Elliptic Curves with Long Arithmetic Progressions Have Large Rank

International Mathematics Research Notices ◽

10.1093/imrn/rnaa061 ◽

2020 ◽

Author(s):

Natalia Garcia-Fritz ◽

Hector Pasten

Keyword(s):

Elliptic Curves ◽

Arithmetic Progression ◽

Nevanlinna Theory ◽

Uniform Boundedness ◽

Rational Numbers ◽

Rational Points ◽

Arithmetic Progressions ◽

Large Rank ◽

Arithmetic Statistics

Abstract For any family of elliptic curves over the rational numbers with fixed $j$-invariant, we prove that the existence of a long sequence of rational points whose $x$-coordinates form a nontrivial arithmetic progression implies that the Mordell–Weil rank is large, and similarly for $y$-coordinates. We give applications related to uniform boundedness of ranks, conjectures by Bremner and Mohanty, and arithmetic statistics on elliptic curves. Our approach involves Nevanlinna theory as well as Rémond’s quantitative extension of results of Faltings.

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Four rational squares in arithmetic progressions and a family of elliptic curves with positive Mordell-Weil rank

Mathematische Semesterberichte ◽

10.1007/s00591-020-00279-z ◽

2020 ◽

Vol 67 (2) ◽

pp. 213-236

Author(s):

Hagen Knaf ◽

Erich Selder ◽

Karlheinz Spindler

Keyword(s):

Elliptic Curves ◽

Arithmetic Progression ◽

Rational Numbers ◽

Rational Points ◽

Arithmetic Progressions ◽

Theoretical Problem ◽

Theoretical Results

Abstract We study the question at which relative distances four squares of rational numbers can occur as terms in an arithmetic progression. This number-theoretical problem is seen to be equivalent to finding rational points on certain elliptic curves. Both number-theoretical results and results concerning the associated elliptic curves are derived; i.e., the correspondence between rational squares in arithmetic progressions and elliptic curves is exploited both ways.

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Heron quadrilaterals with sides in arithmetic or geometric progression

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700032883 ◽

1999 ◽

Vol 59 (2) ◽

pp. 263-269 ◽

Cited By ~ 9

Author(s):

R.H. Buchholz ◽

J.A. MacDougall

Keyword(s):

Elliptic Curves ◽

Arithmetic Progression ◽

Infinite Family ◽

Rational Points ◽

Arithmetic Progressions ◽

Geometric Progression ◽

Complete Characterisation

We study triangles and cyclic quadrilaterals which have rational area and whose sides form geometric or arithmetic progressions. A complete characterisation is given for the infinite family of triangles with sides in arithmetic progression. We show that there are no triangles with sides in geometric progression. We also show that apart from the square there are no cyclic quadrilaterals whose sides form either a geometric or an arithmetic progression. The solution of both quadrilateral cases involves searching for rational points on certain elliptic curves.

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Rational Points in Arithmetic Progressions on y2 = xn + k

Canadian Mathematical Bulletin ◽

10.4153/cmb-2011-058-1 ◽

2012 ◽

Vol 55 (1) ◽

pp. 193-207 ◽

Cited By ~ 4

Author(s):

Maciej Ulas

Keyword(s):

Elliptic Curve ◽

Arithmetic Progression ◽

Hyperelliptic Curve ◽

Rational Points ◽

Arithmetic Progressions ◽

Multiple Roots

AbstractLet C be a hyperelliptic curve given by the equation y2 = f(x) for f ∈ ℤ[x] without multiple roots. We say that points Pi = (xi, yi) ∈ C(ℚ) for i = 1, 2, … , m are in arithmetic progression if the numbers xi for i = 1, 2, … , m are in arithmetic progression.In this paper we show that there exists a polynomial k ∈ ℤ[t] with the property that on the elliptic curve ε′ : y2 = x3+k(t) (defined over the field ℚ(t)) we can find four points in arithmetic progression that are independent in the group of all ℚ(t)-rational points on the curve Ε′. In particular this result generalizes earlier results of Lee and Vélez. We also show that if n ∈ ℕ is odd, then there are infinitely many k's with the property that on curves y2 = xn + k there are four rational points in arithmetic progressions. In the case when n is even we can find infinitely many k's such that on curves y2 = xn +k there are six rational points in arithmetic progression.

