3 research outputs found

    On abelian 22-ramification torsion modules of quadratic fields

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    For a number field FF and a prime number pp, the Zp\mathbb{Z}_p-torsion module of the Galois group of the maximal abelian pro-pp extension of FF unramified outside pp over FF, denoted as Tp(F)\mathcal{T}_p(F), is an important subject in abelian pp-ramification theory. In this paper we study the group T2(F)=T2(m)\mathcal{T}_2(F)=\mathcal{T}_2(m) of the quadratic field F=Q(m)F=\mathbb{Q}(\sqrt{ m}). Firstly, assuming m>0m>0, we prove an explicit 44-rank formula for T2(βˆ’m)\mathcal{T}_2(-m). Furthermore, applying this formula and exploring the connection of T2(βˆ’m)\mathcal{T}_2(-m) to the ideal class group of Q(βˆ’m)\mathbb{Q}(\sqrt{-m}) and the tame kernel of Q(m)\mathbb{Q}(\sqrt{m}), we obtain the 44-rank density of T2\mathcal{T}_2-groups of imaginary quadratic fields. Secondly, for ll an odd prime, we obtain results about the 22-divisibility of orders of T2(Β±l)\mathcal{T}_2(\pm l) and T2(Β±2l)\mathcal{T}_2(\pm 2l). In particular we find that #T2(l)≑2#T2(2l)≑h2(βˆ’2l)β€Šmodβ€Š16\#\mathcal{T}_2(l)\equiv 2\# \mathcal{T}_2(2l)\equiv h_2(-2l)\bmod{16} if l≑7β€Šmodβ€Š8l\equiv 7\bmod{8} where h2(βˆ’2l)h_2(-2l) is the 22-class number of Q(βˆ’2l)\mathbb{Q}(\sqrt{-2l}). We then obtain density results for T2(Β±l)\mathcal{T}_2(\pm l) and T2(Β±2l)\mathcal{T}_2(\pm 2l). Finally, based on our density results and numerical data, we propose distribution conjectures about Tp(F)\mathcal{T}_p(F) when FF varies over real or imaginary quadratic fields for any prime pp, and about T2(Β±l)\mathcal{T}_2(\pm l) and T2(Β±2l)\mathcal{T}_2(\pm 2 l) when ll varies, in the spirit of Cohen-Lenstra heuristics. Our conjecture in the T2(l)\mathcal{T}_2(l) case is closely connected to Shanks-Sime-Washington's speculation on the distributions of the zeros of 22-adic LL-functions and to the distributions of the fundamental units.Comment: 23 page

    Heuristics in direction of a p-adic Brauer--Siegel theorem

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    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) ≀\le 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

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    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
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