ترغب بنشر مسار تعليمي؟ اضغط هنا

A note on asymptotically good extensions in which infinitely many primes split completely

145   0   0.0 ( 0 )
 نشر من قبل Christian Maire
 تاريخ النشر 2020
  مجال البحث
والبحث باللغة English
 تأليف Oussama Hamza




اسأل ChatGPT حول البحث

Let p be a prime number, and let K be a number field. For p=2, assume moreover K totally imaginary. In this note we prove the existence of asymptotically good extensions L{K of cohomological dimension 2 in which infinitely many primes split completely. Our result is inspired by a recent work of Hajir, Maire, and Ramakrishna [7].



قيم البحث

اقرأ أيضاً

112 - Jori Merikoski 2020
Analogously to primes in arithmetic progressions to large moduli, we can study primes that are totally split in extensions of $mathbb{Q}$ of high degree. Motivated by a question of Kowalski we focus on the extensions $mathbb{Q}(E[d])$ obtained by adj oining the coordinates of $d$-torsion points of a non-CM elliptic curve $E/mathbb{Q}$. A prime $p$ is said to be an outside prime of $E$ if it is totally split in $mathbb{Q}(E[d])$ for some $d$ with $p<|text{Gal}(mathbb{Q}(E[d])/mathbb{Q})| = d^{4-o(1)}$ (so that $p$ is not accounted for by the expected main term in the Chebotarev Density Theorem). We show that for almost all integers $d$ there exists a non-CM elliptic curve $E/mathbb{Q}$ and a prime $p<|text{Gal}(mathbb{Q}(E[d])/mathbb{Q})|$ which is totally split in $mathbb{Q}(E[d])$. Furthermore, we prove that for almost all $d$ that factorize suitably there exists a non-CM elliptic curve $E/mathbb{Q}$ and a prime $p$ with $p^{0.2694} < d$ which is totally split in $mathbb{Q}(E[d])$. To show this we use work of Kowalski to relate the question to the distribution of primes in certain residue classes modulo $d^2$. Hence, the barrier $p < d^4$ is related to the limit in the classical Bombieri-Vinogradov Theorem. To break past this we make use of the assumption that $d$ factorizes conveniently, similarly as in the works on primes in arithmetic progression to large moduli by Bombieri, Friedlander, Fouvry, and Iwaniec, and in the more recent works of Zhang, Polymath, and the author. In contrast to these works we do not require any of the deep exponential sum bounds (ie. sums of Kloosterman sums or Weil/Deligne bound). Instead, we only require the classical large sieve for multiplicative characters. We use Harmans sieve method to obtain a combinatorial decomposition for primes.
This paper concerns towers of curves over a finite field with many rational points, following Garcia--Stichtenoth and Elkies. We present a new method to produce such towers. A key ingredient is the study of algebraic solutions to Fuchsian differentia l equations modulo $p$. We apply our results to towers of modular curves, and find new asymptotically good towers.
180 - R. C. Baker , A. J. Irving 2015
By combining a sieve method of Harman with the work of Maynard and Tao we show that $$liminf_{nrightarrow infty}(p_{n+m}-p_n)ll exp(3.815m).$$
Let $p>3$ be a prime, and let $(frac{cdot}p)$ be the Legendre symbol. Let $binmathbb Z$ and $varepsilonin{pm 1}$. We mainly prove that $$left|left{N_p(a,b): 1<a<p text{and} left(frac apright)=varepsilonright}right|=frac{3-(frac{-1}p)}2,$$ where $N_p( a,b)$ is the number of positive integers $x<p/2$ with ${x^2+b}_p>{ax^2+b}_p$, and ${m}_p$ with $minmathbb{Z}$ is the least nonnegative residue of $m$ modulo $p$.
69 - Kummari Mallesham 2017
We obtain an upper bound for the number of pairs $ (a,b) in {Atimes B} $ such that $ a+b $ is a prime number, where $ A, B subseteq {1,...,N }$ with $|A||B| , gg frac{N^2}{(log {N})^2}$, $, N geq 1$ an integer. This improves on a bound given by Balog, Rivat and Sarkozy.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا