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Quelques propriétés arithmétiques des points de 3-division de la jacobienne de \(y^ 2=x^ 5-1\). (Some arithmetic properties of 3- division points of the Jacobian of \(y^ 2=x^ 5-1\)). (French) Zbl 0766.14019

Let \(C\) be the smooth projective irreducible curve, defined over \(\mathbb{Q}\), obtained by adjunction of a point \(\infty\) to the plane affine curve with equation \(y^ 2=x^ 5-1\). Let \(J\) be the Jacobian of \(C\), let \(C^{(2)}\) be the symmetric square of \(C\) and let \(\psi:C^{(2)}\to J\) be the birational morphism defined by \(\psi(\xi+\eta)=[\xi]+[\eta]- 2[\infty]\). Let \(u,v\) and \(f\) be the rational functions on \(J\) defined on \(C^{(2)}\) by \(u(\xi+\eta)=x(\xi)+x(\eta)\), \(v(\xi+\eta)=x(\xi)x(\eta)\) and \(f=-u+v+1\). These functions are regular outside of the origin of \(J\). It is shown that their values at the (nonzero) 3-torsion points \(P\) of \(J\) are algebraic integers whose fifth powers belong to the maximal real subfield of the field of 15-th roots of unity. Moreover, the \(f(P)/\sqrt 5\) are units. Finally, explicit approximate values for the \(f(P)\), as well as their minimal polynomial, are obtained.

MSC:

14H40 Jacobians, Prym varieties
14H52 Elliptic curves
14G05 Rational points
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References:

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