scholarly journals ON THE EULER CHARACTERISTICS OF SIGNED SELMER GROUPS

2019 ◽  
Vol 101 (2) ◽  
pp. 238-246
Author(s):  
SUMAN AHMED ◽  
MENG FAI LIM
Keyword(s):  
Elliptic Curve ◽  
Prime Number ◽  
Number Field ◽  
Selmer Groups ◽  
Good Reduction ◽  
Euler Characteristics ◽  
Definition Of

Let $p$ be an odd prime number and $E$ an elliptic curve defined over a number field $F$ with good reduction at every prime of $F$ above $p$. We compute the Euler characteristics of the signed Selmer groups of $E$ over the cyclotomic $\mathbb{Z}_{p}$-extension. The novelty of our result is that we allow the elliptic curve to have mixed reduction types for primes above $p$ and mixed signs in the definition of the signed Selmer groups.

scholarly journals Mordell–Weil ranks and Tate–Shafarevich groups of elliptic curves with mixed-reduction type over cyclotomic extensions

2021 ◽  
pp. 1-28
Author(s):  
Antonio Lei ◽  
Meng Fai Lim
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Finite Extension ◽  
Selmer Groups ◽  
Good Reduction ◽  
Cyclotomic Extensions ◽  
Iwasawa Algebra

Let [Formula: see text] be an elliptic curve defined over a number field [Formula: see text] where [Formula: see text] splits completely. Suppose that [Formula: see text] has good reduction at all primes above [Formula: see text]. Generalizing previous works of Kobayashi and Sprung, we define multiply signed Selmer groups over the cyclotomic [Formula: see text]-extension of a finite extension [Formula: see text] of [Formula: see text] where [Formula: see text] is unramified. Under the hypothesis that the Pontryagin duals of these Selmer groups are torsion over the corresponding Iwasawa algebra, we show that the Mordell–Weil ranks of [Formula: see text] over a subextension of the cyclotomic [Formula: see text]-extension are bounded. Furthermore, we derive an aysmptotic formula of the growth of the [Formula: see text]-parts of the Tate–Shafarevich groups of [Formula: see text] over these extensions.

scholarly journals On the cohomology of Kobayashi’s plus/minus norm groups and applications

2021 ◽  
pp. 1-24
Author(s):  
MENG FAI LIM
Keyword(s):  
Elliptic Curve ◽  
Formal Group ◽  
Selmer Groups ◽  
Good Reduction ◽  
Characteristic Element ◽  
Definition Of ◽  
Integrality Property

Abstract The plus and minus norm groups are constructed by Kobayashi as subgroups of the formal group of an elliptic curve with supersingular reduction, and they play an important role in Kobayashi’s definition of the signed Selmer groups. In this paper, we study the cohomology of these plus and minus norm groups. In particular, we show that these plus and minus norm groups are cohomologically trivial. As an application of our analysis, we establish certain (quasi-)projectivity properties of the non-primitive mixed signed Selmer groups of an elliptic curve with good reduction at all primes above p. We then build on these projectivity results to derive a Kida formula for the signed Selmer groups under a slight weakening of the usual µ = 0 assumption, and study the integrality property of the characteristic element attached to the signed 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 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 CRITÈRES D'IRRÉDUCTIBILITÉ POUR LES REPRÉSENTATIONS DES COURBES ELLIPTIQUES

2011 ◽  
Vol 07 (04) ◽  
pp. 1001-1032 ◽  
Author(s):  
NICOLAS BILLEREY
Keyword(s):  
Elliptic Curve ◽  
Prime Number ◽  
Number Field ◽  
Efficient Algorithm ◽  
Complex Multiplication ◽  
Prime Numbers ◽  
Prime Divisors ◽  
Numerical Examples ◽  
Finite Case ◽  
Courbe Elliptique

