相关论文: On the moments of torsion points modulo primes and…

In this paper, we study lower bounds on the K-theory of the maximal $C^*$-algebra of a discrete group based on the amount of torsion it contains. We call this the finite part of the operator K-theory and give a lower bound that is valid for…

Let $G$ denote a compact monothetic group, and let $$\rho (x) = \alpha_k x^k + \ldots + \alpha_1 x + \alpha_0,$$ where $\alpha_0, \ldots , \alpha_k$ are elements of $G$ one of which is a generator of $G$. Let $(p_n)_{n\geq 1}$ denote the…

Let $A/K$ be an absolutely simple abelian surface defined over a number field $K$. We give unconditional upper bounds for the number of prime ideals $\mathfrak{p}$ of $K$ with norm up to $x$ such that $A$ has supersingular reduction at…

For finitely generated subgroups $W_1, \ldots , W_t$ of $\mathbb{Q}^{\times}$, integers $k_1, \ldots , k_t$, a Galois extension $F$ of $\mathbb{Q}$ and a union of conjugacy classes $C \subset \text{Gal}(F/\mathbb{Q})$, we develop methods…

Given a map $\varphi:\mathbb{P}^1\rightarrow \mathbb{P}^1$ of degree greater than 1 defined over a number field $k$, one can define a map $\varphi_\mathfrak{p}:\mathbb{P}^1(\mathfrak{o}_k/\mathfrak{p})\rightarrow…

Let $k$ be an algebraic extension of $\mathbb F_p$ and $K/k$ a regular extension of fields (e.g. $\mathbb F_p(T)/\mathbb F_p$). Let $A$ be a $K$-abelian variety such that all the isogeny factors are neither isotrivial nor of $p$-rank zero.…

In 2014, Juul, Kurlberg, Madhu and Tucker asked the following: given $K$ a number field and $f$ a rational function with coefficients in $K$, if $f_\mathfrak{p}$ denotes the reduction of $f$ modulo a prime ideal $\mathfrak{p}$ in the ring…

Let $X$ be a finite set such that $|X|=n$, and let $k< n/2$. A group is $k$-homogeneous if it has only one orbit on the sets of size $k$. The aim of this paper is to prove some general results on permutation groups and then apply them to…

Let $\alpha_1,\ldots,\alpha_r$ be algebraic numbers in a number field $K$ generating a subgroup of rank $r$ in $K^\times$. We investigate under GRH the number of primes $\mathfrak p$ of $K$ such that each of the orders of…

We give an algorithm to explicitly determine all elements of the $q$-torsion (for $q$ an odd prime) of the Brauer group of an elliptic curve over any base field of characteristic different from $q$, containing a primitive $q$-th root of…

We compute the number of orbits of pairs in a finitely generated torsion module (more generally, a module of bounded order) over a discrete valuation ring. The answer is found to be a polynomial in the cardinality of the residue field whose…

Let $\mathfrak b_n(\mathbb F_q)$ denote the Lie algebra of upper triangular $n\times n$ matrices over the finite field $\mathbb F_q$, and let $\mathfrak u_n(\mathbb F_q)$ be the nilradical of $\mathfrak b_n$. For every $\mathfrak…

We examine the Galois groups of the extensions $K((f'\circ f^n)^{-1}(0))/K$ where $K$ is a number field for polynomials $f(x)\in K[x]$. We use our understanding of this group to study the proportion of primes for which $f$ has a $\mathfrak…

Let $K$ be the number field determined by a monic irreducible polynomial $f(x)$ with integer coefficients. In previous papers we parameterized the prime ideals of $K$ in terms of certain invariants attached to Newton polygons of higher…

We study the action of the Galois group $G$ of a finite extension $K/k$ of number fields on the points on an elliptic curve $E$. For an odd prime $p$, we aim to determine the structure of the $p$-adic completion of the Mordell-Weil group…

Let $K$ be an imaginary quadratic field and $\mathcal{O}_K$ be its ring of integers. Let $h_E$ be the Weber function on certain elliptic curve $E$ with complex multiplication by $\mathcal{O}_K$. We show that if $N$ ($>1$) is an integer…

Let $A$ be the product of an abelian variety and a torus defined over a number field $K$. Fix some prime number $\ell$. If $\alpha \in A(K)$ is a point of infinite order, we consider the set of primes $\mathfrak p$ of $K$ such that the…

Let $q$ be a prime power and $\phi$ a rational function with coefficients in a finite field $\mathbb{F}_q$. For $n \geq 1$, each element of $\mathbb{P}^1(\F_{q^n})$ is either periodic or strictly preperiodic under iteration of $\phi$.…

For any polynomial $P(x)\in\mathbb{Z}[x],$ we study arithmetic dynamical systems generated by $\displaystyle{F_P(n)=\prod_{k\le n}}P(n)(\text{mod}\ p),$ $n\ge 1.$ We apply this to improve the lower bound on the number of distinct quadratic…

The main result asserts: Let $G$ be a reductive, affine algebraic group and let $(\rho ,V)$ be a regular representation of $G$. Let $X$ be an irreducible $\mathbb{C}^{ \times } G$ invariant Zariski closed subset such that $G$ has a closed…