1. CJM Online first
2. CJM Online first
3. CJM Online first
 Betina, Adel

Ramification of the Eigencurve at classical RM points
J.BellaÃ¯che and M.Dimitrov have shown that the $p$adic eigencurve
is smooth but not Ã©tale over the weight space at $p$regular
theta series attached to a character of a real quadratic field
$F$ in which $p$ splits. In this paper we prove the existence
of an isomorphism between the subring fixed by the AtkinLehner
involution of the completed local ring of the eigencurve at these
points and an universal ring representing a pseudodeformation
problem. Additionally, we give a precise criterion for which
the ramification index is exactly $2$.
We finish this paper by proving the smoothness of the nearly
ordinary and ordinary Hecke algebras for Hilbert modular forms
over $F$ at the overconvergent cuspidal Eisenstein points, being
the base change lift for $\operatorname{GL}(2)_{/F}$ of these theta series.
Our approach uses deformations and pseudodeformations of reducible
Galois representations.
Keywords:weight one RM modular form, eigencurve, pseudodeformation, deformation of reducible representation Categories:11F80, 11F33, 11R23 

4. CJM Online first
 Bosser, Vincent; Gaudron, Éric

Logarithmes des points rationnels des variÃ©tÃ©s abÃ©liennes
Nous dÃ©montrons une gÃ©nÃ©ralisation
du thÃ©orÃ¨me des pÃ©riodes de Masser et WÃ¼stholz
oÃ¹ la pÃ©riode est remplacÃ©e par un logarithme non
nul $u$ d'un point rationnel $p$ d'une variÃ©tÃ© abÃ©lienne
dÃ©finie sur un corps de nombres. Nous en dÃ©duisons des
minorations explicites de la norme de $u$ et de la hauteur de
NÃ©ronTate de $p$ qui dÃ©pendent des invariants classiques
du problÃ¨me dont la dimension et la hauteur de Faltings de
la variÃ©tÃ© abÃ©lienne. Les dÃ©monstrations reposent
sur une construction de transcendance du type Gel'fondBaker
de la thÃ©orie des formes linÃ©aires de logarithmes dans
laquelle se greffent des formules explicites provenant de la
thÃ©orie des pentes d'Arakelov.
Keywords:periods theorem, abelian variety, logarithm, Gel'fondBaker method, slope theory, NÃ©ronTate height, interpolation lemma Categories:11J86, 11J95, 11G10, 11G50, 14G40 

5. CJM 2018 (vol 70 pp. 1373)
 Tuxanidy, Aleksandr; Wang, Qiang

A New Proof of the HansenMullen Irreducibility Conjecture
We give a new proof of the HansenMullen irreducibility conjecture.
The proof relies on an application of a (seemingly new) sufficient
condition for the existence of
elements of degree $n$ in the support of functions on finite
fields.
This connection to irreducible polynomials is made via the least
period of the discrete Fourier transform (DFT) of functions with
values in finite fields.
We exploit this relation and prove, in an elementary fashion,
that a relevant function related to the DFT of characteristic
elementary symmetric functions (which produce the coefficients
of characteristic polynomials)
satisfies a simple requirement on the least period.
This bears a sharp contrast to previous techniques in literature
employed to tackle existence
of irreducible polynomials with prescribed coefficients.
Keywords:irreducible polynomial, primitive polynomial, HansenMullen conjecture, symmetric function, $q$symmetric, discrete Fourier transform, finite field Category:11T06 

6. CJM Online first
 Cahn, Jordan; Jones, Rafe; Spear, Jacob

Powers in orbits of rational functions: cases of an arithmetic dynamical MordellLang conjecture
Let $K$ be a finitely generated field of characteristic
zero. We study, for fixed $m \geq 2$, the rational functions
$\phi$ defined over $K$ that have a $K$orbit containing infinitely
many distinct $m$th powers. For $m \geq 5$ we show the only such
functions are those of the form $cx^j(\psi(x))^m$ with $\psi
\in K(x)$, and for $m \leq 4$ we show the only additional cases
are certain LattÃ¨s maps and four families of rational functions
whose special properties appear not to have been studied before.
With additional analysis, we show that the index set $\{n \geq
0 : \phi^{n}(a) \in \lambda(\mathbb{P}^1(K))\}$ is a union of
finitely many arithmetic progressions, where $\phi^{n}$ denotes
the $n$th iterate of $\phi$ and $\lambda \in K(x)$ is any map
MÃ¶biusconjugate over $K$ to $x^m$. When the index set is infinite,
we give bounds on the number and moduli of the arithmetic progressions
involved. These results are similar in flavor to the dynamical
MordellLang conjecture, and motivate a new conjecture on the
intersection of an orbit with the value set of a morphism.
A key ingredient in our proofs is a study of the curves $y^m
= \phi^{n}(x)$. We describe all $\phi$ for which these curves
have an irreducible component of genus at most 1, and show that
such $\phi$ must have two distinct iterates that are equal in
$K(x)^*/K(x)^{*m}$.
Keywords:arithmetic dynamics, iteration of rational functions, special orbits of rational function, genus of variablesseparated curve, LattÃ¨s map Categories:37P05, 11G05, 37P15 

