Let be a central -curve over a polyquadratic field . In this article we give an upper bound for prime divisors of the order of the -rational torsion subgroup (see Theorems 1.1 and 1.2). The notion of central -curves is a generalization of that of elliptic curves over . Our result is a generalization of Theorem 2 of Mazur [12], and it is a precision of the upper bounds of Merel [15] and Oesterlé [17].
We generalize the concept of reduced Arakelov divisors and define C-reduced divisors for a given number C ≥ 1. These C-reduced divisors have remarkable properties, similar to the properties of reduced ones. We describe an algorithm to test whether an Arakelov divisor of a real quadratic field F is C-reduced in time polynomial in with the discriminant of F. Moreover, we give an example of a cubic field for which our algorithm does not work.
In this paper it is discus a relation between -density and -density. A generalization of Šalát’s result concerning this relation in the case of asymptotic density is proved.