1. CJM 2016 (vol 68 pp. 571)
 Gras, Georges

Les $\theta$rÃ©gulateurs locaux d'un nombre algÃ©brique : Conjectures $p$adiques
Let $K/\mathbb{Q}$ be Galois and let $\eta\in K^\times$ be such that
$\operatorname{Reg}_\infty (\eta) \ne 0$.
We define the local $\theta$regulators $\Delta_p^\theta(\eta)
\in \mathbb{F}_p$
for the $\mathbb{Q}_p\,$irreducible characters $\theta$ of
$G=\operatorname{Gal}(K/\mathbb{Q})$. A linear representation ${\mathcal L}^\theta\simeq \delta \,
V_\theta$ is associated with
$\Delta_p^\theta (\eta)$ whose nullity is equivalent to $\delta
\geq 1$.
Each $\Delta_p^\theta (\eta)$ yields $\operatorname{Reg}_p^\theta (\eta)$
modulo $p$ in the factorization
$\prod_{\theta}(\operatorname{Reg}_p^\theta (\eta))^{\varphi(1)}$ of
$\operatorname{Reg}_p^G (\eta) := \frac{ \operatorname{Reg}_p(\eta)}{p^{[K : \mathbb{Q}\,]}
}$ (normalized $p$adic regulator).
From $\operatorname{Prob}\big (\Delta_p^\theta(\eta) = 0 \ \& \ {\mathcal
L}^\theta \simeq \delta \, V_\theta\big )
\leq p^{ f \delta^2}$ ($f \geq 1$ is a residue degree) and the
BorelCantelli heuristic,
we conjecture that, for $p$ large enough, $\operatorname{Reg}_p^G (\eta)$
is a $p$adic unit or that
$p^{\varphi(1)} \parallel \operatorname{Reg}_p^G (\eta)$ (a single $\theta$
with $f=\delta=1$); this obstruction may be lifted assuming the
existence of a binomial probability law
confirmed through numerical studies
(groups $C_3$, $C_5$, $D_6$).
This conjecture would imply that, for all $p$ large enough,
Fermat quotients, normalized $p$adic
regulators are $p$adic units and that
number fields are $p$rational.
We recall some deep cohomological results that
may strengthen such conjectures.
Keywords:$p$adic regulators, LeopoldtJaulent conjecture, Frobenius group determinants, characters, Fermat quotient, Abelian $p$ramification, probabilistic number theory Categories:11F85, 11R04, 20C15, 11C20, 11R37, 11R27, 11Y40 

2. CJM 2016 (vol 68 pp. 655)
 Klartag, Bo'az; Kozma, Gady; Ralli, Peter; Tetali, Prasad

Discrete Curvature and Abelian Groups
We study a natural discrete Bochnertype inequality on graphs,
and explore its merit as a notion of ``curvature'' in discrete
spaces.
An appealing feature of this discrete version of the socalled
$\Gamma_2$calculus (of BakryÃmery) seems to be that it is
fairly
straightforward to compute this notion of curvature parameter
for
several specific graphs of interest  particularly, abelian
groups, slices of the hypercube, and the symmetric group under
various sets of generators.
We further develop this notion by deriving Busertype inequalities
(Ã la Ledoux), relating functional and isoperimetric constants
associated with a graph.
Our derivations provide a tight bound on the Cheeger constant
(i.e., the edgeisoperimetric constant) in terms of
the spectral gap, for graphs with nonnegative curvature, particularly,
the class of abelian Cayley graphs  a result of independent
interest.
Keywords:Ricci curvature, graph theory, abelian groups Categories:53C21, 57M15 

3. CJM 2015 (vol 68 pp. 44)
 Fernández Bretón, David J.

Strongly Summable Ultrafilters, Union Ultrafilters, and the Trivial Sums Property
We answer two questions of Hindman, SteprÄns and Strauss,
namely we prove that every
strongly summable
ultrafilter on an abelian group is sparse and has the trivial
sums property. Moreover we
show that in most
cases the sparseness of the given ultrafilter is a
consequence of its being isomorphic to a union ultrafilter. However,
this does not happen
in all cases:
we also construct (assuming Martin's Axiom for countable partial
orders, i.e.
$\operatorname{cov}(\mathcal{M})=\mathfrak c$), on the
Boolean group, a strongly summable ultrafilter that
is not additively isomorphic to any union ultrafilter.
Keywords:ultrafilter, StoneCech compactification, sparse ultrafilter, strongly summable ultrafilter, union ultrafilter, finite sum, additive isomorphism, trivial sums property, Boolean group, abelian group Categories:03E75, 54D35, 54D80, 05D10, 05A18, 20K99 

4. CJM 2015 (vol 67 pp. 654)
 Lim, Meng Fai; Murty, V. Kumar

Growth of Selmer groups of CM Abelian varieties
Let $p$ be an odd prime. We study the variation of the $p$rank of
the Selmer group of Abelian varieties with complex multiplication in
certain towers of number fields.
Keywords:Selmer group, Abelian variety with complex multiplication, $\mathbb{Z}_p$extension, $p$Hilbert class tower Categories:11G15, 11G10, 11R23, 11R34 

