On torsion points of elliptic and hyperelliptic curves with applications

Sei f(x) aus K[x] ein normiertes Polynom vom Grad 2g+1 oder 2g+2 ohne mehrfache Nullstellen über einem Zahlkörper K. Die Kurve C : y^2 = f(x) ist eine elliptische Kurve für g=1 bzw. eine hyperelliptische Kurve für g > 1. Es werden (für g > 1) parametrisierte Kurvenfamilien berechnet, deren Jacobische Varietäten eine Divisorenklasse der Ordnung l=5,7,10 besitzen. Außerdem werden für 2 1. We construct (for g>1) parametric families of curves, whose Jacobians have a divisor class of order l=5,7,10. Further for 2<l<11 and l=12 we construct families of number fields with galois group the diedral group with 2l elements. For these fields we compute systems of fundamental units for l=4,5 and signature (2,1) and (1,2). For 2<l<7 we give subfamilies of number fields with cyclic galois group of order l. We start with a polynomial equation P^2 - Q^2 F = constant. We show that the Jacobian of the curve y^2 = F(x) has a divisor class whose order divides the degree of P. Families of hyperelliptic curves are constructed which have divisor classes of order 5, 7 or 10 in their Jacobians. We show that the families contain modular curves X_0(N) for N=22,29,31,41. All hyperelliptic modular curves X_0(N) with N a prime number (not 37) fit into the described strategy. The construction of parametric number fields with diedral galois group uses a torsion point A of order l of parametric families of elliptic curves. The x-coordinates of the points P+iA for a non rational point P give the polynomial F_l as a product of the factors (x-x(P+iA)) for 0<i<l+1. F_l has diedral galois group. For l=4 with signature (2,1) and for l=5 with signature (1,2) we compute systems of fundamental units for the equation orders generated by F_l. The proof of the fundamentality in case l=5 is done by approximations of the complex roots of F_5. The construction of families of number fields with cyclic galois group is based on indicator functions. These are linked to certain quotient curves. We construct such number fields for 2<l