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 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.

A Titchmarsh divisor problem for elliptic curves

2015 ◽  
Vol 160 (1) ◽  
pp. 167-189 ◽  
Author(s):  
PAUL POLLACK
Keyword(s):  
Elliptic Curve ◽  
Complex Multiplication ◽  
Zeta Functions ◽  
Dirichlet Character ◽  
Good Reduction ◽  
Heuristic Argument ◽  
Mean Values ◽  
Average Size ◽  
Image Position ◽  
Ordinary Reduction

AbstractLet E/Q be an elliptic curve with complex multiplication. We study the average size of τ(#E(Fp)) as p varies over primes of good ordinary reduction. We work out in detail the case of E: y2 = x3 − x, where we prove that $$\begin{equation} \sum_{\substack{p \leq x \\p \equiv 1\pmod{4}}} \tau(\#E({\bf{F}}_p)) \sim \left(\frac{5\pi}{16} \prod_{p > 2} \frac{p^4-\chi(p)}{p^2(p^2-1)}\right)x, \quad\text{as $x\to\infty$}. \end{equation}$$ Here χ is the nontrivial Dirichlet character modulo 4. The proof uses number field analogues of the Brun–Titchmarsh and Bombieri–Vinogradov theorems, along with a theorem of Wirsing on mean values of nonnegative multiplicative functions.Now suppose that E/Q is a non-CM elliptic curve. We conjecture that the sum of τ(#E(Fp)), taken over p ⩽ x of good reduction, is ~cEx for some cE > 0, and we give a heuristic argument suggesting the precise value of cE. Assuming the Generalized Riemann Hypothesis for Dedekind zeta functions, we prove that this sum is ≍Ex. The proof uses combinatorial ideas of Erdős.

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 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 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.

On the Étale K-Theory of an Elliptic Curve with Complex Multiplication for Regular Primes

1990 ◽  
Vol 33 (2) ◽  
pp. 145-150 ◽  
Author(s):  
Kay Wingberg
Keyword(s):  
Elliptic Curve ◽  
Prime Number ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Imaginary Quadratic Field ◽  
K Theory ◽  
Ordinary Reduction

AbstractGeneralizing a result of Soulé we prove that for an elliptic curve E defined over an imaginary quadratic field K with complex multiplication having good ordinary reduction at the prime number p > 3 which is regular for E and the extension F of K contained in K(Ep) the dimensions of the étale K-groups are equal to the numbers predicted by Bloch and Beilinson, i.e.,

On the Vanishing of μ-Invariants of Elliptic Curves over ℚ

2005 ◽  
Vol 57 (4) ◽  
pp. 812-843
Author(s):  
Mak Trifković
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Torsion Point ◽  
Selmer Group ◽  
Group Schemes ◽  
The Core ◽  
Explicit Evaluation ◽  
Rings Of Integers ◽  
Greenberg’S Conjecture ◽  
Ordinary Reduction

AbstractLet E/ℚ be an elliptic curve with good ordinary reduction at a prime p > 2. It has a welldefined Iwasawa μ-invariant μ(E)p which encodes part of the information about the growth of the Selmer group ) as Kn ranges over the subfields of the cyclotomic Zp-extension K∞/ℚ. Ralph Greenberg has conjectured that any such E is isogenous to a curve E′ with μ(E′)p = 0. In this paper we prove Greenberg's conjecture for infinitely many curves E with a rational p-torsion point, p = 3 or 5, no two of our examples having isomorphic p-torsion. The core of our strategy is a partial explicit evaluation of the global duality pairing for finite flat group schemes over rings of integers.

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