数论

We study real numbers defined by multidimensional automatic arrays weighted by multiplicatively independent bases. Let $a_1, \dots, a_r\geq 2$ be integers such that $\log a_1, \dots, \log a_r$ are $\mathbb Q$-linearly independent. Given…

In the previous paper of the author, motivated by the categorical $p$-adic local Langlands correspondence, the author studied families of $G_K$-equivariant vector bundles over the Fargues-Fontaine curve parametrized by algebraic-affinoid…

Let $p\geq5$ be a prime number. Let $L$ be a finite unramified extension of $\mathbb{Q}_p$ with ring of integers $\mathcal{O}_L$ and residue field $\mathbb{F}_q$. Given two Serre weights for $\mathrm{GL}_3(\mathbb{F}_q)$, we prove that in…

In this paper, we prove the simultaneous non-vanishing of four Dirichlet $L$-functions at any point on the critical line. More precisely, let $\chi_1,\ldots,\chi_4$ be even Dirichlet characters modulo $D_1,\ldots, D_4$ respectively, where…

We introduce and study the filtration on the space of automorphic functions (in the everywhere unramified situation for the function field case) obtained by transferring the filtration on the spectral side of the classical Langlands…

The Chabauty--Kim method was developed with the aim of approaching effective Faltings', the problem of explicitly determining the finite set of rational points on a hyperbolic curve. This method has seen success with the more particular…

We establish the asymptotic formula for the number of integral points in non-compact symmetric homogeneous spaces of semi-simple simply connected algebraic groups over global function fields, given by the sum of the products of local…

We attach Hecke polynomials $P_n(F;x)$ to weak Hecke eigenforms $F$ of weight $2-k$ and show that, for large $n$, every zero is simple and lies in $[0,1728]$. The construction pulls back a weakly holomorphic Hecke combination of $F$ along…

The Chabauty--Coleman--Kim method in depth two describes the rational points on a curve in terms of a generalisation of Nekov\'a\v{r}'s $p$-adic height pairing which replaces $\mathbb{G}_m$ with a higher Chow group. It is unclear both what…

An ideal setting to exhibit infinite sets of primes $p$ relative to which an integer is a primitive root $\pmod p$ is provided by the B\'ezout subdomain $\widetilde{\mathbb{B}}:=\mathbb{Z}^{\mathbb{P}}/\mathfrak{U}$ of the valuation domain…

In this paper, we classify all continuous Galois representations $\rho:\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\to \mathrm{GL}_2(\overline{\mathbf{Q}}_p)$ which are unramified outside $\{p,\infty\}$ and locally induced at $p$, under…

In this paper, a strong multiplicity one theorem for Katz modular forms is studied. We show that a cuspidal Katz eigenform which admits an irreducible Galois representation is in the level and weight old space of a uniquely associated Katz…

For a finite field $\mathbf{F}_{p^k}$ and a prime $\ell \neq p$, consider the graph $G$ of $\ell$-isogenies between ordinary elliptic curves over $\mathbf{F}_{p^k}$. Kohel proved that the connected components of $G$ have a remarkable…

We start with $n$-torsions in the Jacobian of an $m$-gonal curve and produce $n$-torsions in the class group of certain number field $K$.

We prove a structural upper bound on the $p$-adic valuation of denominators of rationals belonging to a missing-digit set $K_{m,D}$, generalizing a key step in recent work of Lin, Wu, and Yang [arXiv:2603.24614] on reciprocals of…

Let $\Gamma\subseteq PSL(2, \mathbb R)$ be a finite volume Fuchsian group. The hyperbolic circle problem is the estimation of the number of elements of the $\Gamma$-orbit of $z$ in a hyperbolic circle around $w$ of radius $R$, where $z$ and…

In this article, we provide an explicit constant $C$ such that there is no regular ternary sum of generalized $m$-gonal numbers for any integer $m$ greater than $C$.

Let $\pi$ be a cuspidal automorphic representation for $\mathrm{GL}(n)$ over a number field. We establish a conditional upper bound on the number of cuspidal isobaric summands in the symmetric $k$-th power lift of $\pi$, assuming that the…

We give upper and lower bounds on the Chevalley-Bass number of a field of characteristic zero, whenever this quantity is well-defined. We also describe an algorithm which computes the Chevalley-Bass number of a field, provided its maximal…

Given a finite extension $K/k$ of number fields and a smooth quasi-projective variety $X$ over $K$. If the abelianized fundamental group of $X$ is trivial, we prove that there is a natural identification between Brauer-Manin sets of $X$ and…