Geodesic
Solutions of Specific Diophantine Equations and their Relationship to Complex Multiplication
Canadian mathematical bulletin, Tome 32 (1989) no. 2, pp. 223-229

Voir la notice de l'article provenant de la source Cambridge University Press

In this paper we establish a relationship between the rational solutions (x(t), y(t)), over C(t), of the diophantine equation: and the solutions which parametrize the elliptic curve E, y 2 = 4x 3 — g 2 x — g 3 admitting complex multiplication by λ. We first characterize the form of all rational solutions of diophantine equation (1). The rational solutions are derivable from the subsititutions in which μ = 0,ω1,ω2,ω1 + ω2 = ω3. Using techniques established in elliptic function theory, we prove that the complex multiplier λ, associated with a unique rational solution (x(t), y(t)), must be of a certain form. Next we construct all rational solutions of diophantine equation (1) by using the addition theorems valid for the Weierstrass function, Specific examples are finally worked out for the cases
DOI : 10.4153/CMB-1989-033-6
Mots-clés : 11G05 
Wajngurt, Clara. Solutions of Specific Diophantine Equations and their Relationship to Complex Multiplication. Canadian mathematical bulletin, Tome 32 (1989) no. 2, pp. 223-229. doi: 10.4153/CMB-1989-033-6
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