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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:

[1] Bost, J.-B., Mestre, J.-F., Moret-Bailly, L., Calcul explicite en genre 2, dans Séminaire sur les pinceaux de courbes elliptiques, édité par L Szpiro. Astérisque183 (1990).
[2] Fay, J.D., Theta Functions on Riemann Surfaces, 352, Springer- Verlag (1973). · Zbl 0281.30013
[3] Grant, D., Formal Groups in Genus Two, J. Reine und Angew. Math. 441 (1990), 96-121. · Zbl 0702.14025
[4] Grant, D., A Generalisation of a Formula of Eisenstein, Proc. London Math. Soc., (3) 62 (1991), 121-132. · Zbl 0738.14019
[5] Mumford, D., Tata Lectures on Theta II., Progress in Math.43, Birkhäuser, (1984). · Zbl 0549.14014
[6] Shimura, G., Tanayama, Y., Complex Multiplication of Abelian Varieties and Its Application to Number Theory, Mathematical Society of Japan (1961). · Zbl 0112.03502
[7] Weil, A., On the Theory of Complex Multiplication, Proc. International Symposium on Algebraic Number Theory, Tokyo- Nikko (1955), 9-12. · Zbl 0074.26802
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