1. CJM Online first
 Brasca, Riccardo

Eigenvarieties for cuspforms over PEL type Shimura varieties with dense ordinary locus
Let $p \gt 2$ be a prime and let $X$ be a compactified PEL Shimura
variety of type (A) or (C) such that $p$ is an unramified prime
for the PEL datum and such that the ordinary locus is dense in
the reduction of $X$. Using the geometric approach of Andreatta,
Iovita, Pilloni, and Stevens we define the notion of families
of overconvergent locally analytic $p$adic modular forms of
Iwahoric level for $X$. We show that the system of eigenvalues
of any finite slope cuspidal eigenform of Iwahoric level can
be deformed to a family of systems of eigenvalues living over
an open subset of the weight space. To prove these results, we
actually construct eigenvarieties of the expected dimension that
parameterize finite slope systems of eigenvalues appearing in
the space of families of cuspidal forms.
Keywords:$p$adic modular forms, eigenvarieties, PELtype Shimura varieties Categories:11F55, 11F33 

2. CJM Online first
 Sugiyama, Shingo; Tsuzuki, Masao

Existence of Hilbert cusp forms with nonvanishing $L$values
We develop a derivative version of the relative trace formula
on $\operatorname{PGL}(2)$ studied in our previous work,
and derive an asymptotic formula of an average of central values
(derivatives)
of automorphic $L$functions for Hilbert cusp forms.
As an application, we prove the existence of Hilbert cusp forms
with nonvanishing central values (derivatives)
such that the absolute degrees of their Hecke fields are arbitrarily
large.
Keywords:automorphic representations, relative trace formulas, central $L$values, derivatives of $L$functions Categories:11F67, 11F72 

3. CJM Online first
 Kohen, Daniel; Pacetti, Ariel

Heegner points on Cartan nonsplit curves
Let $E/\mathbb{Q}$ be an elliptic curve of conductor
$N$, and
let $K$ be an imaginary quadratic field such that the root
number of
$E/K$ is $1$. Let $\mathscr{O}$ be an order in $K$ and assume that
there
exists an odd prime $p$, such that $p^2 \mid\mid N$, and $p$
is inert in
$\mathscr{O}$. Although there are no Heegner points on $X_0(N)$
attached to $\mathscr{O}$, in this article we construct such points on
Cartan nonsplit curves. In order to do that we
give a method to compute Fourier expansions for forms on Cartan
nonsplit curves, and prove that the constructed points form a
Heegner system as in the classical case.
Keywords:Cartan curves, Heegner points Categories:11G05, 11F30 

4. CJM Online first
 Levin, Aaron; Wang, Julie TzuYueh

On nonArchimedean curves omitting few components and their arithmetic analogues
Let $\mathbf{k}$ be an algebraically closed field complete with respect
to a nonArchimedean absolute value of arbitrary characteristic.
Let $D_1,\dots, D_n$ be effective nef divisors intersecting
transversally in an $n$dimensional nonsingular projective variety
$X$.
We study the degeneracy of nonArchimedean analytic maps from
$\mathbf{k}$ into $X\setminus \cup_{i=1}^nD_i$ under various geometric
conditions. When $X$ is a rational ruled surface and $D_1$ and
$D_2$ are ample, we obtain a necessary and sufficient condition
such that
there is no nonArchimedean analytic map from $\mathbf{k}$ into $X\setminus
D_1 \cup D_2$.
Using the dictionary between nonArchimedean Nevanlinna theory
and Diophantine approximation that originated in
earlier work with T. T. H. An, %
we also study arithmetic analogues of these problems, establishing
results on integral points on these varieties over $\mathbb{Z}$
or the ring of integers of an imaginary quadratic field.
Keywords:nonArchimedean Picard theorem, nonArchimedean analytic curves, integral points Categories:11J97, 32P05, 32H25 

