scholarly journals The 2-Selmer group of a number field and heuristics for narrow class groups and signature ranks of units

2018 ◽  
Vol 117 (4) ◽  
pp. 682-726 ◽  
Author(s):  
David S. Dummit ◽  
John Voight
Keyword(s):  
Number Field ◽  
Selmer Group ◽  
Class Groups ◽  
Narrow Class

scholarly journals Algebraic functional equations and completely faithful Selmer groups

2015 ◽  
Vol 11 (04) ◽  
pp. 1233-1257
Author(s):  
Tibor Backhausz ◽  
Gergely Zábrádi
Keyword(s):  
Functional Equation ◽  
Elliptic Curve ◽  
Galois Group ◽  
Number Field ◽  
Functional Equations ◽  
Complex Multiplication ◽  
Selmer Groups ◽  
Selmer Group ◽  
Torsion Module ◽  
Ordinary Reduction

Let E be an elliptic curve — defined over a number field K — without complex multiplication and with good ordinary reduction at all the primes above a rational prime p ≥ 5. We construct a pairing on the dual p∞-Selmer group of E over any strongly admissible p-adic Lie extension K∞/K under the assumption that it is a torsion module over the Iwasawa algebra of the Galois group G = Gal(K∞/K). Under some mild additional hypotheses, this gives an algebraic functional equation of the conjectured p-adic L-function. As an application, we construct completely faithful Selmer groups in case the p-adic Lie extension is obtained by adjoining the p-power division points of another non-CM elliptic curve A.

scholarly journals A Density of Ramified Primes

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Stephanie Chan ◽  
Christine McMeekin ◽  
Djordjo Milovic
Keyword(s):  
Number Field ◽  
Class Number ◽  
Joint Distribution ◽  
Number Fields ◽  
Character Sums ◽  
Prime Ideals ◽  
Odd Degree ◽  
Narrow Class

AbstractLet K be a cyclic number field of odd degree over $${\mathbb {Q}}$$ Q with odd narrow class number, such that 2 is inert in $$K/{\mathbb {Q}}$$ K / Q . We define a family of number fields $$\{K(p)\}_p$$ { K ( p ) } p , depending on K and indexed by the rational primes p that split completely in $$K/{\mathbb {Q}}$$ K / Q , in which p is always ramified of degree 2. Conditional on a standard conjecture on short character sums, the density of such rational primes p that exhibit one of two possible ramified factorizations in $$K(p)/{\mathbb {Q}}$$ K ( p ) / Q is strictly between 0 and 1 and is given explicitly as a formula in terms of the degree of the extension $$K/{\mathbb {Q}}$$ K / Q . Our results are unconditional in the cubic case. Our proof relies on a detailed study of the joint distribution of spins of prime ideals.

scholarly journals Higher genus theory

2020 ◽  
Author(s):  
Peter Koymans ◽  
Carlo Pagano
Keyword(s):  
Number Field ◽  
Quadratic Forms ◽  
Class Group ◽  
Quadratic Extension ◽  
Number Fields ◽  
Explicit Description ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Genus Theory ◽  
Narrow Class

Abstract In $1801$, Gauss found an explicit description, in the language of binary quadratic forms, for the $2$-torsion of the narrow class group and dual narrow class group of a quadratic number field. This is now known as Gauss’s genus theory. In this paper, we extend Gauss’s work to the setting of multi-quadratic number fields. To this end, we introduce and parametrize the categories of expansion groups and expansion Lie algebras, giving an explicit description for the universal objects of these categories. This description is inspired by the ideas of Smith [ 16] in his recent breakthrough on Goldfeld’s conjecture and the Cohen–Lenstra conjectures. Our main result shows that the maximal unramified multi-quadratic extension $L$ of a multi-quadratic number field $K$ can be reconstructed from the set of generalized governing expansions supported in the set of primes that ramify in $K$. This provides a recursive description for the group $\textrm{Gal}(L/\mathbb{Q})$ and a systematic procedure to construct the field $L$. A special case of our main result gives an upper bound for the size of $\textrm{Cl}^{+}(K)[2]$.

Capitulation in the cyclotomic ℤ2 extension of CM number fields

2018 ◽  
Vol 166 (2) ◽  
pp. 371-380
Author(s):  
KATHARINA MÜLLER
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Class Groups ◽  
Definition Of

AbstractLet 𝕂n be the intermediate steps in the cyclotomic ℤp-extension of a CM number field 𝕂. For p ⧧ 2 the minus part of the p-class groups is given by An− = $\frac{1}{2}$(1 − j)An. We will give a new definition of the minus part for p = 2 and will prove that there is no finite submodule in lim∞←nAn−. Furthermore we will show that μ = 0 if and only if μ− = 0 in this new definition.

scholarly journals The Selmer groups and the ambiguous ideal class groups of cubic fields

1996 ◽  
Vol 54 (2) ◽  
pp. 267-274
Author(s):  
Yen-Mei J. Chen
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Upper Bound ◽  
Selmer Groups ◽  
Ideal Class ◽  
Selmer Group ◽  
Class Groups ◽  
Ideal Class Groups ◽  
Cubic Fields ◽  
Rational Isogeny

In this paper, we study a family of elliptic curves with CM by which also admits a ℚ-rational isogeny of degree 3. We find a relation between the Selmer groups of the elliptic curves and the ambiguous ideal class groups of certain cubic fields. We also find some bounds for the dimension of the 3-Selmer group over ℚ, whose upper bound is also an upper bound of the rank of the elliptic curve.