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Quadratic sequences of powers and Mohanty’s conjecture

International Journal of Number Theory ◽

10.1142/s1793042118500306 ◽

2018 ◽

Vol 14 (02) ◽

pp. 479-507 ◽

Cited By ~ 2

Author(s):

Natalia Garcia-Fritz

Keyword(s):

Elliptic Curves ◽

Arithmetic Progression ◽

General Type ◽

Function Fields ◽

Hyperelliptic Curves ◽

Arithmetic Progressions ◽

Surfaces Of General Type ◽

Genus Zero ◽

Curves Of Low Genus

We prove under the Bombieri–Lang conjecture for surfaces that there is an absolute bound on the length of sequences of integer squares with constant second differences, for sequences which are not formed by the squares of integers in arithmetic progression. This answers a question proposed in 2010 by Browkin and Brzezinski, and independently by Gonzalez-Jimenez and Xarles. We also show that under the Bombieri–Lang conjecture for surfaces, for every [Formula: see text] there is an absolute bound on the length of sequences formed by [Formula: see text]th powers with constant second differences. This gives a conditional result on one of Mohanty’s conjectures on arithmetic progressions in Mordell’s elliptic curves [Formula: see text]. Moreover, we obtain an unconditional result regarding infinite families of such arithmetic progressions. We also study the case of hyperelliptic curves of the form [Formula: see text]. These results are proved by unconditionally finding all curves of genus zero or one on certain surfaces of general type. Moreover, we prove the unconditional analogues of these arithmetic results for function fields by finding all the curves of low genus on these surfaces.

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POWERS FROM PRODUCTS OF CONSECUTIVE TERMS IN ARITHMETIC PROGRESSION

Proceedings of the London Mathematical Society ◽

10.1112/s0024611505015625 ◽

2006 ◽

Vol 92 (2) ◽

pp. 273-306 ◽

Cited By ~ 22

Author(s):

M. A. BENNETT ◽

N. BRUIN ◽

K. GYÖRY ◽

L. HAJDU

Keyword(s):

Positive Integer ◽

Arithmetic Progression ◽

Diophantine Equations ◽

Galois Representations ◽

Rational Points ◽

Arithmetic Progressions ◽

Superelliptic Curves

We show that if $k$ is a positive integer, then there are, under certain technical hypotheses, only finitely many coprime positive $k$-term arithmetic progressions whose product is a perfect power. If $4 \leq k \leq 11$, we obtain the more precise conclusion that there are, in fact, no such progressions. Our proofs exploit the modularity of Galois representations corresponding to certain Frey curves, together with a variety of results, classical and modern, on solvability of ternary Diophantine equations. As a straightforward corollary of our work, we sharpen and generalize a theorem of Sander on rational points on superelliptic curves.

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Searching for simultaneous arithmetic progressions on elliptic curves

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700038429 ◽

2005 ◽

Vol 71 (3) ◽

pp. 417-424 ◽

Cited By ~ 2

Author(s):

Irene García-Selfa ◽

José M. Tornero

Keyword(s):

Elliptic Curves ◽

Rational Points ◽

Arithmetic Progressions ◽

Constructive Method

We look for elliptic curves featuring rational points whose coordinates form two arithmetic progressions, one for each coordinate. A constructive method for creating such curves is shown, for lengths up to 5.

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ARITHMETIC PROGRESSIONS ON CONIC SECTIONS

International Journal of Number Theory ◽

10.1142/s1793042113500322 ◽

2013 ◽

Vol 09 (06) ◽

pp. 1379-1393

Author(s):

ALEJANDRA ALVARADO ◽

EDRAY HERBER GOINS

Keyword(s):

Elliptic Curve ◽

Arithmetic Progression ◽

Rational Points ◽

Arithmetic Progressions ◽

Torsion Point ◽

Conic Sections ◽

Rational Map ◽

Perfect Squares

The set {1, 25, 49} is a 3-term collection of integers which forms an arithmetic progression of perfect squares. We view the set {(1, 1), (5, 25), (7, 49)} as a 3-term collection of rational points on the parabola y = x2 whose y-coordinates form an arithmetic progression. In this exposition, we provide a generalization to 3-term arithmetic progressions on arbitrary conic sections [Formula: see text] with respect to a linear rational map [Formula: see text]. We explain how this construction is related to rational points on the universal elliptic curve Y2 + 4XY + 4kY = X3 + kX2 classifying those curves possessing a rational 4-torsion point.

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