Integral Points on Elliptic Curves Defined by

Integral Points on Elliptic Curves Defined by Simplest Cubic Fields Sylvain Duquesne

CONTENTS Introduction 1. Elliptic Curves Defined by Simplest Cubic Fields 2. 3. 4. 5.

Linear Forms in Elliptic Logarithms Computation of Integral Points Tables of Results General Results about Integral Points on the Elliptic Curves y2 = x3 + mx2 (m+3)x + 1 References

Let f(X) be a cubic polynomial defining a simplest cubic field in the sense of Shanks. We study integral points on elliptic curves of the form Y2 = f(X). We compute the complete list of integral points on these curves for the values of the parameter below 1000. We prove that this list is exhaustive by using the methods of Tzanakis and de Weger, together with bounds on linear forms in elliptic logarithms due to S. David. Finally, we analyze this list and we prove in the general case the phenomena that we have observed. In particular, we find all integral points on the curve when the rank is equal to 1.

INTRODUCTION

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Duquesne: Integral Points on Elliptic Curves Defined by Simplest Cubic Fields

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100


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7= + ) H 9 Z Q ONIM} S + «) H W H( #69

7 #69 . Q ONIM} S + ) H « W H(

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2

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(5–2)

5E. About the Special Point [0, 1] /2N 1 ljJ : gVF 6 mw N\=Zn~gfU5%k ]5d_f45 L6l.6el?35mM: 39 Theorem 5.7. Y7; ´ %} N ] "!=KOD F>=IT">J#I>=HH* Proof.

3

=H4> !I3HH"!$#H4 Q Jb%FH= / F%=2 }TFI/2 S FI"!B#/74"!M"!. =O ^#%GF=I9 7 }%N ] !V"!V%

Duquesne: Integral Points on Elliptic Curves Defined by Simplest Cubic Fields

% ! F=>OD%!=9u Q ] " e# B K6#I!!% S !h"x"!%Bw N\=Zn~ ) 9^yN!K"! FO7- ] D 4 Q ` S 4HH Q ` S . H>4D Q ` S 9c?!ODaGL#p"!% 4#H !I!X">a"F!I Q w N\=Zn~ S Q w N\=Zn~ S + ) Q S ´

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HpJH 22 W

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U + H;6@ Q }eZ S L ´

+

U + +

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Theorem 5.8. 5 3 l4jJ% : fhg m gVKl?i

+

´

/21Nlb: 6Fk3JgV6Fm laL 3[5%kjO:%gV6Lm8fb: 6aC qNkm gzjeklKf : > m1NljO:%gV6Lmpw N\=Zn~'5 3l '

w N\=Zn~g > gf l.Kl.6 9 w N\=Zn~ 5 6Lv4#w N\=Zn~g >

1gso.1

gfU:=vve9

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