7. CJM Online first
 Furuya, Jun; Minamide, Makoto; Tanigawa, Yoshio

Titchmarsh's method for the approximate functional equations for $\zeta^{\prime}(s)^{2}$, $\zeta(s)\zeta^{\prime\prime}(s)$ and $\zeta^{\prime}(s)\zeta^{\prime\prime}(s)$
Let $\zeta (s)$ be the Riemann zeta function. In 1929, Hardy
and Littlewood proved the approximate
functional equation for $\zeta^2(s)$ with error term $O(x^{1/2\sigma}((x+y)/t)^{1/4}\log
t)$
where $1/2\lt \sigma\lt 3/2$, $x,y \geq 1$, $xy=(t/2\pi)^2$. Later,
in 1938, Titchmarsh improved the error term by removing
the factor $((x+y)/t)^{1/4}$. In 1999, Hall showed the approximate
functional equations for $\zeta'(s)^2, \zeta(s)\zeta''(s) $
and $\zeta'(s)\zeta''(s)$ (in the range $0\lt \sigma\lt 1$) whose error
terms contain the factor $((x+y)/t)^{1/4}$.
In this paper we remove this factor from these three error terms
by using the method of Titchmarsh.
Keywords:derivative of the Riemann zeta function, approximate functional equation, exponential sum Category:11M06 

8. CJM Online first
 Mihara, Tomoki

Cohomological Approach to Class Field Theory in Arithmetic Topology
We establish class field theory for $3$dimensional manifolds
and knots. For this purpose, we formulate analogues of the multiplicative
group, the idÃ¨le class group, and ray class groups in a cocycletheoretic
way. Following the arguments in abstract class field theory,
we construct reciprocity maps and verify the existence theorems.
Keywords:arithmetic topology, class field theory, branched covering, knots and prime numbers Categories:11Z05, 18F15, 55N20, 57P05 

9. CJM 2018 (vol 70 pp. 1096)
 Müllner, Clemens

The RudinShapiro Sequence and Similar Sequences are Normal Along Squares
We prove that digital sequences modulo $m$ along squares are
normal,
which covers some prominent sequences like the sum of digits
in base $q$ modulo $m$, the RudinShapiro sequence and some generalizations.
This gives, for any base, a class of explicit normal numbers
that can be efficiently generated.
Keywords:RudinShapiro sequence, digital sequence, normality, exponential sum Categories:11A63, 11B85, 11L03, 11N60, 60F05 

10. CJM Online first
 Salazar, Daniel Barrera; Williams, Chris

$P$adic $L$functions for GL$_2$
Since Rob Pollack and Glenn Stevens used overconvergent
modular symbols to construct $p$adic $L$functions for noncritical
slope rational modular forms, the theory has been extended to
construct $p$adic $L$functions for noncritical slope automorphic
forms over totally real and imaginary quadratic fields by the
first and second authors respectively. In this paper, we give
an analogous construction over a general number field. In particular,
we start by proving a control theorem stating that the specialisation
map from overconvergent to classical modular symbols is an isomorphism
on the small slope subspace. We then show that if one takes the
modular symbol attached to a small slope cuspidal eigenform,
then one can construct a ray class distribution from the corresponding
overconvergent symbol, that moreover interpolates critical values
of the $L$function of the eigenform. We prove that this distribution
is independent of the choices made in its construction. We define
the $p$adic $L$function of the eigenform to be this distribution.
Keywords:automorphic form, GL(2), padic Lfunction, Lfunction, modular symbol, overconvergent, cohomology, automorphic cycle, control theorem, Lvalue, distribution Categories:11F41, 11F67, 11F85, 11S40, 11M41 

