## 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

### Keywords:

3-divison points; Jacobian
Full Text:

### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.