p-adic Verification of Class Number Computations

Abstract

The aim of this thesis is to determine if it is possible, using p-adic techniques, to unconditionally evaluate the p-valuation of the class number h of an algebraic number field K. This is important in many areas of number theory, especially Iwasawa theory. The class group ClK of an algebraic number field K is the group of fractional ideals of K modulo principal ideals. Its cardinality (the class number h) is directly linked to the existence of unique factorisation in K, and hence the class group is of core importance to almost all multiplicative problems concerning number fields. The explicit computation of ClK (and h) is a fundamental task in computational number theory. Despite its importance, existing algorithms cannot obtain the class group unconditionally in a reasonable amount of time if the field has a large discriminant. Although faster, specialised algorithms (focused only on calculating the p-valuation) are limited in the cases with which they can deal. We present two algorithms to verify the p-valuation of h for any totally real abelian number field, with no restrictions on p. Both algorithms are based on the p-adic class number formula and work by computing p-adic L-functions Lp(s,χ) at the value of s = 1. These algorithms came about from two different ways of computing Lp(1,χ), using either a closed or a convergent series formula. We prove that our algorithms compare favourably against existing class group algorithms, with superior complexity for number fields of degree 5 or higher. We also demonstrate that our algorithms are faster in practice. Finally, we present some open questions arising from the algorithms.