11. CJM Online first
 Gurevich, Nadya; Segal, Avner

Poles of the Standard $\mathcal{L}$function of $G_2$ and the RallisSchiffmann lift
We characterize the cuspidal representations of $G_2$ whose standard
$\mathcal{L}$function admits a pole at $s=2$ as the image of the RallisSchiffmann
lift for the commuting pair $(\widetilde{SL_2}, G_2)$ in $\widetilde{Sp_{14}}$.
The image consists of nontempered representations.
The main tool is the recent construction, by the second author,
of a family of RankinSelberg integrals representing the standard
$\mathcal{L}$function.
Keywords:automorphic representation, exceptional thetalift, SiegelWeil identity Categories:11F70, 11F27, 11F66 

12. CJM Online first
 BarySoroker, Lior; Stix, Jakob M.

Cubic twin prime polynomials are counted by a modular form
We present the geometry lying behind counting twin prime polynomials
in $\mathbb{F}_q[T]$ in general.
We compute cohomology and explicitly count points by means of
a twisted Lefschetz trace formula
applied to these parametrizing varieties for cubic twin
prime polynomials.
The elliptic curve $X^3 = Y(Y1)$ occurs in the geometry, and
thus counting cubic twin prime polynomials involves the associated
modular form. In theory, this approach can be extended to higher
degree twin primes, but the computations become harder.
The formula we get in degree $3$ is compatible with the HardyLittlewood
heuristic on average, agrees with the prediction for $q \equiv
2 \pmod 3$
but shows anomalies for $q \equiv 1 \pmod 3$.
Keywords:twin primes, finite field, polynomial Categories:11T55, 11G25 

13. CJM Online first
 Hartl, Urs; Singh, Rajneesh Kumar

Local Shtukas and Divisible Local Anderson Modules
We develop the analog of crystalline DieudonnÃ© theory for $p$divisible
groups in the arithmetic of function fields. In our theory $p$divisible
groups are replaced by divisible local Anderson modules, and
DieudonnÃ© modules are replaced by local shtukas. We show that
the categories of divisible local Anderson modules and of effective
local shtukas are antiequivalent over arbitrary base schemes.
We also clarify their relation with formal Lie groups and with
global objects like Drinfeld modules, Anderson's abelian $t$modules
and $t$motives, and Drinfeld shtukas. Moreover, we discuss the
existence of a Verschiebung map and apply it to deformations
of local shtukas and divisible local Anderson modules. As a tool
we use Faltings's and Abrashkin's theory of strict modules, which
we review to some extent.
Keywords:local shtuka, formal Drinfeld module, formal tmodule Categories:11G09, 13A35, 14L05 

14. CJM Online first
 Knightly, Andrew; Reno, Caroline

Weighted distribution of lowlying zeros of $\operatorname{GL}(2)$ $L$functions
We show that if the zeros of an automorphic $L$function are
weighted by the central value of the $L$function
or a quadratic imaginary base change,
then for certain families of holomorphic $\operatorname{GL}(2)$ newforms,
it has the effect of changing the distribution type of lowlying
zeros from orthogonal to symplectic, for test functions whose
Fourier
transforms have sufficiently restricted support.
However, if the $L$value is twisted by a nontrivial quadratic
character, the distribution type remains orthogonal.
The proofs involve two vertical equidistribution results for
Hecke
eigenvalues weighted by central twisted $L$values. One of
these
is due to Feigon
and Whitehouse, and the other
is new and involves
an asymmetric probability
measure that has not appeared in previous equidistribution
results for $\operatorname{GL}(2)$.
Keywords:low lying zero, Lfunction Categories:11M41, 11F11, 11M26 

15. CJM 2018 (vol 70 pp. 1390)
 Xiao, Stanley Yao

Squarefree Values of Decomposable Forms
In this paper we prove that decomposable forms,
or homogeneous polynomials $F(x_1, \cdots, x_n)$ with integer
coefficients which split completely into linear factors over
$\mathbb{C}$, take on infinitely many squarefree values subject
to simple necessary conditions and $\deg f \leq 2n + 2$ for all
irreducible factors $f$ of $F$. This work generalizes a theorem
of Greaves.
Keywords:squarefree value, decomposable form, Selberg sieve Category:11B05 

