Fine Selmer groups, Heegner points and anticyclotomic ℤp-extensions

2018 ◽  
Vol 14 (05) ◽  
pp. 1279-1304
Author(s):  
Ahmed Matar
Keyword(s):  
Elliptic Curve ◽  
Selmer Groups ◽  
Selmer Group ◽  
Heegner Points ◽  
Imaginary Field

Let [Formula: see text] be an elliptic curve, [Formula: see text] a prime and [Formula: see text] the anticyclotomic [Formula: see text]-extension of a quadratic imaginary field [Formula: see text] satisfying the Heegner hypothesis. In this paper, we make a conjecture about the fine Selmer group over [Formula: see text]. We also make a conjecture about the structure of the module of Heegner points in [Formula: see text] where [Formula: see text] is the union of the completions of the fields [Formula: see text] at a prime of [Formula: see text] above [Formula: see text]. We prove that these conjectures are equivalent. When [Formula: see text] has supersingular reduction at [Formula: see text] we also show that these conjectures are equivalent to the conjecture in our earlier work. Assuming these conjectures when [Formula: see text] has supersingular reduction at [Formula: see text], we prove various results about the structure of the Selmer group over [Formula: see text].

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.

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 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 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 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 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 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.

scholarly journals Global divisibility of Heegner points and Tamagawa numbers

2008 ◽  
Vol 144 (4) ◽  
pp. 811-826 ◽  
Author(s):  
Dimitar Jetchev
Keyword(s):  
Elliptic Curve ◽  
Upper Bound ◽  
Heegner Points ◽  
Tate Group ◽  
Rank One ◽  
Imaginary Field ◽  
Tamagawa Numbers

AbstractWe improve Kolyvagin’s upper bound on the order of the p-primary part of the Shafarevich–Tate group of an elliptic curve of rank one over a quadratic imaginary field. In many cases, our bound is precisely that predicted by the Birch and Swinnerton-Dyer conjectural formula.

scholarly journals Akashi series of Selmer groups

2011 ◽  
Vol 151 (2) ◽  
pp. 229-243 ◽  
Author(s):  
SARAH LIVIA ZERBES
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Selmer Groups ◽  
Selmer Group ◽  
Correction Terms

AbstractWe study the Selmer group of an elliptic curve over an admissible p-adic Lie extension of a number field F. We give a formula for the Akashi series attached to this module, in terms of the corresponding objects for the cyclotomic ℤp-extension and certain correction terms. This extends our earlier work [16], in particular since it applies to elliptic curves having split multiplicative reduction at some primes above p, in which case the Akashi series can have additional zeros.

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