scholarly journals Self-duality of Selmer groups

2009 ◽  
Vol 146 (2) ◽  
pp. 257-267 ◽  
Author(s):  
TIM DOKCHITSER ◽  
VLADIMIR DOKCHITSER
Keyword(s):  
Galois Group ◽  
Abelian Variety ◽  
Number Field ◽  
Galois Extension ◽  
Selmer Groups ◽  
Selmer Group ◽  
Self Duality ◽  
Tamagawa Numbers

AbstractThe first part of the paper gives a new proof of self-duality for Selmer groups: if A is an abelian variety over a number field K, and F/K is a Galois extension with Galois group G, then the $\Q_p$G-representation naturally associated to the p∞-Selmer group of A/F is self-dual. The second part describes a method for obtaining information about parities of Selmer ranks from the local Tamagawa numbers of A in intermediate extensions of F/K.

SELMER GROUPS AND GENERALIZED CLASS FIELD TOWERS

2012 ◽  
Vol 08 (04) ◽  
pp. 881-909 ◽  
Author(s):  
AHMED MATAR
Keyword(s):  
Galois Group ◽  
Abelian Variety ◽  
Number Field ◽  
Selmer Groups ◽  
Selmer Group ◽  
Class Field ◽  
Finite Set ◽  
Class Field Towers

This paper proves a control theorem for the p-primary Selmer group of an abelian variety with respect to extensions of the form: Maximal pro-p extension of a number field unramified outside a finite set of primes R which does not include any primes dividing p in which another finite set of primes S splits completely. When the Galois group of the extension is not p-adic analytic, the control theorem gives information about p-ranks of Selmer and Tate–Shafarevich groups of the abelian variety. The paper also discusses what can be said in regards to a control theorem when the set R contains all the primes of the number field dividing p.

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 Large Selmer groups over number fields

2009 ◽  
Vol 148 (1) ◽  
pp. 73-86 ◽  
Author(s):  
ALEX BARTEL
Keyword(s):  
Positive Integer ◽  
Elliptic Curve ◽  
Galois Group ◽  
Prime Number ◽  
Galois Extension ◽  
Number Fields ◽  
Selmer Groups ◽  
Selmer Group ◽  
Quadratic Number ◽  
Quadratic Number Field

AbstractLet p be a prime number and M a quadratic number field, M ≠ ℚ() if p ≡ 1 mod 4. We will prove that for any positive integer d there exists a Galois extension F/ℚ with Galois group D2p and an elliptic curve E/ℚ such that F contains M and the p-Selmer group of E/F has size at least pd.

ON FINITE GALOIS STABLE SUBGROUPS OF GLn IN SOME RELATIVE EXTENSIONS OF NUMBER FIELDS

2009 ◽  
Vol 08 (04) ◽  
pp. 493-503 ◽  
Author(s):  
H.-J. BARTELS ◽  
D. A. MALININ
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Galois Extension ◽  
Commutator Subgroup ◽  
Finite Extension ◽  
Number Fields ◽  
Maximal Order ◽  
Linear Groups ◽  
Stable Linear

Let K/ℚ be a finite Galois extension with maximal order [Formula: see text] and Galois group Γ. For finite Γ-stable subgroups [Formula: see text] it is known [4], that they are generated by matrices with coefficients in [Formula: see text], Kab the maximal abelian subextension of K over ℚ. This note gives a contribution to the corresponding question in the case of a relative Galois extension K/R, where R is a finite extension of the rationals ℚ. It turns out, that in this relative situation the answer to the corresponding question depends heavily on the arithmetic of the number field R, more precisely on the ramification behavior of primes in K/R. Due to the possibility of unramified extensions of R for certain number fields R there exist examples of Galois stable linear groups [Formula: see text] which are not fixed elementwise by the commutator subgroup of Gal (K/R).

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 On the pseudo-nullity of the dual fine Selmer groups

2015 ◽  
Vol 11 (07) ◽  
pp. 2055-2063 ◽  
Author(s):  
Meng Fai Lim
Keyword(s):  
Galois Group ◽  
Related Question ◽  
Selmer Groups ◽  
Selmer Group ◽  
Galois Module

In this paper, we will study the pseudo-nullity of the dual fine Selmer group and its related question. We investigate certain situations, where one can deduce the pseudo-nullity of the dual fine Selmer group of a general Galois module over an admissible p-adic Lie extension F∞ from the knowledge of the pseudo-nullity of the Galois group of the maximal abelian unramified pro-p extension of F∞ at which every prime of F∞ above p splits completely. In particular, this gives us a way to construct examples of the pseudo-nullity of the dual fine Selmer group of a Galois module that is unramified outside p. We will illustrate our results with many examples.

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.

FUNCTIONAL EQUATION OF CHARACTERISTIC ELEMENTS OF ABELIAN VARIETIES OVER FUNCTION FIELDS (ℓ ≠ p)

2014 ◽  
Vol 10 (03) ◽  
pp. 705-735
Author(s):  
APRAMEYO PAL
Keyword(s):  
Functional Equation ◽  
Number Field ◽  
Function Field ◽  
Functional Equations ◽  
Function Fields ◽  
Growth Behavior ◽  
Iwasawa Theory ◽  
Selmer Groups ◽  
Selmer Group ◽  
Field Case

In this paper we apply methods from the number field case of Perrin-Riou [20] and Zábrádi [32] in the function field setup. In ℤℓ- and GL2-cases (ℓ ≠ p), we prove algebraic functional equations of the Pontryagin dual of Selmer group which give further evidence of the main conjectures of Iwasawa theory. We also prove some parity conjectures in commutative and non-commutative cases. As a consequence, we also get results on the growth behavior of Selmer groups in commutative and non-commutative extension of function fields.

scholarly journals On central extensions of a Galois extension of algebraic number fields

1984 ◽  
Vol 93 ◽  
pp. 133-148 ◽  
Author(s):  
Katsuya Miyake
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Algebraic Number ◽  
Galois Extension ◽  
Central Extension ◽  
Algebraic Closure ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Algebraic Number Fields ◽  
Central Extensions

Let k be an algebraic number field of finite degree, and K a finite Galois extension of k. A central extension L of K/k is an algebraic number field which contains K and is normal over k, and whose Galois group over K is contained in the center of the Galois group Gal(L/k). We denote the maximal abelian extensions of k and K in the algebraic closure of k by kab and Kab respectively, and the maximal central extension of K/k by MCK/k. Then we have Kab⊃MCK/k⊃kab·K.

scholarly journals A norm residue map for central extensions of an algebraic number field

1984 ◽  
Vol 93 ◽  
pp. 61-69 ◽  
Author(s):  
Yoshiomi Furuta
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Algebraic Number ◽  
Galois Extension ◽  
Algebraic Number Field ◽  
Abelian Extension ◽  
Factor Group ◽  
Central Extensions ◽  
Norm Residue ◽  
Genus Field

Let K be a finite Galois extension of an algebraic number field k with G = Gal (K/k), and M be a Galois extension of k containing K. We denote by resp. the genus field resp. the central class field of K with respect to M/k. By definition, the field is the composite of K and the maximal abelian extension over k contained in M. The field is the maximal Galois extension of k contained in M satisfying the condition that the Galois group over K is contained in the center of that over k. Then it is well known that Gal is isomorphic to a factor group of the Schur multiplicator H-3(G, Z), and is isomorphic to H-3(G, Z) when M is sufficiently large. In this case we call M abundant for K/k (See Heider [3, § 4] and Miyake [6, Theorem 5]).

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