16. CJM 2018 (vol 70 pp. 1173)
 Viada, Evelina

An Explicit ManinDem'janenko Theorem in Elliptic Curves
Let $\mathcal{C}$ be a curve of genus at least $2$ embedded in $E_1
\times \cdots \times E_N$ where the $E_i$ are elliptic curves
for $i=1,\dots, N$. In this article we give an explicit sharp
bound for the NÃ©ronTate height of the points of $\mathcal{C}$ contained
in the union of all algebraic subgroups of dimension
$\lt \max(r_\mathcal{C}t_\mathcal{C},t_\mathcal{C})$
where $t_\mathcal{C}$, respectively $r_\mathcal{C}$, is the minimal dimension
of a translate, respectively of a torsion variety, containing
$\mathcal{C}$.
As a corollary, we give an explicit bound for the height of
the rational points of special curves, proving new cases of
the explicit Mordell Conjecture and in particular making explicit
(and slightly more general in the CM case) the ManinDem'janenko
method in products of elliptic curves.
Keywords:height, elliptic curve, explicit Mordell conjecture, explicit ManinDemjanenko theorem, rational points on a curve Categories:11G50, 14G40 

17. CJM Online first
 Green, Ben Joseph; Lindqvist, Sofia

Monochromatic solutions to $x + y = z^2$
Suppose that $\mathbb{N}$ is $2$coloured. Then there are infinitely
many monochromatic solutions to $x + y = z^2$. On the other hand,
there is a $3$colouring of $\mathbb{N}$ with only finitely many monochromatic solutions to this equation.
Keywords:additive combinatorics, Ramsey theory Categories:11B75, 05D10 

18. CJM 2018 (vol 70 pp. 683)
 Matringe, Nadir; Offen, Omer

Gamma Factors, Root Numbers, and Distinction
We study a relation between distinction and special values of
local invariants for representations of the general linear group
over a quadratic extension of $p$adic fields.
We show that the local RankinSelberg root number of any pair
of distinguished representation is trivial and as a corollary
we obtain an analogue for the global root number of any pair
of distinguished cuspidal representations. We further study the
extent to which the gamma factor at $1/2$ is trivial for distinguished
representations as well as the converse problem.
Keywords:distinguished representation, local constant Category:11F70 

19. CJM 2017 (vol 70 pp. 1319)
20. CJM 2017 (vol 70 pp. 451)
 Zhang, Chao

EkedahlOort strata for good reductions of Shimura varieties of Hodge type
For a Shimura variety of Hodge type with hyperspecial level
structure at a prime~$p$, Vasiu and Kisin constructed a smooth
integral model (namely the integral canonical model) uniquely
determined by a certain extension property. We define and study
the EkedahlOort stratifications on the special fibers of those
integral canonical models when $p\gt 2$. This generalizes
EkedahlOort stratifications defined and studied by Oort on moduli
spaces of principally polarized abelian varieties and those
defined and studied by Moonen, Wedhorn and Viehmann on good
reductions of Shimura varieties of PEL type. We show that the
EkedahlOort strata are parameterized by certain elements $w$ in
the Weyl group of the reductive group in the Shimura datum. We
prove that the stratum corresponding to $w$ is smooth of dimension
$l(w)$ (i.e. the length of $w$) if it is nonempty. We also
determine the closure of each stratum.
Keywords:Shimura variety, Fzip Categories:14G35, 11G18 

21. CJM 2017 (vol 70 pp. 142)
 Hajir, Farshid; Maire, Christian

On the invariant factors of class groups in towers of number fields
For a finite abelian $p$group $A$ of rank $d=\dim A/pA$, let
$\mathbb{M}_A := \log_p A^{1/d}$ be its
\emph{(logarithmic) mean exponent}. We study the behavior of
the mean exponent of $p$class groups in pro$p$ towers $\mathrm{L}/K$
of number fields. Via a combination of results from analytic
and algebraic number theory, we construct infinite tamely
ramified pro$p$ towers in which the mean exponent of $p$class
groups remains bounded. Several explicit
examples are given with $p=2$. Turning to group theory, we
introduce an invariant $\underline{\mathbb{M}}(G)$ attached to a finitely generated
pro$p$ group $G$; when $G=\operatorname{Gal}(\mathrm{L}/\mathrm{K})$, where $\mathrm{L}$ is the Hilbert
$p$class field tower of a number field $K$, $\underline{\mathbb{M}}(G)$ measures
the asymptotic behavior of the mean exponent of $p$class groups
inside $\mathrm{L}/\mathrm{K}$. We compare and contrast the behavior of this
invariant in analytic versus nonanalytic groups. We exploit
the interplay of grouptheoretical and numbertheoretical perspectives
on this invariant and explore some open questions that arise
as a result, which may be of independent interest in group theory.
Keywords:class field tower, ideal class group, prop group, padic analytic group, BrauerSiegel Theorem Categories:11R29, 11R37 

