The infrastructure of a global field of arbitrary unit rank

Abstract

Abstract. In this paper, we show a general way to interpret the infrastructure of a global field of arbitrary unit rank. This interpretation generalizes the prior concepts of the giant-step operation and f -representations, and makes it possible to relate the infrastructure to the (Arakelov) divisor class group of the global field. In the case of global function fields, we present results that establish that effective implementation of the presented methods is indeed possible, and we show how Shanks' baby-step giant-step method can be generalized to this situation.

On the probability of generating a lattice

“…For the sake of completeness, we now describe the reduction in more detail. The infrastructure of a number field is isomorphic to a torus T = R d /M, where M is a fullrank lattice in R d and the coefficients of all non-trivial vectors of M are transcendental numbers [Fon11]. This forces one to work with approximations, which is ultimately responsible for the poor performance when the dimension d increases.…”

Infrastructure: Structure Inside the Class Group of a Real Quadratic Field

“…The result is a cleaner and simpler ideal arithmetic, avoiding the need for numerical approximations as discussed in the previous section. Moreover, interestingly, integer distances make it possible to provide an explicit and effective embedding of the principal infrastructure into a finite cyclic group whose order is the regulator, as was discovered independently by Fontein [12] and Mireles Morales [22]. Such an embedding into a finite cyclic group does not exist in the number field setting, although embedding the infrastructure into an infinite cyclic group is possible (see [20]).…”

Computing the unit group, class group, and compact representations in algebraic function fields

“…Fontein [17] showed that it is possible to compute in a finite abelian group which he denotes Rep f * (ᏻ) and which is isomorphic to ‫ޚ‬ n / . We discuss his approach in the next section.…”

“…Minima and reduced ideals in function fields. We now give the definitions of minima and reduced ideals and define Rep f * (ᏻ) (see [17]). In the following, by an ideal of ᏻ we will always mean a fractional ideal of ᏻ.…”

“…Our work, by contrast, requires the different representation using the infrastructure of Fontein [17] in order to compute in the unit group and the class group, rather than only in the divisor class group Pic 0 (K ). The infrastructure also allows us to show the existence of compact representations.…”