5. CJM 2015 (vol 68 pp. 24)
 Bonfanti, Matteo Alfonso; van Geemen, Bert

Abelian Surfaces with an Automorphism and Quaternionic Multiplication
We construct one dimensional families of Abelian surfaces with
quaternionic multiplication
which also have an automorphism of order three or four. Using Barth's
description of the moduli space of $(2,4)$polarized Abelian surfaces,
we find the Shimura curve parametrizing these Abelian surfaces in a
specific case.
We explicitly relate these surfaces to the Jacobians of genus two
curves studied by Hashimoto and Murabayashi.
We also describe a (Humbert) surface in Barth's moduli space which
parametrizes Abelian surfaces with real multiplication by
$\mathbf{Z}[\sqrt{2}]$.
Keywords:abelian surfaces, moduli, shimura curves Categories:14K10, 11G10, 14K20 

6. CJM 2014 (vol 67 pp. 198)
 Murty, V. Kumar; Patankar, Vijay M.

Tate Cycles on Abelian Varieties with Complex Multiplication
We consider Tate cycles on an Abelian variety $A$ defined over
a sufficiently large number field $K$ and having complex
multiplication. We show that
there is an effective bound $C = C(A,K)$ so that
to check whether a given cohomology class is a Tate class on
$A$, it suffices to check the action of
Frobenius elements at primes $v$ of norm $ \leq C$.
We also show that for a set of primes $v$ of $K$ of density
$1$, the space of Tate cycles on the special fibre $A_v$ of the
NÃ©ron model of $A$ is isomorphic to the space of Tate cycles
on $A$ itself.
Keywords:Abelian varieties, complex multiplication, Tate cycles Categories:11G10, 14K22 

7. CJM 2012 (vol 66 pp. 170)
 Guitart, Xavier; Quer, Jordi

Modular Abelian Varieties Over Number Fields
The main result of this paper is a characterization of the abelian
varieties $B/K$ defined over Galois number fields with the
property that the $L$function $L(B/K;s)$ is a product of
$L$functions of nonCM newforms over $\mathbb Q$ for congruence
subgroups of the form $\Gamma_1(N)$. The characterization involves the
structure of $\operatorname{End}(B)$, isogenies between the Galois conjugates of
$B$, and a Galois cohomology class attached to $B/K$.
We call the varieties having this property strongly modular.
The last section is devoted to the study of a family of abelian surfaces with quaternionic
multiplication.
As an illustration of the ways in which the general results of the paper can be applied
we prove the strong modularity of some particular abelian surfaces belonging to that family, and
we show how to find nontrivial examples of strongly modular varieties by twisting.
Keywords:Modular abelian varieties, $GL_2$type varieties, modular forms Categories:11G10, 11G18, 11F11 

8. CJM 2012 (vol 65 pp. 403)
 Van Order, Jeanine

On the Dihedral Main Conjectures of Iwasawa Theory for Hilbert Modular Eigenforms
We construct a bipartite Euler system in the sense of Howard for Hilbert modular eigenforms of parallel
weight two over totally real fields, generalizing works of BertoliniDarmon, Longo, Nekovar, PollackWeston
and others. The construction has direct applications to Iwasawa main conjectures. For instance, it implies
in many cases one divisibility of the associated dihedral or anticyclotomic main conjecture, at the same
time reducing the other divisibility to a certain nonvanishing criterion for the associated $p$adic $L$functions.
It also has applications to cyclotomic main conjectures for Hilbert modular forms over CM fields via the technique
of Skinner and Urban.
Keywords:Iwasawa theory, Hilbert modular forms, abelian varieties Categories:11G10, 11G18, 11G40 

9. CJM 2012 (vol 65 pp. 195)
 Penegini, Matteo; Polizzi, Francesco

Surfaces with $p_g=q=2$, $K^2=6$, and Albanese Map of Degree $2$
We classify minimal surfaces of general type with $p_g=q=2$ and
$K^2=6$ whose Albanese map is a generically finite double cover.
We show that the corresponding moduli space is the disjoint union
of three generically smooth irreducible components
$\mathcal{M}_{Ia}$, $\mathcal{M}_{Ib}$, $\mathcal{M}_{II}$ of
dimension $4$, $4$, $3$, respectively.
Keywords:surface of general type, abelian surface, Albanese map Categories:14J29, 14J10 

10. CJM 2011 (vol 63 pp. 1058)
 Easton, Robert W.

$S_3$covers of Schemes
We analyze flat $S_3$covers of schemes, attempting to create
structures parallel to those found in the abelian and triple cover
theories. We use an initial local analysis as a guide in finding a
global description.
Keywords:nonabelian groups, permutation group, group covers, schemes Category:14L30 
