution of Reed-Solomon codes. This is equivalent to the problem of counting multivariate polynomials over a…

The harmonic numbers are those $H_n=\sum_{0<k\le n}\frac1k\ (n=0,1,2,\ldots)$. In this paper we confirm over ten conjectural series identities with summands involving the binomial coefficient $\binom{4k}k$ and harmonic numbers. For example,…

For fixed natural numbers $r$ and $s$, where $2\leq s \leq r$, we consider a representation of numbers from the interval $[0;\frac{r}{s-1}]$ obtained by encoding numbers by means of the alphabet $A=\{0,1,...,r\}$ via the expansion…

We investigate the Hasse principle for Diophantine systems consisting of one diagonal form of degree $k$ and one general form of degree $k-1$. By refining the method of Brandes and Parsell (arXiv:2003.04350) in this specific setting, we…

We define and study "tautological classes" in the cohomology of moduli stacks of shtukas, pursuing two directions of applications. First, we prove a formula relating the "arithmetic volume" of tautological classes to higher derivatives of…

There exists a set $A$ of positive integers such that the number of representations of a large positive integer $m$ as a sum of two elements of $A$ grows with a lower bound of order $\log m$, but for which there is no subset $D$ of $A$…

The classical Brauer-Siegel conjecture describes the asymptotic behaviour of the product of the class number and the regulator in families of number fields. All known cases of the conjecture rely on reducing the problem, via group theoretic…

Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $\mathfrak{m}_K$ be its maximal ideal. The image of the group of principal units $1+\mathfrak{m}_K$ under $p$-adic logarithm plays important role in several areas of number theory. In…

In 2013, Sun conjectured that the partition function $p(n)$ is never a perfect power for $n \geq 2$. Building on this, Merca, Ono, and Tsai recently observed that for any fixed integers $d \geq 0$ and $k \geq 2$, there appear to be only…

The Landau--Gonek Theorem evaluates $X^\rho$ summed over the non-trivial zeros of the Riemann zeta function. Their result shows great sensitivity to the arithmetic nature of $X$. We prove a related result concerning the sum of $\chi(\rho)…

We make further progress towards a Kneser-type generalization of Pollard's Theorem to general abelian groups. For two sets $A$ and $B$ in an abelian group $G$, the \emph{$t$-popular sumset} of $A$ and $B$, denoted by $A+_t B$, is the set of…

We propose an efficient computational method for finding all solutions $n\leq U$ to the Diophantine equation $a\sigma(n) = bn + c$, where integer coefficient $a,b,c$ and an upper bound $U$ are given. Our method is implemented in SageMath…

The L-function $ L(\rho_\lambda, s) $ of an almost everywhere unramified $ \lambda $-adic representation $ \rho_\lambda $ of a global function field $ \mathbb{F}_q(C) $ is known to be a rational function in $ q^{-s} $ satisfying a…

Primitive points on the tower of modular curves $X_1(n)$ provide a finite "certificate set" for detecting isolated points above a fixed $j$-invariant: for a non-CM elliptic curve $E/\mathbb{Q}$, $j(E)$ arises from an isolated point on some…

In this article, we study the local behaviour of the multiple polylogarithm functions at integer points, in the $s$-aspect. This is done by writing a Laurent type expansion at integer points, involving certain power series and rational…

We develop an explicit theory of formal modular forms over arbitrary number fields $K$, as functions of modular points. We define modular points for $\Gamma_0({\mathfrak n})$ and $\Gamma_1({\mathfrak n})$, where the level ${\mathfrak n}$ is…

The classical modular polynomial for $j$-invariants describes the relation between two elliptic curves connected by isogenies. This polynomial has been applied to various algorithms in computational number theory, such as point counting on…