Related papers: Primes in the Chebotarev density theorem for all n…

Under GRH, any element in the multiplicative group of a number field $K$ that is globally primitive (i.e., not a perfect power in $K^*$) is a primitive root modulo a set of primes of $K$ of positive density. For elliptic curves $E/K$ that…

Let $N$ be a sufficiently large, odd integer. We prove an asymptotic formula for the number of representations of $N$ as the sum of three primes, one of which is smaller than a given $U$. By inserting the currently best zero-density…

Let $p\geq 7$ be a prime and $n>1$ be a natural number. We show that there exist infinitely many Galois representations $\varrho:Gal(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow GL_{n}(\mathbb{Z}_p)$ which are unramified outside $\{p, \infty\}$…

Given a finite transitive permutation group $G$, we investigate number fields $F/\mathbb{Q}$ of Galois group $G$ whose discriminant is only divisible by small prime powers. This generalizes previous investigations of number fields with…

Let $p$ be an odd prime and $F$ be a number field whose $p$-class group is cyclic. Let $F_{\{\mathfrak{q}\}}$ be the maximal pro-$p$ extension of $F$ which is unramified outside a single non-$p$-adic prime ideal $\mathfrak{q}$ of $F$. In…

Let T be a free Z_p-module of finite rank equipped with a continuous Z_p-linear action of the absolute Galois group of a number field K satisfying certain conditions. In this article, by using a Selmer group corresponding to T, we give a…

Let $p$ be a prime number and let $k\geq 2$ be an integer. In this article we study the semi-simple reductions modulo $p$ of two-dimensional irreducible crystalline $p$-adic Galois representations with Hodge-Tate weights $0$ and $k-1$ and…

Let $n>1$, $e\geq 0$ and a prime number $p\geq 2^{n+2+2e}+3$, such that the index of regularity of $p$ is $\leq e$. We show that there are infinitely many irreducible Galois representations $\rho: Gal(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow…

Let $E(X)$ denote the number of even integers below $X$ which are not a sum of two primes. We prove the bound $E(X)=O(X^{\frac{7}{10}})$, where the implicit constant is ineffective. The method applied here also leads to $P(q)=O(q^5)$, where…

Let $K$ be a totally real number field. For all prime number $p\geq 5$, let us denote by $F_p$ the Fermat curve of equation $x^p+y^p+z^p=0$. Under the assumption that $2$ is totally ramified in $K$, we establish some results about the set…

In a previous paper of the authors, we showed that for any polynomials $P_1,\dots,P_k \in \Z[\mathbf{m}]$ with $P_1(0)=\dots=P_k(0)$ and any subset $A$ of the primes in $[N] = \{1,\dots,N\}$ of relative density at least $\delta>0$, one can…

Let $p$ be an irregular prime. Let $K=\Q(\zeta)$ be the $p$-cyclotomic field. From Kummer and class field theory, there exist Galois extensions $S/\Q$ of degree $p(p-1)$ such that $S/K$ is a cyclic unramified extension of degree $[S:K]=p$.…

Given an abelian algebraic group $A$ over a global field $F$, $\alpha \in A(F)$, and a prime $\ell$, the set of all preimages of $\alpha$ under some iterate of $[\ell]$ generates an extension of $F$ that contains all $\ell$-power torsion…

For a class of polynomials $f \in \mathbb{Z}[X]$, which in particular includes all quadratic polynomials, and also trinomials of some special form, we show that, under some natural conditions (necessary for quadratic polynomials), the set…

Given a polynomial f(z) = z^d + c over a global field K and a_0 in K, we study the density of prime ideals of K dividing at least one element of the orbit of a_0 under f. The density of such sets for linear polynomials has attracted much…

Let $k$ be a field, $K/k$ finitely generated and $L/K$ a finite, separable extension. We show that the existence of a $k$-valuation on $L$ which ramifies in $L/K$ implies the existence of a normal model $X$ of $K$ and a prime divisor $D$ on…

Let $F$ be a number field. Let $p$ be a prime number. Washington proved the $\ell$-part of the class numbers in cyclotomic $\mathbb{Z}_p$ extension of $F$ is bounded when $F$ is an abelian number field and $\ell\neq p$ is a prime. By class…

In the recent preprint [3], Goldston, Pintz, and Y{\i}ld{\i}r{\i}m established, among other things, $$ \liminf_{n\to\infty}{p_{n+1}-p_n\over\log p_n}=0,\leqno(0) $$ with $p_n$ the $n$th prime. In the present article, which is essentially…

For each prime $p$, let $n(p)$ denote the least quadratic nonresidue modulo $p$. Vinogradov conjectured that $n(p) = O(p^\eps)$ for every fixed $\eps>0$. This conjecture follows from the generalised Riemann hypothesis, and is known to hold…

It is known that there are infinitely-many prime numbers which take the form of a polynomial of degree one with integer coefficients, this is Dirichlet's theorem. We use an elementary sieving argument together with bounds on the prime…