scholarly journals A remark on the 𝔐H(G)-conjecture and Akashi series

2014 ◽  
Vol 11 (01) ◽  
pp. 269-297 ◽  
Author(s):  
Meng Fai Lim
Keyword(s):  
Modular Form ◽  
Elliptic Curve ◽  
Selmer Groups ◽  
Selmer Group ◽  
Ordinary Reduction

In this paper, we give a criterion for the dual Selmer group of an elliptic curve which has either good ordinary reduction or multiplicative reduction at every prime above p to satisfy the 𝔐H(G)-conjecture. As a by-product of our calculations, we are able to define the Akashi series of the dual Selmer groups assuming the conjectures of Mazur and Schneider. Previously, the Akashi series are defined under the stronger assumption that the dual Selmer group satisfies the 𝔐H(G)-conjecture. We then establish a criterion for the vanishing of the dual Selmer groups using the Akashi series. We will apply this criterion to prove some results on the characteristic elements of the dual Selmer groups. Our methods in this paper are inspired by the work of Coates–Schneider–Sujatha and can be extended to the Greenberg Selmer groups attached to other ordinary representations, for instance, those coming from a p-ordinary modular form.

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 THE PLUS/MINUS SELMER GROUPS FOR SUPERSINGULAR PRIMES

2013 ◽  
Vol 95 (2) ◽  
pp. 189-200 ◽  
Author(s):  
BYOUNG DU KIM
Keyword(s):  
Elliptic Curve ◽  
Galois Group ◽  
Finite Index ◽  
Selmer Groups ◽  
Selmer Group ◽  
Supersingular Primes ◽  
Reduction Case ◽  
Ordinary Reduction ◽  
Iwasawa Algebra

AbstractSuppose that an elliptic curve $E$ over $ \mathbb{Q} $ has good supersingular reduction at $p$. We prove that Kobayashi’s plus/minus Selmer group of $E$ over a ${ \mathbb{Z} }_{p} $-extension has no proper $\Lambda $-submodule of finite index under some suitable conditions, where $\Lambda $ is the Iwasawa algebra of the Galois group of the ${ \mathbb{Z} }_{p} $-extension. This work is analogous to Greenberg’s result in the ordinary reduction case.

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.

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 On the Refined Conjectures on Fitting Ideals of Selmer Groups of Elliptic Curves with Supersingular Reduction

2019 ◽  
Author(s):  
Chan-Ho Kim ◽  
Masato Kurihara
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Error Term ◽  
Iwasawa Theory ◽  
Selmer Groups ◽  
Main Conjecture ◽  
Finite Layer ◽  
Ordinary Reduction ◽  
Explicit Comparison

AbstractIn this paper, we study the Fitting ideals of Selmer groups over finite subextensions in the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ of an elliptic curve over $\mathbb{Q}$. Especially, we present a proof of the “weak main conjecture” à la Mazur and Tate for elliptic curves with good (supersingular) reduction at an odd prime $p$. We also prove the “strong main conjecture” suggested by the second named author under the validity of the $\pm $-main conjecture and the vanishing of a certain error term. The key idea is the explicit comparison among “finite layer objects”, “$\pm $-objects”, and “fine objects” in Iwasawa theory. The case of good ordinary reduction is also treated.

scholarly journals Computing the Cassels–Tate pairing on the 3-Selmer group of an elliptic curve

2014 ◽  
Vol 10 (07) ◽  
pp. 1881-1907 ◽  
Author(s):  
Tom Fisher ◽  
Rachel Newton
Keyword(s):  
Elliptic Curve ◽  
Upper Bound ◽  
Selmer Groups ◽  
Selmer Group ◽  
Tate Pairing

We extend the method of Cassels for computing the Cassels–Tate pairing on the 2-Selmer group of an elliptic curve, to the case of 3-Selmer groups. This requires significant modifications to both the local and global parts of the calculation. Our method is practical in sufficiently small examples, and can be used to improve the upper bound for the rank of an elliptic curve obtained by 3-descent.

scholarly journals CM cycles on Kuga–Sato varieties over Shimura curves and Selmer groups

2018 ◽  
Vol 30 (2) ◽  
pp. 321-346
Author(s):  
Yara Elias ◽  
Carlos de Vera-Piquero
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Cohomology Class ◽  
Quadratic Field ◽  
Galois Cohomology ◽  
Euler System ◽  
Shimura Curves ◽  
Selmer Groups ◽  
Selmer Group ◽  
Shimura Curve

AbstractGiven a modular form{{f}}of even weight larger than two and an imaginary quadratic field{{K}}satisfying a relaxed Heegner hypothesis, we construct a collection of CM cycles on a Kuga–Sato variety over a suitable Shimura curve which gives rise to a system of Galois cohomology classes attached to{{f}}enjoying the compatibility properties of an Euler system. Then we use Kolyvagin’s method [21], as adapted by Nekovář [28] to higher weight modular forms, to bound the size of the relevant Selmer group associated to{{f}}and{{K}}and prove the finiteness of the (primary part) of the Shafarevich–Tate group, provided that a suitable cohomology class does not vanish.

scholarly journals Selmer Groups and Anticyclotomic Zp-extensions

2016 ◽  
Vol 161 (3) ◽  
pp. 409-433 ◽  
Author(s):  
AHMED MATAR
Keyword(s):  
Elliptic Curve ◽  
Selmer Groups ◽  
Heegner Points ◽  
Imaginary Field ◽  
Ordinary Reduction

AbstractLet E/Q be an elliptic curve, p a prime and K∞/K the anticyclotomic Zp-extension of a quadratic imaginary field K satisfying the Heegner hypothesis. In this paper we give a new proof to a theorem of Bertolini which determines the value of the Λ-corank of Selp∞(E/K∞) in the case where E has ordinary reduction at p. In the case where E has supersingular reduction at p we make a conjecture about the structure of the module of Heegner points mod p. Assuming this conjecture we give a new proof to a theorem of Ciperiani which determines the value of the Λ-corank of Selp∞(E/K∞) in the case where E has supersingular reduction at p.

ON THE SELMER GROUP OF A CERTAIN -ADIC LIE EXTENSION

2019 ◽  
Vol 100 (2) ◽  
pp. 245-255
Author(s):  
AMALA BHAVE ◽  
LACHIT BORA
Keyword(s):  
Elliptic Curve ◽  
Complex Multiplication ◽  
Quadratic Extension ◽  
Selmer Group ◽  
Ordinary Reduction ◽  
Iwasawa Module

Let $E$ be an elliptic curve over $\mathbb{Q}$ without complex multiplication. Let $p\geq 5$ be a prime in $\mathbb{Q}$ and suppose that $E$ has good ordinary reduction at $p$. We study the dual Selmer group of $E$ over the compositum of the $\text{GL}_{2}$ extension and the anticyclotomic $\mathbb{Z}_{p}$-extension of an imaginary quadratic extension as an Iwasawa module.

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.

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