Bounding the Iwasawa invariants of Selmer groups

2020 ◽  
pp. 1-33
Author(s):  
Sören Kleine
Keyword(s):  
Number Field ◽  
Analogous Result ◽  
Abelian Varieties ◽  
Fixed Number ◽  
Selmer Groups ◽  
Iwasawa Invariants ◽  
Ordinary Reduction

Abstract We study the growth of p-primary Selmer groups of abelian varieties with good ordinary reduction at p in ${{Z}}_p$ -extensions of a fixed number field K. Proving that in many situations the knowledge of the Selmer groups in a sufficiently large number of finite layers of a ${{Z}}_p$ -extension over K suffices for bounding the over-all growth, we relate the Iwasawa invariants of Selmer groups in different ${{Z}}_p$ -extensions of K. As applications, we bound the growth of Mordell–Weil ranks and the growth of Tate-Shafarevich groups. Finally, we derive an analogous result on the growth of fine Selmer groups.

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 On the Selmer groups of abelian varieties over function fields of characteristic p > 0

2009 ◽  
Vol 146 (1) ◽  
pp. 23-43 ◽  
Author(s):  
TADASHI OCHIAI ◽  
FABIEN TRIHAN
Keyword(s):  
Number Field ◽  
Function Field ◽  
Function Fields ◽  
Abelian Varieties ◽  
Iwasawa Theory ◽  
Selmer Groups ◽  
Characteristic P ◽  
Field Case ◽  
New Phenomena ◽  
Geometric Analogue

AbstractWe study a (p-adic) geometric analogue for abelian varieties over a function field of characteristic p of the cyclotomic Iwasawa theory and the non-commutative Iwasawa theory for abelian varieties over a number field initiated by Mazur and Coates respectively. We will prove some analogue of the principal results obtained in the case over a number field and we study new phenomena which did not happen in the case of number field case. We also propose a conjecture (Conjecture 1.6) which might be considered as a counterpart of the principal conjecture in the case over a number field.

scholarly journals $3$ -Isogeny Selmer groups and ranks of Abelian varieties in quadratic twist families over a number field

2019 ◽  
Vol 168 (15) ◽  
pp. 2951-2989
Author(s):  
Manjul Bhargava ◽  
Zev Klagsbrun ◽  
Robert J. Lemke Oliver ◽  
Ari Shnidman
Keyword(s):  
Number Field ◽  
Abelian Varieties ◽  
Selmer Groups ◽  
Quadratic Twist

Generalised Iwasawa invariants and the growth of class numbers

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Sören Kleine
Keyword(s):  
Number Field ◽  
Fixed Number ◽  
Class Numbers ◽  
Open Neighbourhood ◽  
Finitely Generated ◽  
Asymptotic Growth ◽  
Iwasawa Invariants ◽  
Locally Maximal ◽  
Global Boundedness ◽  
The Impact

AbstractWe study the generalised Iwasawa invariants of {\mathbb{Z}_{p}^{d}}-extensions of a fixed number field K. Based on an inequality between ranks of finitely generated torsion {\mathbb{Z}_{p}[\kern-2.133957pt[T_{1},\dots,T_{d}]\kern-2.133957pt]}-modules and their corresponding elementary modules, we prove that these invariants are locally maximal with respect to a suitable topology on the set of {\mathbb{Z}_{p}^{d}}-extensions of K, i.e., that the generalised Iwasawa invariants of a {\mathbb{Z}_{p}^{d}}-extension {\mathbb{K}} of K bound the invariants of all {\mathbb{Z}_{p}^{d}}-extensions of K in an open neighbourhood of {\mathbb{K}}. Moreover, we prove an asymptotic growth formula for the class numbers of the intermediate fields in certain {\mathbb{Z}_{p}^{2}}-extensions, which improves former results of Cuoco and Monsky. We also briefly discuss the impact of generalised Iwasawa invariants on the global boundedness of Iwasawa λ-invariants.

scholarly journals Level structures on abelian varieties and Vojta’s conjecture

2017 ◽  
Vol 153 (2) ◽  
pp. 373-394 ◽  
Author(s):  
Dan Abramovich ◽  
Anthony Várilly-Alvarado
Keyword(s):  
Positive Integer ◽  
Recent Work ◽  
Abelian Variety ◽  
Number Field ◽  
Abelian Varieties ◽  
Fixed Number ◽  
Vojta's Conjecture

Assuming Vojta’s conjecture, and building on recent work of the authors, we prove that, for a fixed number field $K$ and a positive integer $g$, there is an integer $m_{0}$ such that for any $m>m_{0}$ there is no principally polarized abelian variety $A/K$ of dimension $g$ with full level-$m$ structure. To this end, we develop a version of Vojta’s conjecture for Deligne–Mumford stacks, which we deduce from Vojta’s conjecture for schemes.

IWASAWA TYPE FORMULA FOR COVERS OF A LINK IN A RATIONAL HOMOLOGY SPHERE

2008 ◽  
Vol 17 (10) ◽  
pp. 1199-1221 ◽  
Author(s):  
TERUHISA KADOKAMI ◽  
YASUSHI MIZUSAWA
Keyword(s):  
Number Field ◽  
Class Number ◽  
Homology Sphere ◽  
Rational Homology ◽  
Homology Groups ◽  
Cyclic Covers ◽  
Class Number Formula ◽  
Number Formula ◽  
Iwasawa Invariants ◽  
Rational Homology Sphere

Based on the analogy between links and primes, we present an analogue of the Iwasawa's class number formula in a Zp-extension for the p-homology groups of pn-fold cyclic covers of a link in a rational homology 3-sphere. We also describe the associated Iwasawa invariants precisely for some examples and discuss analogies with the number field case.

scholarly journals ON SELMER RANK PARITY OF TWISTS

2016 ◽  
Vol 102 (3) ◽  
pp. 316-330 ◽  
Author(s):  
MAJID HADIAN ◽  
MATTHEW WEIDNER
Keyword(s):  
Elliptic Curves ◽  
Abelian Variety ◽  
Number Field ◽  
Arbitrary Number ◽  
Hyperelliptic Curve ◽  
Abelian Varieties ◽  
Hyperelliptic Curves ◽  
Torsion Subgroup ◽  
Tate Conjecture ◽  
Quadratic Twists

In this paper we study the variation of the $p$-Selmer rank parities of $p$-twists of a principally polarized Abelian variety over an arbitrary number field $K$ and show, under certain assumptions, that this parity is periodic with an explicit period. Our result applies in particular to principally polarized Abelian varieties with full $K$-rational $p$-torsion subgroup, arbitrary elliptic curves, and Jacobians of hyperelliptic curves. Assuming the Shafarevich–Tate conjecture, our result allows one to classify the rank parities of all quadratic twists of an elliptic or hyperelliptic curve after a finite calculation.

Sur les invariants d'Iwasawa des tours cyclotomiques

2003 ◽  
Vol 46 (2) ◽  
pp. 178-190 ◽  
Author(s):  
Jean-François Jaulent ◽  
Christian Maire
Keyword(s):  
Number Field ◽  
Iwasawa Invariants

AbstractWe carry out the computation of the Iwasawa invariants associated to abelian T-ramified S-decomposed ℓ-extensions over the finite steps Kn of the cyclotomic -extension K∞/K of a number field of CM-type.

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