Théorie d’Iwasawa des unités de Stark et groupe de classes

2017 ◽  
Vol 13 (05) ◽  
pp. 1165-1190 ◽  
Author(s):  
Jilali Assim ◽  
Youness Mazigh ◽  
Hassan Oukhaba
Keyword(s):  
Number Field ◽  
Finite Extension ◽  
Projective Limit ◽  
Ideal Class ◽  
Class Groups ◽  
Euler Systems ◽  
Ideal Class Groups ◽  
Stark Units ◽  
Characteristic Ideal ◽  
The Ideal

Let [Formula: see text] be a number field and let [Formula: see text] be an odd rational prime. Let [Formula: see text] be a [Formula: see text]-extension of [Formula: see text] and let [Formula: see text] be a finite extension of [Formula: see text], abelian over [Formula: see text]. In this paper we extend the classical results, e.g. [16], relating characteristic ideal of the [Formula: see text]-quotient of the projective limit of the ideal class groups to the [Formula: see text]-quotient of the projective limit of units modulo Stark units, in the non-semi-simple case, for some [Formula: see text]-irreductible characters [Formula: see text] of [Formula: see text]. The proof essentially uses the theory of Euler systems.

Algebraic functional equation for Hida family

2014 ◽  
Vol 10 (07) ◽  
pp. 1649-1674
Author(s):  
Somnath Jha ◽  
Aprameyo Pal
Keyword(s):  
Functional Equation ◽  
Number Field ◽  
Modular Forms ◽  
Euler System ◽  
Selmer Groups ◽  
Selmer Group ◽  
Main Conjecture ◽  
Hida Family ◽  
General Number ◽  
The Individual

We prove a functional equation for the characteristic ideal of the "big" Selmer group 𝒳(𝒯ℱ/F cyc ) associated to an ordinary Hida family of elliptic modular forms over the cyclotomic ℤp extension of a general number field F, under the assumption that there is at least one arithmetic specialization whose Selmer group is torsion over its Iwasawa algebra. For a general number field, the two-variable cyclotomic Iwasawa main conjecture for ordinary Hida family is not proved and this can be thought of as an evidence to the validity of the Iwasawa main conjecture. The central idea of the proof is to prove a variant of the result of Perrin-Riou [Groupes de Selmer et accouplements; cas particulier des courbes elliptiques, Doc. Math.2003 (2003) 725–760, Extra Volume: Kazuya Kato's fiftieth birthday] by constructing a generalized pairing on the individual Selmer groups corresponding to the arithmetic points and make use of the appropriate specialization techniques of Ochiai [Euler system for Galois deformations, Ann. Inst. Fourier (Grenoble)55(1) (2005) 113–146].

scholarly journals Reducing number field defining polynomials: an application to class group computations

2016 ◽  
Vol 19 (A) ◽  
pp. 315-331
Author(s):  
Alexandre Gélin ◽  
Antoine Joux
Keyword(s):  
Number Field ◽  
Class Group ◽  
Main Idea ◽  
Class Groups ◽  
Algebraic Integers ◽  
Small Height ◽  
Subexponential Algorithm ◽  
One To One ◽  
The One

In this paper we describe how to compute smallest monic polynomials that define a given number field $\mathbb{K}$. We make use of the one-to-one correspondence between monic defining polynomials of $\mathbb{K}$ and algebraic integers that generate $\mathbb{K}$. Thus, a smallest polynomial corresponds to a vector in the lattice of integers of $\mathbb{K}$ and this vector is short in some sense. The main idea is to consider weighted coordinates for the vectors of the lattice of integers of $\mathbb{K}$. This allows us to find the desired polynomial by enumerating short vectors in these weighted lattices. In the context of the subexponential algorithm of Biasse and Fieker for computing class groups, this algorithm can be used as a precomputation step that speeds up the rest of the computation. It also widens the applicability of their faster conditional method, which requires a defining polynomial of small height, to a much larger set of number field descriptions.

scholarly journals Selmer Groups of Elliptic Curves with Complex Multiplication

2004 ◽  
Vol 56 (1) ◽  
pp. 194-208
Author(s):  
A. Saikia
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Simple Formula ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Selmer Groups ◽  
Selmer Group ◽  
Finitely Generated ◽  
Ring Of Integers

AbstractSuppose K is an imaginary quadratic field and E is an elliptic curve over a number field F with complex multiplication by the ring of integers in K. Let p be a rational prime that splits as in K. Let Epn denote the pn-division points on E. Assume that F(Epn) is abelian over K for all n ≥ 0. This paper proves that the Pontrjagin dual of the -Selmer group of E over F(Ep∞) is a finitely generated free Λ-module, where Λ is the Iwasawa algebra of . It also gives a simple formula for the rank of the Pontrjagin dual as a Λ-module.

scholarly journals Composition of binary quadratic forms over number fields

2021 ◽  
Vol 71 (6) ◽  
pp. 1339-1360
Author(s):  
Kristýna Zemková
Keyword(s):  
Number Field ◽  
Quadratic Forms ◽  
Number Fields ◽  
Binary Quadratic Form ◽  
Base Number ◽  
Binary Quadratic Forms ◽  
Quadratic Number Field ◽  
Totally Positive ◽  
Equivalent Conditions ◽  
Narrow Class

Abstract In this article, the standard correspondence between the ideal class group of a quadratic number field and the equivalence classes of binary quadratic forms of given discriminant is generalized to any base number field of narrow class number one. The article contains an explicit description of the correspondence. In the case of totally negative discriminants, equivalent conditions are given for a binary quadratic form to be totally positive definite.

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