1. 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 

2. CJM Online first
 Bonfanti, Matteo Alfonso; van Geemen,

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 Online first
 Chandee, Vorrapan; David, Chantal; Koukoulopoulos, Dimitris; Smith, Ethan

The frequency of elliptic curve groups over prime finite fields
Letting $p$ vary over all primes and $E$ vary over all elliptic
curves over the finite field $\mathbb{F}_p$, we study the frequency to
which a given group $G$ arises as a group of points $E(\mathbb{F}_p)$.
It is wellknown that the only permissible groups are of the
form $G_{m,k}:=\mathbb{Z}/m\mathbb{Z}\times \mathbb{Z}/mk\mathbb{Z}$.
Given such a candidate group, we let $M(G_{m,k})$ be the frequency
to which the group $G_{m,k}$ arises in this way.
Previously, the second and fourth named authors determined an
asymptotic formula for $M(G_{m,k})$ assuming a conjecture about primes
in short arithmetic progressions. In this paper, we prove several
unconditional bounds for $M(G_{m,k})$, pointwise and on average. In
particular, we show that $M(G_{m,k})$ is bounded above by a constant
multiple of the expected quantity when $m\le k^A$ and that the
conjectured asymptotic for $M(G_{m,k})$ holds for almost all groups
$G_{m,k}$ when $m\le k^{1/4\epsilon}$.
We also apply our methods to study the frequency to which a given
integer $N$ arises as the group order $\#E(\mathbb{F}_p)$.
Keywords:average order, elliptic curves, primes in short intervals Categories:11G07, 11N45, 11N13, 11N36 

4. CJM Online first
 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 

5. CJM Online first
 Stange, Katherine E.

Integral Points on Elliptic Curves and Explicit Valuations of Division Polynomials
Assuming Lang's conjectured lower bound on the heights of nontorsion
points on an elliptic curve, we show that there exists an absolute
constant $C$ such that for any elliptic curve $E/\mathbb{Q}$ and nontorsion
point $P \in E(\mathbb{Q})$, there is at most one integral multiple
$[n]P$ such that $n \gt C$. The proof is a modification of a proof
of Ingram giving an unconditional but not uniform bound. The
new ingredient is a collection of explicit formulae for the
sequence $v(\Psi_n)$ of valuations of the division polynomials.
For $P$ of nonsingular reduction, such sequences are already
well described in most cases, but for $P$ of singular reduction,
we are led to define a new class of sequences called \emph{elliptic
troublemaker sequences}, which measure the failure of the NÃ©ron
local height to be quadratic. As a corollary in the spirit of
a conjecture of Lang and Hall, we obtain a uniform upper bound
on $\widehat{h}(P)/h(E)$ for integer points having two large
integral multiples.
Keywords:elliptic divisibility sequence, Lang's conjecture, height functions Categories:11G05, 11G07, 11D25, 11B37, 11B39, 11Y55, 11G50, 11H52 

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 2013 (vol 66 pp. 826)
 Kim, Byoung Du

SignedSelmer Groups over the $\mathbb{Z}_p^2$extension of an Imaginary Quadratic Field
Let $E$ be an elliptic curve over $\mathbb Q$ which has good supersingular
reduction at $p\gt 3$. We construct what we call the $\pm/\pm$Selmer
groups of $E$ over the $\mathbb Z_p^2$extension of an imaginary quadratic
field $K$ when the prime $p$ splits completely over $K/\mathbb Q$, and
prove they enjoy a property analogous to Mazur's control theorem.
Furthermore, we propose a conjectural connection between the
$\pm/\pm$Selmer groups and Loeffler's twovariable $\pm/\pm$$p$adic
$L$functions of elliptic curves.
Keywords:elliptic curves, Iwasawa theory Category:11Gxx 

8. CJM 2013 (vol 66 pp. 1305)
 Koskivirta, JeanStefan

Congruence Relations for Shimura Varieties Associated with $GU(n1,1)$
We prove the congruence relation for the mod$p$ reduction of Shimura
varieties associated to a unitary similitude group $GU(n1,1)$ over
$\mathbb{Q}$, when $p$ is inert and $n$ odd. The case when $n$
is even was obtained by T. Wedhorn and O. B?ltel, as a special case
of a result of B. Moonen, when the $\mu$ordinary locus of the $p$isogeny
space is dense. This condition fails in our case. We show that every
supersingular irreducible component of the special fiber of $p\textrm{}\mathscr{I}sog$
is annihilated by a degree one polynomial in the Frobenius element
$F$, which implies the congruence relation.
Keywords:Shimura varieties, congruence relation Categories:11G18, 14G35, 14K10 