22. CJM 2017 (vol 70 pp. 117)
 Ha, Junsoo

Smooth Polynomial Solutions to a Ternary Additive Equation
Let $\mathbf{F}_{q}[T]$ be the ring of polynomials over the finite
field of $q$ elements, and $Y$ be a large integer. We say a polynomial
in $\mathbf{F}_{q}[T]$ is $Y$smooth if all of its irreducible
factors
are of degree at most $Y$. We show that a ternary additive equation
$a+b=c$ over $Y$smooth polynomials has many solutions. As an
application,
if $S$ is the set of first $s$ primes in $\mathbf{F}_{q}[T]$ and
$s$ is large, we prove that the $S$unit equation $u+v=1$ has at
least $\exp(s^{1/6\epsilon}\log q)$ solutions.
Keywords:smooth number, polynomial over a finite field, circle method Categories:11T55, 11D04, 11L07, 11T23 

23. CJM 2017 (vol 70 pp. 824)
 Hare, Kathryn; Hare, Kevin; Ng, Michael Ka Shing

Local Dimensions of Measures of Finite Type II  Measures without Full Support and with Nonregular Probabilities
Consider a finite sequence of linear contractions $S_{j}(x)=\varrho
x+d_{j}$ and
probabilities $p_{j}\gt 0$ with $\sum p_{j}=1$. We are interested
in the
selfsimilar measure $\mu =\sum p_{j}\mu \circ S_{j}^{1}$, of
finite type.
In this paper we study the multifractal analysis of such measures,
extending the theory to measures arising from nonregular probabilities
and
whose support is not necessarily an interval.
Under some mild technical assumptions, we prove that there exists
a subset
of supp$\mu $ of full $\mu $ and Hausdorff measure, called the
truly
essential class, for which the set of (upper or lower) local
dimensions is a
closed interval. Within the truly essential class we show that
there exists
a point with local dimension exactly equal to the dimension of
the support.
We give an example where the set of local dimensions is a two
element set,
with all the elements of the truly essential class giving the
same local
dimension. We give general criteria for these measures to be
absolutely
continuous with respect to the associated Hausdorff measure of
their support
and we show that the dimension of the support can be computed
using only
information about the essential class.
To conclude, we present a detailed study of three examples. First,
we show
that the set of local dimensions of the biased Bernoulli convolution
with
contraction ratio the inverse of a simple Pisot number always
admits an
isolated point. We give a precise description of the essential
class of a
generalized Cantor set of finite type, and show that the $kth$
convolution
of the associated Cantor measure has local dimension at $x\in
(0,1)$ tending
to 1 as $k$ tends to infinity. Lastly, we show that within a
maximal loop
class that is not truly essential, the set of upper local dimensions
need
not be an interval. This is in contrast to the case for finite
type measures
with regular probabilities and full interval support.
Keywords:multifractal analysis, local dimension, IFS, finite type Categories:28A80, 28A78, 11R06 

24. CJM 2017 (vol 70 pp. 481)
 Asakura, Masanori; Otsubo, Noriyuki

CM Periods, CM Regulators and Hypergeometric Functions, I
We prove the GrossDeligne conjecture on CM periods for motives
associated with $H^2$ of certain surfaces fibered over the projective
line. Then we prove for the same motives a formula which expresses
the $K_1$regulators in terms of hypergeometric functions ${}_3F_2$,
and obtain a new example of nontrivial regulators.
Keywords:period, regulator, complex multiplication, hypergeometric function Categories:14D07, 19F27, 33C20, 11G15, 14K22 

25. CJM 2017 (vol 70 pp. 241)
 Böcherer, Siegfried; Kikuta, Toshiyuki; Takemori, Sho

Weights of the mod $p$ kernel of the theta operators
Let $\Theta ^{[j]}$ be an analogue of the Ramanujan theta operator
for Siegel modular forms.
For a given prime $p$, we give the weights of elements of mod
$p$ kernel of $\Theta ^{[j]}$,
where the mod $p$ kernel of $\Theta ^{[j]}$ is the set of all
Siegel modular forms $F$ such that $\Theta ^{[j]}(F)$ is congruent
to zero modulo $p$.
In order to construct examples of the mod $p$ kernel of $\Theta
^{[j]}$ from any Siegel modular form,
we introduce new operators $A^{(j)}(M)$ and show the modularity
of $FA^{(j)}(M)$ when $F$ is a Siegel modular form.
Finally, we give some examples of the mod $p$ kernel of $\Theta
^{[j]}$ and the filtrations of some of them.
Keywords:Siegel modular form, congruences for modular forms, Fourier coefficients, Ramanujan's operator, filtration Categories:11F33, 11F46 
