scholarly journals Knots which behave like the prime numbers

2013 ◽  
Vol 149 (8) ◽  
pp. 1235-1244 ◽  
Author(s):  
Curtis T. McMullen
Keyword(s):  
Number Fields ◽  
Prime Numbers ◽  
Density Theorem ◽  
Chebotarev Density Theorem

AbstractThis paper establishes a version of the Chebotarev density theorem in which number fields are replaced by 3-manifolds.

scholarly journals A VARIANT OF THE BARBAN–DAVENPORT– HALBERSTAM THEOREM

2011 ◽  
Vol 07 (08) ◽  
pp. 2203-2218 ◽  
Author(s):  
ETHAN SMITH
Keyword(s):  
Galois Extension ◽  
Error Term ◽  
Number Fields ◽  
Density Theorem ◽  
Prime Ideals ◽  
Congruence Condition ◽  
Chebotarev Density Theorem

Let L/K be a Galois extension of number fields. The problem of counting the number of prime ideals 𝔭 of K with fixed Frobenius class in Gal (L/K) and norm satisfying a congruence condition is considered. We show that the square of the error term arising from the Chebotarëv Density Theorem for this problem is small "on average". The result may be viewed as a variation on the classical Barban–Davenport–Halberstam Theorem.

scholarly journals SMALLEST IRREDUCIBLE OF THE FORM x2-dy2

2009 ◽  
Vol 05 (03) ◽  
pp. 449-456
Author(s):  
SHANSHAN DING
Keyword(s):  
Field Theory ◽  
Finite Field ◽  
Polynomial Ring ◽  
Function Field ◽  
Classical Result ◽  
Prime Numbers ◽  
Density Theorem ◽  
Chebotarev Density Theorem ◽  
Effective Version ◽  
Infinite Set

It is a classical result that prime numbers of the form x2 + ny2 can be characterized via class field theory for an infinite set of n. In this paper, we derive the function field analogue of the classical result. Then, we apply an effective version of the Chebotarev density theorem to bound the degree of the smallest irreducible of the form x2 - dy2, where x, y, and d are elements of a polynomial ring over a finite field.

scholarly journals Explicit counting of ideals and a Brun–Titchmarsh inequality for the Chebotarev density theorem

2019 ◽  
Vol 15 (05) ◽  
pp. 883-905 ◽  
Author(s):  
Korneel Debaene
Keyword(s):  
Arithmetic Progression ◽  
Number Fields ◽  
Field Extension ◽  
Density Theorem ◽  
Explicit Estimate ◽  
Chebotarev Density Theorem ◽  
Set Up ◽  
Selberg's Sieve

We prove a bound on the number of primes with a given splitting behavior in a given field extension. This bound generalizes the Brun–Titchmarsh bound on the number of primes in an arithmetic progression. The proof is set up as an application of Selberg’s Sieve in number fields. The main new ingredient is an explicit counting result estimating the number of integral elements with certain properties up to multiplication by units. As a consequence of this result, we deduce an explicit estimate for the number of ideals of norm up to [Formula: see text].

scholarly journals An effective Chebotarev density theorem for families of number fields, with an application to $$\ell $$-torsion in class groups

2019 ◽  
Vol 219 (2) ◽  
pp. 701-778 ◽  
Author(s):  
Lillian B. Pierce ◽  
Caroline L. Turnage-Butterbaugh ◽  
Melanie Matchett Wood
Keyword(s):  
Number Fields ◽  
Density Theorem ◽  
Class Groups ◽  
Chebotarev Density Theorem

scholarly journals An explicit Chebotarev density theorem under GRH

2019 ◽  
Vol 200 ◽  
pp. 441-485 ◽  
Author(s):  
Loïc Grenié ◽  
Giuseppe Molteni
Keyword(s):  
Density Theorem ◽  
Chebotarev Density Theorem

On the Iwasawa λ-invariant of the cyclotomic ℤ2-extension of ℚ(pq) and the 2-part of the class number of ℚ(pq,2 + 2)

2020 ◽  
pp. 1-28
Author(s):  
Naoki Kumakawa
Keyword(s):  
Upper Bound ◽  
Class Number ◽  
Number Fields ◽  
Prime Numbers

In this paper, we study the Iwasawa [Formula: see text]-invariant of the cyclotomic [Formula: see text]-extension of [Formula: see text], where [Formula: see text] are distinct odd prime numbers satisfying certain arithmetic conditions. Moreover, we obtain an upper bound of the [Formula: see text]-part of the class number of certain quartic number fields by calculating the Sinnott index explicitly.

scholarly journals A bound for the least prime ideal in the Chebotarev Density Theorem

1979 ◽  
Vol 54 (3) ◽  
pp. 271-296 ◽  
Author(s):  
J. C. Lagarias ◽  
H. L. Montgomery ◽  
A. M. Odlyzko
Keyword(s):  
Prime Ideal ◽  
Density Theorem ◽  
Chebotarev Density Theorem

scholarly journals On the moments of torsion points modulo primes and their applications

2020 ◽  
pp. 1-17
Author(s):  
Amir Akbary ◽  
Peng-Jie Wong
Keyword(s):  
Algebraic Group ◽  
Algebraic Groups ◽  
Residue Field ◽  
Density Theorem ◽  
Polynomial Equations ◽  
Absolute Galois Group ◽  
Torsion Points ◽  
Chebotarev Density Theorem ◽  
System Of Polynomial Equations ◽  
Dimension One

Let [Formula: see text] be the group of [Formula: see text]-torsion points of a commutative algebraic group [Formula: see text] defined over a number field [Formula: see text]. For a prime [Formula: see text] of [Formula: see text], we let [Formula: see text] be the number of [Formula: see text]-solutions of the system of polynomial equations defining [Formula: see text] when reduced modulo [Formula: see text]. Here, [Formula: see text] is the residue field at [Formula: see text]. Let [Formula: see text] denote the number of primes [Formula: see text] of [Formula: see text] such that [Formula: see text]. We then, for algebraic groups of dimension one, compute the [Formula: see text]th moment limit [Formula: see text] by appealing to the Chebotarev density theorem. We further interpret this limit as the number of orbits of the action of the absolute Galois group of [Formula: see text] on [Formula: see text] copies of [Formula: see text] by an application of Burnside’s Lemma. These concrete examples suggest a possible approach for determining the number of orbits of a group acting on [Formula: see text] copies of a set.

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