location:  Publications → journals
Search results

Search: All articles in the CJM digital archive with keyword class group

 Expand all        Collapse all Results 1 - 4 of 4

1. CJM Online first

Hajir, Farshid; Maire, Christian
 On the invariant factors of class groups in towers of number fields For a finite abelian $p$-group $A$ of rank $d=\dim A/pA$, let $\mathbb{M}_A := \log_p |A|^{1/d}$ be its \emph{(logarithmic) mean exponent}. We study the behavior of the mean exponent of $p$-class groups in pro-$p$ towers $\mathrm{L}/K$ of number fields. Via a combination of results from analytic and algebraic number theory, we construct infinite tamely ramified pro-$p$ towers in which the mean exponent of $p$-class groups remains bounded. Several explicit examples are given with $p=2$. Turning to group theory, we introduce an invariant $\underline{\mathbb{M}}(G)$ attached to a finitely generated pro-$p$ group $G$; when $G=\operatorname{Gal}(\mathrm{L}/\mathrm{K})$, where $\mathrm{L}$ is the Hilbert $p$-class field tower of a number field $K$, $\underline{\mathbb{M}}(G)$ measures the asymptotic behavior of the mean exponent of $p$-class groups inside $\mathrm{L}/\mathrm{K}$. We compare and contrast the behavior of this invariant in analytic versus non-analytic groups. We exploit the interplay of group-theoretical and number-theoretical perspectives on this invariant and explore some open questions that arise as a result, which may be of independent interest in group theory. Keywords:class field tower, ideal class group, pro-p group, p-adic analytic group, Brauer-Siegel TheoremCategories:11R29, 11R37

2. CJM 2001 (vol 53 pp. 434)

van der Poorten, Alfred J.; Williams, Kenneth S.
 Values of the Dedekind Eta Function at Quadratic Irrationalities: Corrigendum Habib Muzaffar of Carleton University has pointed out to the authors that in their paper [A] only the result $\pi_{K,d}(x)+\pi_{K^{-1},d}(x)=\frac{1}{h(d)}\frac{x}{\log x}+O_{K,d}\Bigl(\frac {x}{\log^2x}\Bigr)$ follows from the prime ideal theorem with remainder for ideal classes, and not the stronger result $\pi_{K,d}(x)=\frac{1}{2h(d)}\frac{x}{\log x}+O_{K,d}\Bigl(\frac {x}{\log^2x}\Bigr)$ stated in Lemma~5.2. This necessitates changes in Sections~5 and 6 of [A]. The main results of the paper are not affected by these changes. It should also be noted that, starting on page 177 of [A], each and every occurrence of $o(s-1)$ should be replaced by $o(1)$. Sections~5 and 6 of [A] have been rewritten to incorporate the above mentioned correction and are given below. They should replace the original Sections~5 and 6 of [A]. Keywords:Dedekind eta function, quadratic irrationalities, binary quadratic forms, form class groupCategories:11F20, 11E45

3. CJM 1999 (vol 51 pp. 176)

van der Poorten, Alfred; Williams, Kenneth S.
 Values of the Dedekind Eta Function at Quadratic Irrationalities Let $d$ be the discriminant of an imaginary quadratic field. Let $a$, $b$, $c$ be integers such that $$b^2 - 4ac = d, \quad a > 0, \quad \gcd (a,b,c) = 1.$$ The value of $\bigl|\eta \bigl( (b + \sqrt{d})/2a \bigr) \bigr|$ is determined explicitly, where $\eta(z)$ is Dedekind's eta function $$\eta (z) = e^{\pi iz/12} \prod^\ty_{m=1} (1 - e^{2\pi imz}) \qquad \bigl( \im(z) > 0 \bigr). %\eqno({\rm im}(z)>0).$$ Keywords:Dedekind eta function, quadratic irrationalities, binary quadratic forms, form class groupCategories:11F20, 11E45

4. CJM 1998 (vol 50 pp. 1253)

López-Bautista, Pedro Ricardo; Villa-Salvador, Gabriel Daniel
 Integral representation of $p$-class groups in ${\Bbb Z}_p$-extensions and the Jacobian variety For an arbitrary finite Galois $p$-extension $L/K$ of $\zp$-cyclotomic number fields of $\CM$-type with Galois group $G = \Gal(L/K)$ such that the Iwasawa invariants $\mu_K^-$, $\mu_L^-$ are zero, we obtain unconditionally and explicitly the Galois module structure of $\clases$, the minus part of the $p$-subgroup of the class group of $L$. For an arbitrary finite Galois $p$-extension $L/K$ of algebraic function fields of one variable over an algebraically closed field $k$ of characteristic $p$ as its exact field of constants with Galois group $G = \Gal(L/K)$ we obtain unconditionally and explicitly the Galois module structure of the $p$-torsion part of the Jacobian variety $J_L(p)$ associated to $L/k$. Keywords:${\Bbb Z}_p$-extensions, Iwasawa's theory, class group, integral representation, fields of algebraic functions, Jacobian variety, Galois module structureCategories:11R33, 11R23, 11R58, 14H40
 top of page | contact us | privacy | site map |