Bestimmung der Idealklassenzahl in gewissen normalen einfachen Algebren.
It is already known that all Pisot numbers are beta numbers, but for Salem numbers this was proved just for the degree 4 case. In 1945, R. Salem showed that for any Pisot number θ we can construct a sequence of Salem numbers which converge to θ. In this short note, we give some results on the beta expansion for infinitely many sequences of Salem numbers obtained by this construction.
In a recent paper we proved that there are at most finitely many complex numbers such that the points and are both torsion on the Legendre elliptic curve defined by . In a sequel we gave a generalization to any two points with coordinates algebraic over the field and even over . Here we reconsider the special case and with complex numbers and .
For any Eichler order of level in an indefinite quaternion algebra of discriminant there is a Fuchsian group and a Shimura curve . We associate to a set of binary quadratic forms which have semi-integer quadratic coefficients, and we develop a classification theory, with respect to , for primitive forms contained in . In particular, the classification theory of primitive integral binary quadratic forms by is recovered. Explicit fundamental domains for allow the characterization...
Let , with a positive integer, be a pure cubic number field. We show that the elements whose squares have the form for rational numbers form a group isomorphic to the group of rational points on the elliptic curve . This result will allow us to construct unramified quadratic extensions of pure cubic number fields .
A bibliography of recent papers on lower bounds for discriminants of number fields and related topics is presented. Some of the main methods, results, and open problems are discussed.
Let be an odd prime and a cyclic -extension of number fields. We give a lower bound for the order of the kernel and cokernel of the natural extension map between the even étale -groups of the ring of -integers of , where is a finite set of primes containing those which are -adic.