3 research outputs found
On abelian -ramification torsion modules of quadratic fields
For a number field and a prime number , the -torsion
module of the Galois group of the maximal abelian pro- extension of
unramified outside over , denoted as , is an important
subject in abelian -ramification theory. In this paper we study the group
of the quadratic field . Firstly, assuming , we prove an explicit -rank formula for
. Furthermore, applying this formula and exploring the
connection of to the ideal class group of
and the tame kernel of , we
obtain the -rank density of -groups of imaginary quadratic
fields. Secondly, for an odd prime, we obtain results about the
-divisibility of orders of and . In particular we find that if where
is the -class number of . We then obtain
density results for and .
Finally, based on our density results and numerical data, we propose
distribution conjectures about when varies over real or
imaginary quadratic fields for any prime , and about
and when varies, in the spirit of Cohen-Lenstra
heuristics. Our conjecture in the case is closely connected
to Shanks-Sime-Washington's speculation on the distributions of the zeros of
-adic -functions and to the distributions of the fundamental units.Comment: 23 page
Heuristics in direction of a p-adic Brauer--Siegel theorem
Let p be a fixed prime number. Let K be a totally real number field of
discriminant D\_K and let T\_K be the torsion group of the Galois group of the
maximal abelian p-ramified pro-p-extension of K (under Leopoldt's conjecture).
We conjecture the existence of a constant C\_p>0 such that log(\#T\_K)
C\_p log(\sqrt(D\_K)) when K varies in some specified families (e.g., fields of
fixed degree). In some sense, we suggest the existence of a p-adic analogue, of
the classical Brauer--Siegel Theorem, wearing here on the valuation of the
residue at s=1 (essentially equal to \#T\_K) of the p-adic zeta-function
zeta\_p(s) of K.We shall use a different definition that of Washington, given
in the 1980's, and approach this question via the arithmetical study of T\_K
since p-adic analysis seems to fail because of possible abundant "Siegel zeros"
of zeta\_p(s), contrary to the classical framework.We give extensive numerical
verifications for quadratic and cubic fields (cyclic or not) and publish the
PARI/GP programs directly usable by the reader for numerical improvements. Such
a conjecture (if exact) reinforces our conjecture that any fixed number field K
is p-rational (i.e., T\_K=1) for all p >> 0 .Comment: Improvements of numerical results - New arguments for the
conjecture(34 pages+PARI/GP programs
ZEROS OF 2-ADIC L-FUNCTIONS AND CONGRUENCES FOR CLASS NUMBERS AND FUNDAMENTAL UNITS
Abstract. We study the imaginary quadratic fields such that the Iwasawa Ξ»2-invariant equals 1, obtaining information on zeros of 2-adic L-functions and relating this to congruences for fundamental units and class numbers. This paper explores the interplay between zeros of 2-adic L-functions and congruences for fundamental units and class numbers of quadratic fields. An underlying motivation was to study the distribution of zeros of 2-adic L-functions, the basic philosophy being that the location of the zeros causes restrictions on the 2-adic behavior of the class numbers and fundamental units of real quadratic fields. Though the predicted restrictions involved the unit and class number together, numerical computations (we used PARI) revealed definite patterns for the unit and class number separately, which we were then able to prove. Several of these congruences are classical, but some of them seem to be new. We use the information obtained to study the distribution of the zeros, in particular their distances from 1 and 0. In a previous paper [14], one of us showed that, if (2 p +1)/3 is prime infinitely often, then it is possible to have zeros of 2-adic Lfunction