9. 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 

10. CJM 2013 (vol 66 pp. 1167)
 Rotger, Victor; de VeraPiquero, Carlos

Galois Representations Over Fields of Moduli and Rational Points on Shimura Curves
The purpose of this note is introducing a method for proving the
existence of no rational points on a coarse moduli space $X$ of abelian varieties
over a given number field $K$, in cases where the moduli problem is not fine and
points in $X(K)$ may not be represented by an abelian variety (with additional structure)
admitting a model over the field $K$. This is typically the case when the abelian
varieties that are being classified have even dimension. The main idea, inspired on
the work of Ellenberg and Skinner on the modularity of $\mathbb{Q}$curves, is that to a
point $P=[A]\in X(K)$ represented by an abelian variety $A/\bar K$ one may still
attach a Galois representation of $\operatorname{Gal}(\bar K/K)$ with values in the quotient
group $\operatorname{GL}(T_\ell(A))/\operatorname{Aut}(A)$, provided
$\operatorname{Aut}(A)$ lies in the centre of $\operatorname{GL}(T_\ell(A))$.
We exemplify our method in the cases where $X$ is a Shimura curve over an imaginary
quadratic field or an AtkinLehner quotient over $\mathbb{Q}$.
Keywords:Shimura curves, rational points, Galois representations, Hasse principle, BrauerManin obstruction Categories:11G18, 14G35, 14G05 

11. 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 

12. 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 

13. CJM 2011 (vol 64 pp. 1248)
 Gärtner, Jérôme

Darmon's Points and Quaternionic Shimura Varieties
In this paper, we generalize a conjecture due to Darmon and Logan in
an adelic setting. We study the relation between our construction and
Kudla's works on cycles on orthogonal Shimura varieties. This relation
allows us to conjecture a GrossKohnenZagier theorem for Darmon's
points.
Keywords:elliptic curves, StarkHeegner points, quaternionic Shimura varieties Categories:11G05, 14G35, 11F67, 11G40 

14. CJM 2011 (vol 64 pp. 588)
15. CJM 2011 (vol 64 pp. 282)
16. CJM 2011 (vol 64 pp. 301)
17. CJM 2011 (vol 64 pp. 151)
 Miller, Steven J.; Wong, Siman

Moments of the Rank of Elliptic Curves
Fix an elliptic curve $E/\mathbb{Q}$ and assume the Riemann Hypothesis
for the $L$function $L(E_D, s)$ for every quadratic twist $E_D$ of
$E$ by $D\in\mathbb{Z}$. We combine Weil's
explicit formula with techniques of HeathBrown to derive an asymptotic
upper bound for the weighted moments of the analytic rank of $E_D$. We
derive from this an upper bound for the density of lowlying zeros of
$L(E_D, s)$ that is compatible with the random matrix models of Katz and
Sarnak. We also show that for any unbounded increasing function $f$ on $\mathbb{R}$,
the analytic rank and (assuming in addition the Birch and SwinnertonDyer
conjecture)
the number of integral points of $E_D$ are less than $f(D)$
for almost all $D$.
Keywords:elliptic curve, explicit formula, integral point, lowlying zeros, quadratic twist, rank Categories:11G05, 11G40 

18. CJM 2011 (vol 63 pp. 992)
 Bruin, Nils; Doerksen, Kevin

The Arithmetic of Genus Two Curves with (4,4)Split Jacobians
In this paper we study genus $2$ curves whose Jacobians admit a
polarized $(4,4)$isogeny to a product of elliptic curves. We consider
base fields of characteristic different from $2$ and $3$, which we do
not assume to be algebraically closed.
We obtain a full classification of all principally polarized abelian
surfaces that can arise from gluing two elliptic curves along their
$4$torsion, and we derive the relation their absolute invariants
satisfy.
As an intermediate step, we give a general description of Richelot
isogenies between Jacobians of genus $2$ curves, where previously only
Richelot isogenies with kernels that are pointwise defined over the
base field were considered.
Our main tool is a Galois theoretic characterization of genus $2$
curves admitting multiple Richelot isogenies.
Keywords:Genus 2 curves, isogenies, split Jacobians, elliptic curves Categories:11G30, 14H40 