5. CJM Online first
 Garibaldi, Skip; Nakano, Daniel K.

Bilinear and quadratic forms on rational modules of split reductive groups
The representation theory of semisimple algebraic groups over
the complex numbers (equivalently, semisimple complex Lie algebras
or Lie groups, or real compact Lie groups) and the question of
whether a
given complex representation is symplectic or orthogonal has
been solved since at least the 1950s. Similar results for Weyl
modules of split reductive groups over fields of characteristic
different from 2 hold by
using similar proofs. This paper considers analogues of these
results for simple, induced and tilting modules of split reductive
groups over fields of prime characteristic as well as a complete
answer for Weyl modules over fields of characteristic 2.
Keywords:orthogonal representations, symmetric tensors, alternating forms, characteristic 2, split reductive groups Categories:20G05, 11E39, 11E88, 15A63, 20G15 

6. 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 

7. CJM 2015 (vol 67 pp. 1046)
 Dubickas, Arturas; Sha, Min; Shparlinski, Igor

Explicit Form of Cassels' $p$adic Embedding Theorem for Number Fields
In this paper, we mainly give a general explicit form of Cassels'
$p$adic embedding theorem for number fields. We also give its
refined form in the case of cyclotomic fields. As a byproduct,
given an irreducible polynomial $f$ over $\mathbb{Z}$, we give a general
unconditional upper bound for the smallest prime number $p$ such
that $f$ has a simple root modulo $p$.
Keywords:number field, $p$adic embedding, height, polynomial, cyclotomic field Categories:11R04, 11S85, 11G50, 11R09, 11R18 

8. CJM Online first
 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 

9. CJM Online first
 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 

10. 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 

11. 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 

12. 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 

13. CJM 2015 (vol 67 pp. 597)
 Drappeau, Sary

Sommes friables d'exponentielles et applications
An integer is said to be $y$friable if its greatest prime factor is less than $y$.
In this paper, we obtain estimates for exponential sums
over $y$friable numbers up to $x$ which are nontrivial
when $y \geq \exp\{c \sqrt{\log x} \log \log x\}$. As a consequence,
we obtain an asymptotic formula for the
number of $y$friable solutions to the equation $a+b=c$ which is valid
unconditionnally under the same assumption.
We use a contour integration argument based on the saddle point
method, as developped in the context of friable numbers by Hildebrand
and Tenenbaum,
and used by Lagarias, Soundararajan and Harper to study exponential and character sums over friable numbers.
Keywords:thÃ©orie analytique des nombres, entiers friables, mÃ©thode du col Categories:12N25, 11L07 

14. 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 

15. CJM Online first
 Takeda, Shuichiro

Metaplectic Tensor Products for Automorphic Representation of $\widetilde{GL}(r)$
Let $M=\operatorname{GL}_{r_1}\times\cdots\times\operatorname{GL}_{r_k}\subseteq\operatorname{GL}_r$ be a Levi
subgroup of $\operatorname{GL}_r$, where $r=r_1+\cdots+r_k$, and $\widetilde{M}$ its metaplectic preimage
in the $n$fold metaplectic cover $\widetilde{\operatorname{GL}}_r$ of $\operatorname{GL}_r$. For automorphic
representations $\pi_1,\dots,\pi_k$ of $\widetilde{\operatorname{GL}}_{r_1}(\mathbb{A}),\dots,\widetilde{\operatorname{GL}}_{r_k}(\mathbb{A})$,
we construct (under a certain
technical assumption, which is always satisfied when $n=2$) an
automorphic representation $\pi$
of $\widetilde{M}(\mathbb{A})$ which can be considered as the ``tensor product'' of the
representations $\pi_1,\dots,\pi_k$. This is
the global analogue of the metaplectic tensor product
defined by P. Mezo in the sense that locally at each place $v$,
$\pi_v$ is equivalent to the local metaplectic tensor product of
$\pi_{1,v},\dots,\pi_{k,v}$ defined by Mezo. Then we show that if all
of $\pi_i$ are cuspidal (resp. squareintegrable modulo center), then
the metaplectic tensor product is cuspidal (resp. squareintegrable
modulo center). We also show that (both
locally and globally) the metaplectic tensor product behaves in the
expected way under the action of a Weyl group element, and show the
compatibility with parabolic inductions.
Keywords:automorphic forms, representations of covering groups Category:11F70 

