scholarly journals Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks

2016 ◽  
Vol 19 (A) ◽  
pp. 351-370 ◽  
Author(s):  
Jennifer S. Balakrishnan ◽  
Wei Ho ◽  
Nathan Kaplan ◽  
Simon Spicer ◽  
William Stein ◽  
...  
Keyword(s):  
Elliptic Curves ◽  
Upper Bounds ◽  
Selmer Groups ◽  
Selmer Group ◽  
Average Rank ◽  
Recent Developments ◽  
Theoretical Results

Most systematic tables of data associated to ranks of elliptic curves order the curves by conductor. Recent developments, led by work of Bhargava and Shankar studying the average sizes of $n$-Selmer groups, have given new upper bounds on the average algebraic rank in families of elliptic curves over $\mathbb{Q}$, ordered by height. We describe databases of elliptic curves over $\mathbb{Q}$, ordered by height, in which we compute ranks and $2$-Selmer group sizes, the distributions of which may also be compared to these theoretical results. A striking new phenomenon that we observe in our database is that the average rank eventually decreases as height increases.

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.

Large families of elliptic curves ordered by conductor

2021 ◽  
Vol 157 (7) ◽  
pp. 1538-1583
Author(s):  
Ananth N. Shankar ◽  
Arul Shankar ◽  
Xiaoheng Wang
Keyword(s):  
Elliptic Curves ◽  
Upper Bounds ◽  
Selmer Groups ◽  
Good Reduction ◽  
Positive Proportion ◽  
The Family ◽  
Large Families ◽  
Average Size

In this paper we study the family of elliptic curves $E/{{\mathbb {Q}}}$ , having good reduction at $2$ and $3$ , and whose $j$ -invariants are small. Within this set of elliptic curves, we consider the following two subfamilies: first, the set of elliptic curves $E$ such that the quotient $\Delta (E)/C(E)$ of the discriminant divided by the conductor is squarefree; and second, the set of elliptic curves $E$ such that the Szpiro quotient $\beta _E:=\log |\Delta (E)|/\log (C(E))$ is less than $7/4$ . Both these families are conjectured to contain a positive proportion of elliptic curves, when ordered by conductor. Our main results determine asymptotics for both these families, when ordered by conductor. Moreover, we prove that the average size of the $2$ -Selmer groups of elliptic curves in the first family, again when these curves are ordered by their conductors, is $3$ . The key new ingredients necessary for the proofs are ‘uniformity estimates’, namely upper bounds on the number of elliptic curves with bounded height, whose discriminants are divisible by high powers of primes.

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 On the Asymptotic Growth of Bloch-Kato-Shafarevich-Tate Groups of Modular Forms Over Cyclotomic Extensions

2017 ◽  
Vol 69 (4) ◽  
pp. 826-850 ◽  
Author(s):  
Antonio Lei ◽  
David Loeffler ◽  
Zerbes Sarah Livia
Keyword(s):  
Asymptotic Behaviour ◽  
Modular Form ◽  
Elliptic Curves ◽  
Modular Forms ◽  
Upper Bounds ◽  
Selmer Groups ◽  
Asymptotic Growth ◽  
Supersingular Elliptic Curves ◽  
Cyclotomic Extensions ◽  
Iwasawa Invariants

AbstractWe study the asymptotic behaviour of the Bloch-Kato-Shafarevich-Tate group of a modular form f over the cyclotomic ℤp-extension of ℚ under the assumption that f is non-ordinary at p. In particular, we give upper bounds of these groups in terms of Iwasawa invariants of Selmer groups defined using p-adic Hodge Theory. These bounds have the same form as the formulae of Kobayashi, Kurihara, and Sprung for supersingular elliptic curves.

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.

Anomalous Heat Conduction, Diffusion and Heat Rectification in Nanoscale Structures

2009 ◽  
Author(s):  
Gang Zhang ◽  
Nuo Yang ◽  
Gang Wu ◽  
Baowen Li
Keyword(s):  
Thermal Conductivity ◽  
Heat Conduction ◽  
Heat Transport ◽  
Nano Materials ◽  
Fourier Law ◽  
Nanoscale Structures ◽  
Recent Developments ◽  
Theoretical Results ◽  
Good Agreement ◽  
Anomalous Heat

In this paper, we report the recent developments in the study of heat transport in nano materials. First of all, we show that phonon transports in nanotube super-diffusively which leads to a length dependence thermal conductivity, thus breaks down the Fourier law. Then we discuss how the introduction of isotope doping can reduce the thermal conductivity efficiently. The theoretical results are in good agreement with experimental ones. Finally, we will demonstrate that nanoscale structures are promising candidates for heat rectification.

On the Surjectivity of the Galois Representations Associated to Non-CM Elliptic Curves

2005 ◽  
Vol 48 (1) ◽  
pp. 16-31 ◽  
Author(s):  
Alina Carmen Cojocaru ◽  
Ernst Kani
Keyword(s):  
Positive Integer ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Galois Representations ◽  
Complex Multiplication ◽  
Galois Representation ◽  
Upper Bounds

AbstractLet E be an elliptic curve defined over ℚ, of conductor N and without complex multiplication. For any positive integer l, let ϕl be the Galois representation associated to the l-division points of E. From a celebrated 1972 result of Serre we know that ϕl is surjective for any sufficiently large prime l. In this paper we find conditional and unconditional upper bounds in terms of N for the primes l for which ϕl is not surjective.

Growth of Fine Selmer Groups in Infinite Towers

2020 ◽  
Vol 63 (4) ◽  
pp. 921-936 ◽  
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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