Soit E une courbe elliptique définie sur un corps de nombres K. On dit qu'un nombre premier p est réductible pour le couple (E, K) si E admet une p-isogénie définie sur K. L'ensemble de tous ces nombres premiers est fini si et seulement si E n'a pas de multiplication complexe définie sur K. Dans cet article, on montre que l'ensemble des nombres premiers réductibles pour le couple (E, K) est contenu dans l'ensemble des diviseurs premiers d'une liste explicite d'entiers (dépendant de E et de K) dont une infinité d'entre eux est non nulle. Cela fournit un algorithme efficace de calcul dans le cas fini. D'autres critères moins généraux, mais néanmoins utiles sont donnés ainsi que de nombreux exemples numériques. Let E be an elliptic curve defined over a number field K. We say that a prime number p is reducible for (E, K) if E admits a p-isogeny defined over K. The so-called reducible set of all such prime numbers is finite if and only if E does not have complex multiplication over K. In this paper, we prove that the reducible set is included in the set of prime divisors of an explicit list of integers (depending on E and K), infinitely many of them being non-zero. It provides an efficient algorithm for computing it in the finite case. Other less general but rather useful criteria are given, as well as many numerical examples.

COURBES ELLIPTIQUES SEMI-STABLES SUR LES CORPS DE NOMBRES

2007 ◽  
Vol 03 (04) ◽  
pp. 611-633 ◽  
Author(s):  
ALAIN KRAUS
Keyword(s):  
Elliptic Curve ◽  
Prime Number ◽  
Number Field ◽  
Galois Representation ◽  
Torsion Points

Let K be a number field. In this paper, we are interested in the following problem: does there exist a constant cK, which depends only on K, such that for any semi-stable elliptic curve defined over K, the Galois representation in its p-torsion points is irreducible whenever p is a prime number greater than cK? In case the answer is positive, how can we get such a constant? We prove that if a certain condition is satisfied by K, the answer is positive and we obtain cK explicitly. Furthermore, we prove that this condition is realized in many situations.

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 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.

scholarly journals Criteria for p-ordinarity of families of elliptic curves over infinitely many number fields

2014 ◽  
Vol 11 (01) ◽  
pp. 81-87
Author(s):  
Nuno Freitas ◽  
Panagiotis Tsaknias
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Sufficient Conditions ◽  
Number Fields ◽  
Good Reduction ◽  
Ordinary Reduction

Let Ki be a number field for all i ∈ ℤ>0 and let ℰ be a family of elliptic curves containing infinitely many members defined over Ki for all i. Fix a rational prime p. We give sufficient conditions for the existence of an integer i0 such that, for all i > i0 and all elliptic curve E ∈ ℰ having good reduction at all 𝔭 | p in Ki, we have that E has good ordinary reduction at all primes 𝔭 | p. We illustrate our criteria by applying it to certain Frey curves in [Recipes to Fermat-type equations of the form xr + yr = Czp, to appear in Math. Z.; http://arXiv.org/abs/1203.3371 ] attached to Fermat-type equations of signature (r, r, p).

scholarly journals Primitive divisors of sequences associated to elliptic curves with complex multiplication

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Matteo Verzobio
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Complex Multiplication ◽  
Dedekind Domain ◽  
Torsion Point ◽  
Integral Ideal ◽  
Primitive Divisors ◽  
Primitive Divisor ◽  
The Ideal

AbstractLet P and Q be two points on an elliptic curve defined over a number field K. For $$\alpha \in {\text {End}}(E)$$ α ∈ End ( E ) , define $$B_\alpha $$ B α to be the $$\mathcal {O}_K$$ O K -integral ideal generated by the denominator of $$x(\alpha (P)+Q)$$ x ( α ( P ) + Q ) . Let $$\mathcal {O}$$ O be a subring of $${\text {End}}(E)$$ End ( E ) , that is a Dedekind domain. We will study the sequence $$\{B_\alpha \}_{\alpha \in \mathcal {O}}$$ { B α } α ∈ O . We will show that, for all but finitely many $$\alpha \in \mathcal {O}$$ α ∈ O , the ideal $$B_\alpha $$ B α has a primitive divisor when P is a non-torsion point and there exist two endomorphisms $$g\ne 0$$ g ≠ 0 and f so that $$f(P)= g(Q)$$ f ( P ) = g ( Q ) . This is a generalization of previous results on elliptic divisibility sequences.

Sign in / Sign up

Export Citation Format

Share Document