TORSION POINTS ON ELLIPTIC CURVES WITH COMPLEX MULTIPLICATION (WITH AN APPENDIX BY ALEX RICE)

We present seven theorems on the structure of prime order torsion points on CM elliptic curves defined over number fields. The first three results refine bounds of Silverberg and Prasad–Yogananda by taking into account the class number of the CM order and the splitting of the prime in the CM field. In many cases we can show that our refined bounds are optimal or asymptotically optimal. We also derive asymptotic upper and lower bounds on the least degree of a CM-point on X1(N). Upon comparison to bounds for the least degree for which there exist infinitely many rational points on X1(N), we deduce that, for sufficiently large N, X1(N) will have a rational CM point of degree smaller than the degrees of at least all but finitely many non-CM points.

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Galois groups of number fields generated by torsion points of elliptic curves

Nagoya Mathematical Journal ◽

10.1017/s0027763000022662 ◽

1986 ◽

Vol 104 ◽

pp. 43-53 ◽

Cited By ~ 2

Author(s):

Kay Wingberg

Keyword(s):

Elliptic Curve ◽

Elliptic Curves ◽

Galois Group ◽

Number Field ◽

Quadratic Field ◽

Complex Multiplication ◽

Number Fields ◽

Galois Groups ◽

Torsion Points ◽

Special Case

Coates and Wiles [1] and B. Perrin-Riou (see [2]) study the arithmetic of an elliptic curve E defined over a number field F with complex multiplication by an imaginary quadratic field K by using p-adic techniques, which combine the classical descent of Mordell and Weil with ideas of Iwasawa’s theory of Zp-extensions of number fields. In a special case they consider a non-cyclotomic Zp-extension F∞ defined via torsion points of E and a certain Iwasawa module attached to E/F, which can be interpreted as an abelian Galois group of an extension of F∞. We are interested in the corresponding non-abelian Galois group and we want to show that the whole situation is quite analogous to the case of the cyclotomic Zp-extension (which is generated by torsion points of Gm).

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ON THE AVERAGE CASE CIRCUIT DELAY OF DISJUNCTION

Parallel Processing Letters ◽

10.1142/s0129626495000254 ◽

1995 ◽

Vol 05 (02) ◽

pp. 275-280 ◽

Cited By ~ 2

Author(s):

BEATE BOLLIG ◽

MARTIN HÜHNE ◽

STEFAN PÖLT ◽

PETR SAVICKÝ

Keyword(s):

Lower Bounds ◽

Wide Class ◽

Time Complexity ◽

Upper And Lower Bounds ◽

Average Case ◽

Asymptotically Optimal ◽

Circuit Delay ◽

Suitable Measure

For circuits the expected delay is a suitable measure for the average case time complexity. In this paper, new upper and lower bounds on the expected delay of circuits for disjunction and conjunction are derived. The circuits presented yield asymptotically optimal expected delay for a wide class of distributions on the inputs even when the parameters of the distribution are not known in advance.

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New Bounds on the Minimum Average Distance of Binary Codes

Journal of Scientific Research ◽

10.3329/jsr.v2i3.2708 ◽

2010 ◽

Vol 2 (3) ◽

pp. 489

Author(s):

M. Basu ◽

S. Bagchi

Keyword(s):

Lower Bounds ◽

Binary Code ◽

Hamming Distance ◽

Average Distance ◽

Upper And Lower Bounds ◽

Binary Codes ◽

Large N

The minimum average Hamming distance of binary codes of length n and cardinality M is denoted by b(n,M). All the known lower bounds b(n,M) are useful when M is at least of size about 2n-1/n . In this paper, for large n, we improve upper and lower bounds for b(n,M). Keywords: Binary code; Hamming distance; Minimum average Hamming distance. © 2010 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. DOI: 10.3329/jsr.v2i3.2708 J. Sci. Res. 2 (3), 489-493 (2010)

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Torsion points on CM elliptic curves over real number fields

Transactions of the American Mathematical Society ◽

10.1090/tran/6905 ◽

2017 ◽

Vol 369 (12) ◽

pp. 8457-8496 ◽

Cited By ~ 8

Author(s):

Abbey Bourdon ◽

Pete L. Clark ◽

James Stankewicz

Keyword(s):

Real Number ◽

Elliptic Curves ◽

Number Fields ◽

Torsion Points

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Discriminants of Complex Multiplication Fields of Elliptic Curves over Finite Fields

Canadian Mathematical Bulletin ◽

10.4153/cmb-2007-039-2 ◽

2007 ◽

Vol 50 (3) ◽

pp. 409-417 ◽

Cited By ~ 5

Author(s):

Florian Luca ◽

Igor E. Shparlinski

Keyword(s):

Asymptotic Formula ◽

Finite Field ◽

Elliptic Curves ◽

Endomorphism Ring ◽

Complex Multiplication ◽

Number Fields ◽

Quadratic Number ◽

Endomorphism Rings ◽

Quadratic Number Field ◽

Curves Over Finite Fields

AbstractWe show that, for most of the elliptic curves E over a prime finite field p of p elements, the discriminant D(E) of the quadratic number field containing the endomorphism ring of E over p is sufficiently large. We also obtain an asymptotic formula for the number of distinct quadratic number fields generated by the endomorphism rings of all elliptic curves over p.

