1. CJM 2017 (vol 70 pp. 142)
 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 nonanalytic groups. We exploit
the interplay of grouptheoretical and numbertheoretical 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, prop group, padic analytic group, BrauerSiegel Theorem Categories:11R29, 11R37 

2. CJM 2016 (vol 69 pp. 579)
 Lee, Jungyun; Lee, Yoonjin

Regulators of an Infinite Family of the Simplest Quartic Function Fields
We explicitly find regulators of an infinite family $\{L_m\}$
of the simplest quartic function fields
with a parameter $m$ in a polynomial ring $\mathbb{F}_q [t]$, where
$\mathbb{F}_q$
is the finite field of order $q$
with odd characteristic. In fact, this infinite family of the
simplest quartic function fields are
subfields of maximal real subfields of cyclotomic function fields,
where they have the same conductors.
We obtain a lower bound on the class numbers of the family $\{L_m\}$
and some result on the divisibility
of the divisor class numbers of cyclotomic function fields which
contain $\{L_m\}$ as their subfields.
Furthermore, we find an explicit criterion for the characterization
of splitting types of all the primes
of the rational function field $\mathbb{F}_q (t)$ in $\{L_m\}$.
Keywords:regulator, function field, quartic extension, class number Categories:11R29, 11R58 

3. CJM 2012 (vol 65 pp. 1201)
 Cho, Peter J.; Kim, Henry H.

Application of the Strong Artin Conjecture to the Class Number Problem
We construct unconditionally several families of number fields with
the largest possible class numbers. They are number fields of degree 4
and 5 whose Galois closures have the Galois group $A_4, S_4$ and
$S_5$. We first construct families of number fields with smallest
regulators, and by using the strong Artin conjecture and applying zero
density result of KowalskiMichel, we choose subfamilies of
$L$functions which are zero free close to 1.
For these subfamilies, the $L$functions have the extremal value at
$s=1$, and by the class number formula, we obtain the extreme class
numbers.
Keywords:class number, strong Artin conjecture Categories:11R29, 11M41 

4. CJM 2010 (vol 62 pp. 787)
 Landquist, E.; Rozenhart, P.; Scheidler, R.; Webster, J.; Wu, Q.

An Explicit Treatment of Cubic Function Fields with Applications
We give an explicit treatment of cubic function fields of characteristic at least five. This includes an efficient technique for converting such a field into standard form, formulae for the field discriminant and the genus, simple necessary and sufficient criteria for nonsingularity of the defining curve, and a characterization of all triangular integral bases. Our main result is a description of the signature of any rational place in a cubic extension that involves only the defining curve and the order of the base field. All these quantities only require simple polynomial arithmetic as well as a few squarefree polynomial factorizations and, in some cases, square and cube root extraction modulo an irreducible polynomial. We also illustrate why and how signature computation plays an important role in computing the class number of the function field. This in turn has applications to the study of zeros of zeta functions of function fields.
Keywords:cubic function field, discriminant, nonsingularity, integral basis, genus, signature of a place, class number Categories:14H05, 11R58, 14H45, 11G20, 11G30, 11R16, 11R29 

5. CJM 2005 (vol 57 pp. 180)
 Somodi, Marius

On the Size of the Wild Set
To every pair of algebraic number fields with isomorphic Witt rings
one can associate a number, called the {\it minimum number of wild
primes}. Earlier investigations have established lower bounds for this
number. In this paper an analysis is presented that expresses the
minimum number of wild primes in terms of the number of wild dyadic
primes. This formula not only gives immediate upper bounds, but can be
considered to be an exact formula for the minimum number of wild
primes.
Categories:11E12, 11E81, 19F15, 11R29 

6. CJM 2001 (vol 53 pp. 1194)
7. CJM 1998 (vol 50 pp. 794)
 Louboutin, Stéphane

Upper bounds on $L(1,\chi)$ and applications
We give upper bounds on the modulus of the values at $s=1$ of
Artin $L$functions of abelian extensions unramified at all
the infinite places. We also explain how we can compute better
upper bounds and explain how useful such computed bounds are
when dealing with class number problems for $\CM$fields. For
example, we will reduce the determination of all the
nonabelian normal $\CM$fields of degree $24$ with Galois
group $\SL_2(F_3)$ (the special linear group over the finite
field with three elements) which have class number one to the
computation of the class numbers of $23$ such $\CM$fields.
Keywords:Dedekind zeta function, Dirichlet series, $\CM$field, relative class number Categories:11M20, 11R42, 11Y35, 11R29 

8. CJM 1997 (vol 49 pp. 283)
 McCall, Thomas M.; Parry, Charles J.; Ranalli, Ramona R.

The $2$rank of the class group of imaginary bicyclic biquadratic fields
A formula is obtained for the rank of the $2$Sylow subgroup of the
ideal class group of imaginary bicyclic biquadratic fields. This
formula involves the number of primes that ramify in the field, the
ranks of the $2$Sylow subgroups of the ideal class groups of the
quadratic subfields and the rank of a $Z_2$matrix determined by
Legendre symbols involving pairs of ramified primes. As
applications, all subfields with both $2$class and class group
$Z_2 \times Z_2$ are determined. The final results assume the
completeness of D.~A.~Buell's list of imaginary fields with small
class numbers.
Categories:11R16, 11R29, 11R20 
