## $$S$$-integral points on elliptic curves.(English)Zbl 0817.11031

This paper gives an algorithm to determine all “small” $$S$$-integral points on a Weierstraß model of an elliptic curve over $$\mathbb{Q}$$. Here “small” has the following meaning. If $$P$$ is an $$S$$-integral point ($$S$$ is a finite set of $$\mathbb{Q}$$-primes, including infinity), then $$P= \sum_{i=1}^ r q_ i P_ i+T$$, where $$\{P_ 1, \dots, P_ r\}$$ is a complete set of generators of infinite order of the Mordell-Weil group $$E(\mathbb{Q})$$ and $$T\in E(\mathbb{Q} )_{\text{tors}}$$. Now only those $$S$$- integral points $$P$$ are considered for which $$\max_{1\leq i\leq r} | q_ i|$$ is bounded by a (large) constant $$K$$. An example is given with $$K$$ of size $$10^{40}$$. Use is made of the theory of linear forms in real-valued elliptic logarithms. Subsequently, it is shown that this large bound $$K$$ may be reduced to one of size $$\sqrt {(r+1)\ln K}$$ by applying the $$L^ 3$$ algorithm. Usually this reduction process can be repeated several times.

### MSC:

 11G05 Elliptic curves over global fields 14H52 Elliptic curves

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### References:

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