16. CJM 2014 (vol 67 pp. 893)
17. CJM 2014 (vol 67 pp. 848)
 Köck, Bernhard; Tait, Joseph

Faithfulness of Actions on RiemannRoch Spaces
Given a faithful action of a finite group $G$ on an algebraic
curve~$X$ of genus $g_X\geq 2$, we give explicit criteria for
the induced action of~$G$ on the RiemannRoch space~$H^0(X,\mathcal{O}_X(D))$
to be faithful, where $D$ is a $G$invariant divisor on $X$ of
degree at least~$2g_X2$. This leads to a concise answer to the
question when the action of~$G$ on the space~$H^0(X, \Omega_X^{\otimes
m})$ of global holomorphic polydifferentials of order $m$ is
faithful. If $X$ is hyperelliptic, we furthermore provide an
explicit basis of~$H^0(X, \Omega_X^{\otimes m})$. Finally, we
give applications in deformation theory and in coding theory
and we discuss the analogous problem for the action of~$G$ on
the first homology $H_1(X, \mathbb{Z}/m\mathbb{Z})$ if $X$ is a Riemann surface.
Keywords:faithful action, RiemannRoch space, polydifferential, hyperelliptic curve, equivariant deformation theory, Goppa code, homology Categories:14H30, 30F30, 14L30, 14D15, 11R32 

18. CJM 2014 (vol 66 pp. 993)
 BeuzartPlessis, Raphaël

Expression d'un facteur epsilon de paire par une formule intÃ©grale
Let $E/F$ be a quadratic extension of $p$adic fields and
let $d$, $m$ be nonnegative integers of distinct parities. Fix
admissible irreducible tempered representations $\pi$ and $\sigma$ of
$GL_d(E)$ and $GL_m(E)$ respectively. We assume that $\pi$ and
$\sigma$ are conjugatedual. That is to say $\pi\simeq \pi^{\vee,c}$
and $\sigma\simeq \sigma^{\vee,c}$ where $c$ is the non trivial
$F$automorphism of $E$. This implies, we can extend $\pi$ to an
unitary representation $\tilde{\pi}$ of a nonconnected group
$GL_d(E)\rtimes \{1,\theta\}$. Define $\tilde{\sigma}$ the same
way. We state and prove an integral formula for
$\epsilon(1/2,\pi\times \sigma,\psi_E)$ involving the characters of
$\tilde{\pi}$ and $\tilde{\sigma}$. This formula is related to the
local GanGrossPrasad conjecture for unitary groups.
Keywords:epsilon factor, twisted groups Categories:22E50, 11F85 

19. CJM 2014 (vol 67 pp. 795)
 Di Nasso, Mauro; Goldbring, Isaac; Jin, Renling; Leth, Steven; Lupini, Martino; Mahlburg, Karl

On a Sumset Conjecture of ErdÅs
ErdÅs conjectured that for any set $A\subseteq \mathbb{N}$
with positive
lower asymptotic density, there are infinite sets $B,C\subseteq
\mathbb{N}$
such that $B+C\subseteq A$. We verify ErdÅs' conjecture in
the case that $A$ has Banach density exceeding $\frac{1}{2}$.
As a consequence, we prove that, for $A\subseteq \mathbb{N}$
with
positive Banach density (a much weaker assumption than positive
lower density), we can find infinite $B,C\subseteq \mathbb{N}$
such
that $B+C$ is contained in the union of $A$ and a translate of
$A$. Both of the aforementioned
results are generalized to arbitrary countable
amenable groups. We also provide a positive solution to ErdÅs'
conjecture for subsets of the natural numbers that are pseudorandom.
Keywords:sumsets of integers, asymptotic density, amenable groups, nonstandard analysis Categories:11B05, 11B13, 11P70, 28D15, 37A45 

