1. CJM 2016 (vol 68 pp. 361)
 Fité, Francesc; González, Josep; Lario, Joan Carles

Frobenius Distribution for Quotients of Fermat Curves of Prime Exponent
Let $\mathcal{C}$ denote the Fermat curve over $\mathbb{Q}$ of prime
exponent $\ell$. The Jacobian $\operatorname{Jac}(\mathcal{C})$
of~$\mathcal{C}$ splits over $\mathbb{Q}$ as the product of Jacobians
$\operatorname{Jac}(\mathcal{C}_k)$, $1\leq k\leq \ell2$, where
$\mathcal{C}_k$ are curves obtained as quotients of $\mathcal{C}$ by
certain subgroups of automorphisms of $\mathcal{C}$. It is well known
that $\operatorname{Jac}(\mathcal{C}_k)$ is the power of an absolutely
simple abelian variety $B_k$ with complex multiplication. We call
degenerate those pairs $(\ell,k)$ for which $B_k$ has degenerate CM
type. For a nondegenerate pair $(\ell,k)$, we compute the SatoTate
group of $\operatorname{Jac}(\mathcal{C}_k)$, prove the generalized
SatoTate Conjecture for it, and give an explicit method to compute
the moments and measures of the involved distributions. Regardless of
$(\ell,k)$ being degenerate or not, we also obtain Frobenius
equidistribution results for primes of certain residue degrees in the
$\ell$th cyclotomic field. Key to our results is a detailed study of
the rank of certain generalized Demjanenko matrices.
Keywords:SatoTate group, Fermat curve, Frobenius distribution Categories:11D41, 11M50, 11G10, 14G10 

2. 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 

3. 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 

4. 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 

5. CJM 2013 (vol 66 pp. 924)
 Stankewicz, James

Twists of Shimura Curves
Consider a Shimura curve $X^D_0(N)$ over the rational
numbers. We determine criteria for the twist by an AtkinLehner
involution to have points over a local field. As a corollary we give a
new proof of the theorem of JordanLivnÃ© on $\mathbf{Q}_p$ points
when $p\mid D$ and for the first time give criteria for $\mathbf{Q}_p$
points when $p\mid N$. We also give congruence conditions for roots
modulo $p$ of Hilbert class polynomials.
Keywords:Shimura curves, complex multiplication, modular curves, elliptic curves Categories:11G18, 14G35, 11G15, 11G10 

6. 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 

7. 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 

8. CJM 2011 (vol 63 pp. 481)
 Baragar, Arthur

The Ample Cone for a K3 Surface
In this paper, we give several pictorial fractal
representations of the ample or KÃ¤hler cone for surfaces in a
certain class of $K3$ surfaces. The class includes surfaces
described by smooth $(2,2,2)$ forms in ${\mathbb P^1\times\mathbb P^1\times \mathbb P^1}$ defined over a
sufficiently large number field $K$ that have a line parallel to
one of the axes and have Picard number four. We relate the
Hausdorff dimension of this fractal to the asymptotic growth of
orbits of curves under the action of the surface's group of
automorphisms. We experimentally estimate the Hausdorff dimension
of the fractal to be $1.296 \pm .010$.
Keywords:Fractal, Hausdorff dimension, K3 surface, Kleinian groups, dynamics Categories:14J28, , , , 14J50, 11D41, 11D72, 11H56, 11G10, 37F35, 37D05 

9. CJM 2008 (vol 60 pp. 532)
 Clark, Pete L.; Xarles, Xavier

Local Bounds for Torsion Points on Abelian Varieties
We say that an abelian variety over a $p$adic field $K$ has
anisotropic reduction (AR) if the special fiber of its N\'eron minimal
model does not contain a nontrivial split torus. This includes all
abelian varieties with potentially good reduction and, in particular,
those with complex or quaternionic multiplication. We give a bound for
the size of the $K$rational torsion subgroup of a $g$dimensional AR
variety depending only on $g$ and the numerical invariants of $K$ (the
absolute ramification index and the cardinality of the residue
field). Applying these bounds to abelian varieties over a number field
with everywhere locally anisotropic reduction, we get bounds which, as
a function of $g$, are close to optimal. In particular, we determine
the possible cardinalities of the torsion subgroup of an AR abelian
surface over the rational numbers, up to a set of 11 values which are
not known to occur. The largest such value is 72.
Categories:11G10, 14K15 

10. CJM 2007 (vol 59 pp. 372)
 Maisner, Daniel; Nart, Enric

Zeta Functions of Supersingular Curves of Genus 2
We determine which isogeny classes of supersingular abelian
surfaces over a finite field $k$ of characteristic $2$ contain
jacobians. We deal with this problem in a direct way by computing
explicitly the zeta function of all supersingular curves of genus
$2$. Our procedure is constructive, so that we are able to exhibit
curves with prescribed zeta function and find formulas for the
number of curves, up to $k$isomorphism, leading to the same zeta
function.
Categories:11G20, 14G15, 11G10 

11. CJM 2002 (vol 54 pp. 1202)
 Fernández, J.; Lario, JC.; Rio, A.

Octahedral Galois Representations Arising From $\mathbf{Q}$Curves of Degree $2$
Generically, one can attach to a $\mathbf{Q}$curve $C$ octahedral representations
$\rho\colon\Gal(\bar{\mathbf{Q}}/\mathbf{Q})\rightarrow\GL_2(\bar\mathbf{F}_3)$
coming from the Galois action on the $3$torsion of those abelian varieties of
$\GL_2$type whose building block is $C$. When $C$ is defined over a quadratic
field and has an isogeny of degree $2$ to its Galois conjugate, there exist
such representations $\rho$ having image into $\GL_2(\mathbf{F}_9)$. Going
the other way, we can ask which $\mod 3$ octahedral representations $\rho$ of
$\Gal(\bar\mathbf{Q}/\mathbf{Q})$ arise from $\mathbf{Q}$curves in the above
sense. We characterize those arising from quadratic $\mathbf{Q}$curves of
degree $2$. The approach makes use of Galois embedding techniques in
$\GL_2(\mathbf{F}_9)$, and the characterization can be given in terms of a
quartic polynomial defining the $\mathcal{S}_4$extension of $\mathbf{Q}$
corresponding to the projective representation $\bar{\rho}$.
Categories:11G05, 11G10, 11R32 
