1. CJM Online first
 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 

2. CJM 2015 (vol 67 pp. 1326)
 Cojocaru, Alina Carmen; Shulman, Andrew Michael

The Distribution of the First Elementary Divisor of the Reductions of a Generic Drinfeld Module of Arbitrary Rank
Let $\psi$ be a generic Drinfeld module of rank $r \geq 2$. We study
the first elementary divisor
$d_{1, \wp}(\psi)$ of the reduction of $\psi$ modulo a prime $\wp$, as $\wp$ varies.
In particular, we prove the existence of the density of the primes $\wp$ for which $d_{1, \wp} (\psi)$ is fixed. For $r = 2$, we also study the second elementary divisor (the exponent) of the reduction of $\psi$ modulo $\wp$
and prove that, on average, it has a large norm. Our work is motivated by the study of J.P. Serre of an elliptic curve analogue of Artin's Primitive Root Conjecture, and, moreover, by refinements to Serre's study developed by the first author and M.R. Murty.
Keywords:Drinfeld modules, density theorems Categories:11R45, 11G09, 11R58 

3. 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 

4. CJM 1998 (vol 50 pp. 1253)
 LópezBautista, Pedro Ricardo; VillaSalvador, 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 structure Categories:11R33, 11R23, 11R58, 14H40 