20. CJM 2014 (vol 67 pp. 424)
 Samart, Detchat

Mahler Measures as Linear Combinations of $L$values of Multiple Modular Forms
We study the Mahler measures of certain families of Laurent
polynomials in two and three variables. Each of the known Mahler
measure formulas for these families involves $L$values of at most one
newform and/or at most one quadratic character. In this paper, we
show, either rigorously or numerically, that the Mahler measures of
some polynomials are related to $L$values of multiple newforms and
quadratic characters simultaneously. The results suggest that the
number of modular $L$values appearing in the formulas significantly
depends on the shape of the algebraic value of the parameter chosen
for each polynomial. As a consequence, we also obtain new formulas
relating special values of hypergeometric series evaluated at
algebraic numbers to special values of $L$functions.
Keywords:Mahler measures, EisensteinKronecker series, $L$functions, hypergeometric series Categories:11F67, 33C20 

21. CJM 2014 (vol 67 pp. 507)
 Borwein, Peter; Choi, Stephen; Ferguson, Ron; Jankauskas, Jonas

On Littlewood Polynomials with Prescribed Number of Zeros Inside the Unit Disk
We investigate the numbers of complex zeros of Littlewood polynomials
$p(z)$ (polynomials with coefficients $\{1, 1\}$) inside or
on the unit circle $z=1$, denoted by $N(p)$ and $U(p)$, respectively.
Two types of Littlewood polynomials are considered: Littlewood
polynomials with one sign change in the sequence of coefficients
and Littlewood polynomials with one negative coefficient. We
obtain explicit formulas for $N(p)$, $U(p)$ for polynomials $p(z)$
of these types. We show that, if $n+1$ is a prime number, then
for each integer $k$, $0 \leq k \leq n1$, there exists a Littlewood
polynomial $p(z)$ of degree $n$ with $N(p)=k$ and $U(p)=0$. Furthermore,
we describe some cases when the ratios $N(p)/n$ and $U(p)/n$
have limits as $n \to \infty$ and find the corresponding limit
values.
Keywords:Littlewood polynomials, zeros, complex roots Categories:11R06, 11R09, 11C08 

22. CJM 2014 (vol 66 pp. 1078)
 Lanphier, Dominic; Skogman, Howard

Values of Twisted Tensor $L$functions of Automorphic Forms Over Imaginary Quadratic Fields
Let $K$ be a complex quadratic extension of $\mathbb{Q}$ and let $\mathbb{A}_K$
denote the adeles of $K$.
We find special values at all of the critical points of twisted
tensor $L$functions attached to cohomological cuspforms on $GL_2(\mathbb{A}_K)$,
and establish Galois equivariance of the values.
To investigate the values, we determine the archimedean factors
of a class of integral representations of these $L$functions,
thus proving a conjecture due to Ghate. We also investigate
analytic properties of these $L$functions, such as their functional
equations.
Keywords:twisted tensor $L$function, cuspform, hypergeometric series Categories:11F67, 11F37 

23. 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 

24. CJM 2014 (vol 67 pp. 286)
 Bell, Jason P.; Lagarias, Jeffrey C.

A SkolemMahlerLech Theorem for Iterated Automorphisms of $K$algebras
This paper proves a commutative algebraic extension
of a generalized SkolemMahlerLech theorem due to the first
author.
Let $A$ be a finitely generated commutative $K$algebra
over a field of characteristic $0$, and let $\sigma$ be
a $K$algebra automorphism of $A$.
Given ideals $I$ and $J$ of $A$, we show that
the set $S$ of integers $m$ such that
$\sigma^m(I) \supseteq J$ is a finite union of
complete doubly infinite arithmetic progressions in $m$, up to
the addition of a finite set.
Alternatively, this result states that for an affine scheme
$X$ of finite type over $K$,
an automorphism $\sigma \in \operatorname{Aut}_K(X)$, and $Y$ and $Z$
any two closed subschemes of $X$, the set
of integers $m$ with $\sigma^m(Z ) \subseteq Y$ is as above.
The paper presents examples
showing that this result may fail to hold if the affine scheme
$X$ is
not of finite type, or if $X$ is of finite type but the field
$K$ has positive characteristic.
Keywords:automorphisms, endomorphisms, affine space, commutative algebras, SkolemMahlerLech theorem Categories:11D45, 14R10, 11Y55, 11D88 

25. 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 
