Counting rational points on elliptic curves with a rational 2-torsion point

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.

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Computing rational points on rank 1 elliptic curves via $L$-series and canonical heights

Mathematics of Computation ◽

10.1090/s0025-5718-99-01068-6 ◽

1999 ◽

Vol 68 (226) ◽

pp. 835-859 ◽

Cited By ~ 8

Author(s):

Joseph H. Silverman

Keyword(s):

Elliptic Curves ◽

Rational Points

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Construction of elliptic curves with cyclic groups over prime fields

Bulletin of the Australian Mathematical Society ◽

10.1017/s000497270003882x ◽

2006 ◽

Vol 73 (2) ◽

pp. 245-254 ◽

Cited By ~ 1

Author(s):

Naoya Nakazawa

Keyword(s):

Elliptic Curves ◽

Finite Fields ◽

Function Field ◽

Modular Group ◽

Modular Function ◽

Invariant Function ◽

Rational Points ◽

Modular Invariant ◽

Cyclic Groups ◽

Prime Fields

The purpose of this article is to construct families of elliptic curves E over finite fields F so that the groups of F-rational points of E are cyclic, by using a representation of the modular invariant function by a generator of a modular function field associated with the modular group Γ0(N), where N = 5, 7 or 13.

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Galois criterion for torsion points of Drinfeld modules

International Journal of Number Theory ◽

10.1142/s1793042121500901 ◽

2021 ◽

pp. 1-12

Author(s):

Chien-Hua Chen

Keyword(s):

Maximal Ideal ◽

Abelian Varieties ◽

Number Fields ◽

Rational Points ◽

Drinfeld Module ◽

Torsion Point ◽

Drinfeld Modules ◽

Torsion Points ◽

Ideal Formula ◽

Almost All

In this paper, we formulate the Drinfeld module analogue of a question raised by Lang and studied by Katz on the existence of rational points on abelian varieties over number fields. Given a maximal ideal [Formula: see text] of [Formula: see text], the question essentially asks whether, up to isogeny, a Drinfeld module [Formula: see text] over [Formula: see text] contains a rational [Formula: see text]-torsion point if the reduction of [Formula: see text] at almost all primes of [Formula: see text] contains a rational [Formula: see text]-torsion point. Similar to the case of abelian varieties, we show that the answer is positive if the rank of the Drinfeld module is 2, but negative if the rank is 3. Moreover, for rank 3 Drinfeld modules we classify those cases where the answer is positive.

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Elliptic Curves over the Perfect Closure of a Function Field

Canadian Mathematical Bulletin ◽

10.4153/cmb-2010-019-9 ◽

2010 ◽

Vol 53 (1) ◽

pp. 87-94

Author(s):

Dragos Ghioca

Keyword(s):

Elliptic Curve ◽

Elliptic Curves ◽

Function Field ◽

Positive Characteristic ◽

Rational Points ◽

Finitely Generated

AbstractWe prove that the group of rational points of a non-isotrivial elliptic curve defined over the perfect closure of a function field in positive characteristic is finitely generated.

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Kato’s Euler system and rational points on elliptic curves I: A p-adic Beilinson formula

Israel Journal of Mathematics ◽

10.1007/s11856-013-0047-2 ◽

2013 ◽

Vol 199 (1) ◽

pp. 163-188 ◽

Cited By ~ 9

Author(s):

Massimo Bertolini ◽

Henri Darmon

Keyword(s):

Elliptic Curves ◽

Rational Points ◽

Euler System

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Points of low height on elliptic surfaces with torsion

LMS Journal of Computation and Mathematics ◽

10.1112/s1461157009000151 ◽

2010 ◽

Vol 13 ◽

pp. 370-387

Author(s):

Sonal Jain

Keyword(s):

Elliptic Curve ◽

Elliptic Curves ◽

Open Subset ◽

Function Field ◽

Rational Point ◽

Torsion Point ◽

Integral Points ◽

Elliptic Surfaces ◽

Canonical Height

AbstractWe determine the smallest possible canonical height$\hat {h}(P)$for a non-torsion pointPof an elliptic curveEover a function field(t) of discriminant degree 12nwith a 2-torsion point forn=1,2,3, and with a 3-torsion point forn=1,2. For eachm=2,3, we parametrize the set of triples (E,P,T) of an elliptic curveE/with a rational pointPandm-torsion pointTthat satisfy certain integrality conditions by an open subset of2. We recover explicit equations for all elliptic surfaces (E,P,T) attaining each minimum by locating them as curves in our projective models. We also prove that forn=1,2 , these heights are minimal for elliptic curves over a function field of any genus. In each case, the optimal (E,P,T) are characterized by their patterns of integral points.

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Number of rational points of elliptic curves

Asian-European Journal of Mathematics ◽

10.1142/s1793557122500176 ◽

2021 ◽

pp. 2250017

Author(s):

Viliam Ďuriš ◽

Timotej Šumný

Keyword(s):

Elliptic Curve ◽

Elliptic Curves ◽

Current Knowledge ◽

American Mathematical Society ◽

Rational Points ◽

Modern Theory ◽

Cambridge University ◽

Weil Theorem ◽

A Finite Group

In the modern theory of elliptic curves, one of the important problems is the determination of the number of rational points on an elliptic curve. The Mordel–Weil theorem [T. Shioda, On the Mordell–Weil lattices, Comment. Math. University St. Paul. 39(2) (1990) 211–240] points out that the elliptic curve defined above the rational points is generated by a finite group. Despite the knowledge that an elliptic curve has a final number of rational points, it is still difficult to determine their number and the way how to determine them. The greatest progress was achieved by Birch and Swinnerton–Dyer conjecture, which was included in the Millennium Prize Problems [A. Wiles, The Birch and Swinnerton–Dyer conjecture, The Millennium Prize Problems (American Mathematical Society, 2006), pp. 31–44]. This conjecture uses methods of the analytical theory of numbers, while the current knowledge corresponds to the assumptions of the conjecture but has not been proven to date. In this paper, we focus on using a tangent line and the osculating circle for characterizing the rational points of the elliptical curve, which is the greatest benefit of the contribution. We use a different view of elliptic curves by using Minkowki’s theory of number geometry [H. F. Blichfeldt, A new principle in the geometry of numbers, with some applications, Trans. Amer. Math. Soc. 15(3) (1914) 227–235; V. S. Miller, Use of elliptic curves in cryptography, in Proc. Advances in Cryptology — CRYPTO ’85, Lecture Notes in Computer Science, Vol. 218 (Springer, Berlin, Heidelberg, 1985), pp. 417–426; E. Bombieri and W. Gubler, Heights in Diophantine Geometry, Vol. 670, 1st edn. (Cambridge University Press, 2007)].

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