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A. Geometrical construction of 2-dimensional galois representations of A5-type. B. On the realisation of the groups PSL2(1) as galois groups over number fields by means of l-torsion points of elliptic curves

Lecture Notes in Mathematics - On Artin's Conjecture for Odd 2-dimensional Representations ◽

10.1007/bfb0074109 ◽

1994 ◽

pp. 47-58

Author(s):

Martin Kinzelbach

Keyword(s):

Elliptic Curves ◽

Galois Representations ◽

Number Fields ◽

Galois Groups ◽

Type B ◽

Geometrical Construction ◽

Torsion Points

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Extreme residues of Dedekind zeta functions

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004117000019 ◽

2017 ◽

Vol 163 (2) ◽

pp. 369-380 ◽

Cited By ~ 1

Author(s):

PETER J. CHO ◽

HENRY H. KIM

Keyword(s):

Lower Bounds ◽

Zeta Functions ◽

Infinite Family ◽

Number Fields ◽

Upper And Lower Bounds ◽

Zero Set ◽

Density Zero

AbstractIn a family ofSd+1-fields (d= 2, 3, 4), we obtain the conjectured upper and lower bounds of the residues of Dedekind zeta functions except for a density zero set. ForS5-fields, we need to assume the strong Artin conjecture. We also show that there exists an infinite family of number fields with the upper and lower bounds, resp.

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A MODULI INTERPRETATION FOR THE NON-SPLIT CARTAN MODULAR CURVE

Glasgow Mathematical Journal ◽

10.1017/s0017089517000180 ◽

2017 ◽

Vol 60 (2) ◽

pp. 411-434 ◽

Cited By ~ 2

Author(s):

MARUSIA REBOLLEDO ◽

CHRISTIAN WUTHRICH

Keyword(s):

Elliptic Curves ◽

Half Plane ◽

London Math ◽

Number Fields ◽

Arithmetic Geometry ◽

Modular Curves ◽

Torsion Points ◽

Modular Curve ◽

Upper Half Plane ◽

Cartan Subgroups

AbstractModular curves likeX0(N) andX1(N) appear very frequently in arithmetic geometry. While their complex points are obtained as a quotient of the upper half plane by some subgroups of SL2(ℤ), they allow for a more arithmetic description as a solution to a moduli problem. We wish to give such a moduli description for two other modular curves, denoted here byXnsp(p) andXnsp+(p) associated to non-split Cartan subgroups and their normaliser in GL2(𝔽p). These modular curves appear for instance in Serre's problem of classifying all possible Galois structures ofp-torsion points on elliptic curves over number fields. We give then a moduli-theoretic interpretation and a new proof of a result of Chen (Proc. London Math. Soc.(3)77(1) (1998), 1–38;J. Algebra231(1) (2000), 414–448).

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On thep-part of the Birch–Swinnerton-Dyer conjecture for elliptic curves with complex multiplication by the ring of integers of

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004116000852 ◽

2016 ◽

Vol 164 (1) ◽

pp. 67-98 ◽

Cited By ~ 2

Author(s):

YUKAKO KEZUKA

Keyword(s):

Lower Bound ◽

Elliptic Curve ◽

Elliptic Curves ◽

Lower Bounds ◽

Complex Multiplication ◽

Ring Of Integers ◽

Quadratic Twists ◽

The Family ◽

Adic Valuation

AbstractWe study infinite families of quadratic and cubic twists of the elliptic curveE=X0(27). For the family of quadratic twists, we establish a lower bound for the 2-adic valuation of the algebraic part of the value of the complexL-series ats=1, and, for the family of cubic twists, we establish a lower bound for the 3-adic valuation of the algebraic part of the sameL-value. We show that our lower bounds are precisely those predicted by the celebrated conjecture of Birch and Swinnerton-Dyer.

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Hessian matrices, automorphisms of p-groups, and torsion points of elliptic curves

Mathematische Annalen ◽

10.1007/s00208-021-02193-8 ◽

2021 ◽

Author(s):

Mima Stanojkovski ◽

Christopher Voll

Keyword(s):

Elliptic Curves ◽

Finite Fields ◽

Automorphism Groups ◽

Number Fields ◽

Explicit Formulas ◽

Torsion Points ◽

Curves Over Finite Fields

AbstractWe describe the automorphism groups of finite p-groups arising naturally via Hessian determinantal representations of elliptic curves defined over number fields. Moreover, we derive explicit formulas for the orders of these automorphism groups for elliptic curves of j-invariant 1728 given in Weierstrass form. We interpret these orders in terms of the numbers of 3-torsion points (or flex points) of the relevant curves over finite fields. Our work greatly generalizes and conceptualizes previous examples given by du Sautoy and Vaughan-Lee. It explains, in particular, why the orders arising in these examples are polynomial on Frobenius sets and vary with the primes in a nonquasipolynomial manner.

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