橢圓曲線上的整數點

Integral Points on Elliptic Curves

廖宏銘(Hung-Min Laio)

指導教授 ： 紀文鎮

國立臺灣師範大學/理學院/數學系/碩士(2003年)

摘要

我們已經知道，橢圓曲線上的有理點形成一有限生成的交換群，而其上的整數點亦是有限多個。那如何找出所有的整數點呢？在本篇論文裡，假設已知一組基底，那麼透過許多不等式的運用，我們能夠確確實實的找出在其上所有的整數點。

關鍵字

橢圓曲線 ； 整數點

並列摘要

Let E be an elliptic curve over Q. A well-known theorem of Siegel asserts that the
number of integral points on E is finite. So, for a given elliptic curve E over Q, it would
be interesting to find all the integral points. In [Za], Zagier describes several methods for
explicitly computing large integral points on elliptic curves defined over Q. In this thesis,
follow the line of [ST1], we shall discuss a method of computing all the integral points on an
elliptic curve over Q under the hypothesis that a basis for the free part of the Mordell-Weil
group is given.
In [ST1], R. J. Stroeker and N. Tzanakis adopt a natural approach, in which the linear
relation between an integral point and the generators of the free part of the Mordell-Weil
group is directly transformed into a linear form in elliptic logarithms. In order to produce
upper bounds for the coefficients in the original linear relation, we need an effective lower
bound for the linear form in elliptic logarithms. Thanks to S. David [D, Th´eor`eme 2.1], such
an explicit lower bound was established. The upper bound for the linear form in elliptic
logarithms was established in [ST1], where one needs to deduce an upper bound for the
function (see section 2.2) described in [Za].
In section 2, we discuss three main inequalities which are given in [ST1], as well as a special
case of David’s lower bound which is described in the appendix of [ST1]. In section 3, by
combining the main inequalities and David’s lower bound, we obtain an upper bound for the
coefficients in the original linear relation. However, the upper bound obtained in section 3 is
too large to search all the integral points. So, we need to apply the LLL-reduction procedure
to reduce the upper bound of the coefficients. This will constitute section 4. In the final
section, some examples are given.

並列關鍵字

elliptic curves ； integral point

參考文獻

[D] S. David, Minorations de formes lin´eaires de logarithmes elliptiques , Publ. Math. Univ. Pierre et Marie

[GZ] Josef Gebel and Horst G. Zimmer Computing the Mordell-Weil group of an elliptic curve over Q, CRM

Proceedings and Lecture Notes, vol. 4, 1994, 61-83.

[LLL] A. K. Lenstra, H. W. Lenstra Jr. , and L. Lov´asz, Factoring polynomials with rational coefficients,

Math. Ann. 261 (1982), 515-534.