scholarly journals On the Irreducibility of Polynomials Associated with the Complete Residue Systems in any Imaginary Quadratic Fields

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Phitthayathon Phetnun ◽  
Narakorn Rompurk Kanasri ◽  
Patiwat Singthongla
Keyword(s):  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Gaussian Integer ◽  
Ring Of Integers ◽  
Complete Residue System ◽  
Explicit Representations ◽  
Residue Systems

For a Gaussian prime π and a nonzero Gaussian integer β = a + b i ∈ ℤ i with a ≥ 1 and β ≥ 2 + 2 , it was proved that if π = α n β n + α n − 1 β n − 1 + ⋯ + α 1 β + α 0 ≕ f β where n ≥ 1 , α n ∈ ℤ i \ 0 , α 0 , … , α n − 1 belong to a complete residue system modulo β , and the digits α n − 1 and α n satisfy certain restrictions, then the polynomial f x is irreducible in ℤ i x . For any quadratic field K ≔ ℚ m , it is well known that there are explicit representations for a complete residue system in K , but those of the case m ≡ 1   mod 4 are inapplicable to this work. In this article, we establish a new complete residue system for such a case and then generalize the result mentioned above for the ring of integers of any imaginary quadratic field.

scholarly journals NOTE ON THE DIVISIBILITY OF THE CLASS NUMBER OF CERTAIN IMAGINARY QUADRATIC FIELDS

2009 ◽  
Vol 51 (1) ◽  
pp. 187-191 ◽  
Author(s):  
YASUHIRO KISHI
Keyword(s):  
Class Number ◽  
Diophantine Equations ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Positive Integers

AbstractWe prove that the class number of the imaginary quadratic field $\Q(\sqrt{2^{2k}-3^n})$ is divisible by n for any positive integers k and n with 22k < 3n, by using Y. Bugeaud and T. N. Shorey's result on Diophantine equations.

scholarly journals Similar Sublattices of Planar Lattices

2011 ◽  
Vol 63 (6) ◽  
pp. 1220-1537 ◽  
Author(s):  
Michael Baake ◽  
Rudolf Scharlau ◽  
Peter Zeiner
Keyword(s):  
Dirichlet Series ◽  
Generating Functions ◽  
Quadratic Field ◽  
Zeta Functions ◽  
Imaginary Quadratic Field ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Planar Lattice ◽  
Planar Lattices

AbstractThe similar sublattices of a planar lattice can be classified via its multiplier ring. The latter is the ring of rational integers in the generic case, and an order in an imaginary quadratic field otherwise. Several classes of examples are discussed, with special emphasis on concrete results. In particular, we derive Dirichlet series generating functions for the number of distinct similar sublattices of a given index, and relate them to zeta functions of orders in imaginary quadratic fields.

The Square Sieve and the Lang–Trotter Conjecture

2005 ◽  
Vol 57 (6) ◽  
pp. 1155-1177 ◽  
Author(s):  
Alina Carmen Cojocaru ◽  
Etienne Fouvry ◽  
M. Ram Murty
Keyword(s):  
Elliptic Curve ◽  
Riemann Hypothesis ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Upper Bounds ◽  
Imaginary Quadratic Field ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Frobenius Endomorphism ◽  
Square Sieve

AbstractLet E be an elliptic curve defined over ℚ and without complex multiplication. Let K be a fixed imaginary quadratic field. We find nontrivial upper bounds for the number of ordinary primes p ≤ x for which ℚ(πp) = K, where πp denotes the Frobenius endomorphism of E at p. More precisely, under a generalized Riemann hypothesis we show that this number is OE(x17/18 log x), and unconditionally we show that this number is We also prove that the number of imaginary quadratic fields K, with −disc K ≤ x and of the form K = ℚ(πp), is ≫E log log log x for x ≥ x0(E). These results represent progress towards a 1976 Lang–Trotter conjecture.

scholarly journals Euler systems for modular forms over imaginary quadratic fields

2015 ◽  
Vol 151 (9) ◽  
pp. 1585-1625 ◽  
Author(s):  
Antonio Lei ◽  
David Loeffler ◽  
Sarah Livia Zerbes
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Quadratic Field ◽  
Euler System ◽  
Imaginary Quadratic Field ◽  
Quadratic Fields ◽  
Selmer Groups ◽  
Imaginary Quadratic Fields ◽  
Euler Systems

We construct an Euler system attached to a weight 2 modular form twisted by a Grössencharacter of an imaginary quadratic field $K$, and apply this to bounding Selmer groups.

