2,113 research outputs found
p-adic Verification of Class Number Computations
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
Graded structures and differential operators on nearly holomorphic and quasimodular forms on classical groups
We wish to use graded structures [KrVu87], [Vu01] on dffierential operators
and quasimodular forms on classical groups and show that these structures
provide a tool to construct p-adic measures and p-adic L-functions on the
corresponding non-archimedean weight spaces. An approach to constructions of
automorphic L-functions on uni-tary groups and their p-adic avatars is
presented. For an algebraic group G over a number eld K these L functions are
certain Euler products L(s, , r, ). In particular, our constructions
cover the L-functions in [Shi00] via the doubling method of Piatetski-Shapiro
and Rallis. A p-adic analogue of L(s, , r, ) is a p-adic analytic
function L p (s, , r, ) of p-adic arguments s Z p , mod
p rPresented in a talk for the INTERNATIONAL SCIENTIFIC CONFERENCE "GRADED
STRUCTURES IN ALGEBRA AND THEIR APPLICATIONS" dedicated to the memory of Prof.
Marc Krasner on Friday, September 23, 2016, International University Centre
(IUC), Dubrovnik, Croatia
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