P-Adic algorithms and the computation of zeros of p-adic l-functions

Abstract

A fundamental problem of experimental algebraic number theory is that of determining the unitand class group of an algebraic number field K. To solve this problem for large classes of number fields, effective algorithms are needed. Such algorithms mostly rest on methods from geometry of numbers, as this is the case with the method of Pohst and Zassenhaus. They require knowledge of an integral basis for K/~. Such bases of number fields canbe constructed in an efficient manner by applying an algorithm of Ford and Zassenhaus° Here local considerations play an important role. That is why one has to perform calculations not only over the completion of Q with respect to ordinary absolute value, i.e. over the real field IR, but at the same time also over the completions of Q with respect to the p-adic valuations, i.e. over the p-adic fields Up. Furthermore, a p-adic analogue of the complex number field ~ the latter being the ~]g~h~ closure of ~rR, is the completion Cp of the algebraic closure ~p of ~p. In this connection we recall that Hasse, in his fundamental book on algebraic number theory, uses a simultaneous treatment of the completions IR and @p as a foundation of the arithmetic in algebraic number fields.