## 180 Citations

The difference between the Weil height and the canonical height on elliptic curves

- Mathematics
- 1990

Estimates for the difference of the Weil height and the canonical height of points on elliptic curves are used for many purposes, both theoretical and computational. In this note we give an explicit…

ON THE p-TORSION OF ELLIPTIC CURVES AND ELLIPTIC SURFACES IN CHARACTERISTIC p

- Mathematics
- 2004

We study the extension generated by the x-coordinates of the petorsion points of an elliptic curve over a function field of characteristic p. If S → C is a non-isotrivial elliptic surface in…

The second moment of the number of integral points on elliptic curves is bounded

- Mathematics
- 2018

In this paper, we show that the second moment of the number of integral points on elliptic curves over $\mathbb{Q}$ is bounded. In particular, we prove that, for any $0 < s < \log_2 5 = 2.3219…

Bounds for the integral points on elliptic curves over function fields

- Mathematics
- 2017

In this paper we give an upper bound for the number of integral points on an elliptic curve E over F_q[T] in terms of its conductor N and q. We proceed by applying the lower bounds for the canonical…

Lifting Problems, Cross-fiberedness, and Diffusive Properties on Elliptic Surfaces

- Mathematics
- 2015

Lifting Problems, Cross-fiberedness, and Diffusive Properties on Elliptic Surfaces William H. George Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2015 Given an…

The Geometry of Elliptic Curves

- Mathematics
- 2009

Elliptic curves, our principal object of study in this book, are curves of genus one having a specified base point. Our ultimate goal, as the title of the book indicates, is to study the arithmetic…

On the ^{}-torsion of elliptic curves and elliptic surfaces in characteristic

- Mathematics
- 2004

We study the extension generated by the x-coordinates of the pe-torsion points of an elliptic curve over a function field of characteristic p. If S → C is a non-isotrivial elliptic surface in…

Computing S-Integral Points on Elliptic Curves

- MathematicsANTS
- 1996

Borders for the size of the coefficients of integral points on E have been found and can be used only for soloving some particular equations or for treating a special model of elliptic curves, namely Thue curves of degree 3 (see [GSch]).

Computing integral points on elliptic curves

- Mathematics
- 1994

By a famous theorem of Siegel [S], the number of integral points on an elliptic curve E over an algebraic number field K is finite. A conjecture of Lang and Demjanenko [L3] states that, for a…

The filled Julia set of a Drinfeld module and uniform bounds for torsion

- Mathematics
- 2012

If M is a Drinfeld module over a local function field L, we may view M as a dynamical system, and consider its filled Julia set J. If J^0 is the connected component of the identity, relative to the…

## References

SHOWING 1-10 OF 28 REFERENCES

Computing heights on elliptic curves

- Mathematics
- 1988

We describe how to compute the canonical height of points orn elliptic curves. Tate has given a rapidly converging series for Archimedean local heights over R. We describe a modified version of…

Intersection numbers of sections of elliptic surfaces

- Mathematics
- 1979

The theory of elliptic surfaces over C draws on ideas and techniques from arithmetic, geometry and analysis. Let f : X ~ S be a minimal elliptic fibration with non-constant j-invariant, which…

Séminaire sur les pinceaux de courbes de genre au moins deux

- Mathematics
- 1981

PENCILS OF CURVES OF GENUS AT LEAST TWO (a seminar at the E.N.S. organised by L. Szpiro) This seminar contains eigth papers that we can divided int o four groups : Semi-stable réduction (expos é 1 by…

The hyperelliptic equation over function fields

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 1983

Siegel, in a letter to Mordell of 1925(9), proved that the hyper-elliptic equation y2 = g(x) has only finitely many solutions in integers x and y, where g denotes a square-free polynomial of degree…

Modular Functions and Dirichlet Series in Number Theory

- Mathematics
- 1976

This is the second volume of a 2-volume textbook which evolved from a course (Mathematics 160) offered at the California Institute of Technology du ring the last 25 years. The second volume…

A quantitative version of Siegel's theorem: integral points on elliptic curves and Catalan curves.

- Mathematics
- 1987

The two fundamental finiteness theorems in the arithmetic theory of elliptic curves are the Mordell-Weil theorem, which says that the group of rational points is finitely generated, and Siegel's…

The S-unit equation over function fields

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 1984

In the study of integral solutions to Diophantine equations, many problems can be reduced to that of solving the equation in S-units of the given ring. To accomplish this over number fields, the only…

Thue's equation over function fields

- MathematicsJournal of the Australian Mathematical Society
- 1978

Abstract Suppose we are given a “Thue equation” f(x, y) = 1, where f is a binary form with coefficients in a function field K of characteristic zero. A typical result is that if f is of degree at…

ON TATE HEIGHT AND THE REPRESENTATION OF NUMBERS BY BINARY FORMS

- Mathematics
- 1974

We give a decomposition of the Tate height into components possessing quadraticity with respect to a group law, and on the basis of this decomposition we obtain an estimate for the number of K-points…

Heights and Elliptic Curves

- Mathematics
- 1986

Many of the deep results involving heights of abelian varieties become quite transparent in the case of elliptic curves. In this chapter we propose to prove some of these theorems for elliptic curves…