scholarly journals On the Gras Conjecture for Imaginary Quadratic Fields

2013 ◽  
Vol 56 (1) ◽  
pp. 148-160
Author(s):  
Hassan Oukhaba ◽  
Stéphane Viguié
Keyword(s):  
Quadratic Field ◽  
Prime Numbers ◽  
Imaginary Quadratic Field ◽  
Quadratic Fields ◽  
Roots Of Unity ◽  
Imaginary Quadratic Fields ◽  
Abelian Extensions

AbstractIn this paper we extend K. Rubin's methods to prove the Gras conjecture for abelian extensions of a given imaginary quadratic field k and prime numbers p that divide the number of roots of unity in k.

On a conjecture of Chen and Yui: Resultants and discriminants

2020 ◽  
pp. 1-41
Author(s):  
Dongxi Ye
Keyword(s):  
Recent Result ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Prime Decomposition ◽  
Class Polynomials

Abstract In [5], Chen and Yui conjectured that Gross–Zagier type formulas may also exist for Thompson series. In this work, we verify Chen and Yui’s conjecture for the cases for Thompson series $j_{p}(\tau )$ for $\Gamma _{0}(p)$ for p prime, and equivalently establish formulas for the prime decomposition of the resultants of two ring class polynomials associated to $j_{p}(\tau )$ and imaginary quadratic fields and the prime decomposition of the discriminant of a ring class polynomial associated to $j_{p}(\tau )$ and an imaginary quadratic field. Our method for tackling Chen and Yui’s conjecture on resultants can be used to give a different proof to a recent result of Yang and Yin. In addition, as an implication, we verify a conjecture recently raised by Yang, Yin, and Yu.

scholarly journals Averages and moments associated to class numbers of imaginary quadratic fields

2017 ◽  
Vol 153 (11) ◽  
pp. 2287-2309 ◽  
Author(s):  
D. R. Heath-Brown ◽  
L. B. Pierce
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Upper Bounds ◽  
Imaginary Quadratic Field ◽  
Quadratic Fields ◽  
Class Numbers ◽  
Higher Moments ◽  
Imaginary Quadratic Fields

For any odd prime $\ell$, let $h_{\ell }(-d)$ denote the $\ell$-part of the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-d})$. Nontrivial pointwise upper bounds are known only for $\ell =3$; nontrivial upper bounds for averages of $h_{\ell }(-d)$ have previously been known only for $\ell =3,5$. In this paper we prove nontrivial upper bounds for the average of $h_{\ell }(-d)$ for all primes $\ell \geqslant 7$, as well as nontrivial upper bounds for certain higher moments for all primes $\ell \geqslant 3$.

scholarly journals Iwasawa Theory of Elliptic Modular Forms Over Imaginary Quadratic Fields at Non-ordinary Primes

2019 ◽  
Author(s):  
Kâzım Büyükboduk ◽  
Antonio Lei
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Quadratic Field ◽  
Euler System ◽  
Base Change ◽  
Iwasawa Theory ◽  
Imaginary Quadratic Field ◽  
Selmer Groups ◽  
Imaginary Quadratic Fields ◽  
Elliptic Modular Form

AbstractThis article is a continuation of our previous work [7] on the Iwasawa theory of an elliptic modular form over an imaginary quadratic field $K$, where the modular form in question was assumed to be ordinary at a fixed odd prime $p$. We formulate integral Iwasawa main conjectures at non-ordinary primes $p$ for suitable twists of the base change of a newform $f$ to an imaginary quadratic field $K$ where $p$ splits, over the cyclotomic ${\mathbb{Z}}_p$-extension, the anticyclotomic ${\mathbb{Z}}_p$-extensions (in both the definite and the indefinite cases) as well as the ${\mathbb{Z}}_p^2$-extension of $K$. In order to do so, we define Kobayashi–Sprung-style signed Coleman maps, which we use to introduce doubly signed Selmer groups. In the same spirit, we construct signed (integral) Beilinson–Flach elements (out of the collection of unbounded Beilinson–Flach elements of Loeffler–Zerbes), which we use to define doubly signed $p$-adic $L$-functions. The main conjecture then relates these two sets of objects. Furthermore, we show that the integral Beilinson–Flach elements form a locally restricted Euler system, which in turn allow us to deduce (under certain technical assumptions) one inclusion in each one of the four main conjectures we formulate here (which may be turned into equalities in favorable circumstances).

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