On the pseudo-nullity of the dual fine Selmer groups

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.

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SELMER GROUPS AND GENERALIZED CLASS FIELD TOWERS

International Journal of Number Theory ◽

10.1142/s1793042112500522 ◽

2012 ◽

Vol 08 (04) ◽

pp. 881-909 ◽

Cited By ~ 3

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.

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Self-duality of Selmer groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004108001989 ◽

2009 ◽

Vol 146 (2) ◽

pp. 257-267 ◽

Cited By ~ 9

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.

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THE PLUS/MINUS SELMER GROUPS FOR SUPERSINGULAR PRIMES

Journal of the Australian Mathematical Society ◽

10.1017/s1446788713000165 ◽

2013 ◽

Vol 95 (2) ◽

pp. 189-200 ◽

Cited By ~ 6

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.

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Large Selmer groups over number fields

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004109990132 ◽

2009 ◽

Vol 148 (1) ◽

pp. 73-86 ◽

Cited By ~ 3

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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Growth of Fine Selmer Groups in Infinite Towers

Canadian Mathematical Bulletin ◽

10.4153/s0008439520000168 ◽

2020 ◽

Vol 63 (4) ◽

pp. 921-936 ◽

Cited By ~ 1

Author(s):

Debanjana Kundu

Keyword(s):

Sufficient Condition ◽

Selmer Groups ◽

Selmer Group ◽

Class Field ◽

Class Field Tower

AbstractIn this paper, we study the growth of fine Selmer groups in two cases. First, we study the growth of fine Selmer ranks in multiple $\mathbb{Z}_{p}$-extensions. We show that the growth of the fine Selmer group is unbounded in such towers. We recover a sufficient condition to prove the $\unicode[STIX]{x1D707}=0$ conjecture for cyclotomic $\mathbb{Z}_{p}$-extensions. We show that in certain non-cyclotomic $\mathbb{Z}_{p}$-towers, the $\unicode[STIX]{x1D707}$-invariant of the fine Selmer group can be arbitrarily large. Second, we show that in an unramified $p$-class field tower, the growth of the fine Selmer group is unbounded. This tower is non-Abelian and non-$p$-adic analytic.

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The Selmer groups and the ambiguous ideal class groups of cubic fields

Bulletin of the Australian Mathematical Society ◽

10.1017/s000497270001772x ◽

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.

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Algebraic functional equation for Hida family

International Journal of Number Theory ◽

10.1142/s1793042114500493 ◽

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].

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Selmer Groups of Elliptic Curves with Complex Multiplication

Canadian Journal of Mathematics ◽

10.4153/cjm-2004-009-7 ◽

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.

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GENERALIZED EXPLICIT DESCENT AND ITS APPLICATION TO CURVES OF GENUS 3

Forum of Mathematics Sigma ◽

10.1017/fms.2016.1 ◽

2016 ◽

Vol 4 ◽

Cited By ~ 4

Author(s):

NILS BRUIN ◽

BJORN POONEN ◽

MICHAEL STOLL

Keyword(s):

Abelian Varieties ◽

Selmer Groups ◽

Explicit Computation ◽

Selmer Group ◽

Common Generalization ◽

Global Fields

We introduce a common generalization of essentially all known methods for explicit computation of Selmer groups, which are used to bound the ranks of abelian varieties over global fields. We also simplify and extend the proofs relating what is computed to the cohomologically defined Selmer groups. Selmer group computations have been practical for many Jacobians of curves over $\mathbb{Q}$ of genus up to 2 since the 1990s, but our approach is the first to be practical for general curves of genus 3. We show that our approach succeeds on some genus 3 examples defined by polynomials with small coefficients.

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