19. CJM 2011 (vol 63 pp. 826)
 Errthum, Eric

Singular Moduli of Shimura Curves
The $j$function acts as a parametrization of the classical modular
curve. Its values at complex multiplication (CM) points are called
singular moduli and are algebraic integers. A Shimura curve is a
generalization of the modular curve and, if the Shimura curve has
genus~$0$, a rational parameterizing function exists and when
evaluated at a CM point is again algebraic over~$\mathbf{Q}$. This paper shows
that the coordinate maps given by N.~Elkies for the Shimura
curves associated to the quaternion algebras with discriminants $6$
and $10$ are Borcherds lifts of vectorvalued modular forms. This
property is then used to explicitly compute the rational norms of
singular moduli on these curves. This method not only verifies
conjectural values for the rational CM points, but also provides a way
of algebraically calculating the norms of CM points with arbitrarily
large negative discriminant.
Categories:11G18, 11F12 

20. CJM 2011 (vol 63 pp. 616)
 Lee, Edward

A Modular Quintic CalabiYau Threefold of Level 55
In this note we search the parameter space of HorrocksMumford quintic
threefolds and locate a CalabiYau threefold that is modular, in the
sense that the $L$function of its middledimensional cohomology is
associated with a classical modular form of weight 4 and level 55.
Keywords: CalabiYau threefold, nonrigid CalabiYau threefold, twodimensional Galois representation, modular variety, HorrocksMumford vector bundle Categories:14J15, 11F23, 14J32, 11G40 

21. 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 

22. CJM 2010 (vol 62 pp. 1155)
 Young, Matthew P.

Moments of the Critical Values of Families of Elliptic Curves, with Applications
We make conjectures on the moments of the central values of the family
of all elliptic curves and on the moments of the first derivative of
the central values of a large family of positive rank curves. In both
cases the order of magnitude is the same as that of the moments of the
central values of an orthogonal family of $L$functions. Notably, we
predict that the critical values of all rank $1$ elliptic curves is
logarithmically larger than the rank $1$ curves in the positive rank
family.
Furthermore, as arithmetical applications, we make a conjecture on the
distribution of $a_p$'s amongst all rank $2$ elliptic curves and
show how the Riemann hypothesis can be deduced from sufficient
knowledge of the first moment of the positive rank family (based on an
idea of Iwaniec)
Categories:11M41, 11G40, 11M26 

23. CJM 2010 (vol 62 pp. 1060)
 Darmon, Henri; Tian, Ye

Heegner Points over Towers of Kummer Extensions
Let $E$ be an elliptic curve, and let $L_n$ be the Kummer extension
generated by a primitive $p^n$th root of unity and a $p^n$th root of
$a$ for a fixed $a\in \mathbb{Q}^\times\{\pm 1\}$. A detailed case study
by Coates, Fukaya, Kato and Sujatha and V. Dokchitser has led these
authors to predict unbounded and strikingly regular growth for the
rank of $E$ over $L_n$ in certain cases. The aim of this note is to
explain how some of these predictions might be accounted for by
Heegner points arising from a varying collection of Shimura curve
parametrisations.
Categories:11G05, 11R23, 11F46 

24. 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 

25. CJM 2010 (vol 62 pp. 668)
 Vollaard, Inken

The Supersingular Locus of the Shimura Variety for GU(1,s)
In this paper we study the supersingular locus of the reduction modulo $p$ of the Shimura variety for $GU(1,s)$ in the case of an inert prime $p$. Using DieudonnÃ© theory we define a stratification of the corresponding moduli space of $p$divisible groups. We describe the incidence relation of this stratification in terms of the BruhatTits building of a unitary group. In the case of $GU(1,2)$, we show that the supersingular locus is equidimensional of dimension 1 and is of complete intersection. We give an explicit description of the irreducible components and their intersection behaviour.
Categories:14G35, 11G18, 14K10 
