1. CJM Online first
 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 

2. CJM Online first
 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 

3. 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 

4. CJM Online first
 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 

5. CJM Online first
 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 

6. CJM Online first
 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 

7. CJM Online first
 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 

8. CJM Online first
 Martin, Kimball

Congruences for modular forms mod 2 and quaternionic $S$ideal classes
We prove many simultaneous congruences mod 2 for elliptic and
Hilbert modular forms
among forms with different AtkinLehner eigenvalues. The proofs
involve the notion of quaternionic $S$ideal classes and the
distribution of AtkinLehner signs among
newforms.
Keywords:modular forms, congruences, quaternion algebras Categories:11F33, 11R52 

9. CJM Online first
 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 

10. CJM Online first
 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 

11. CJM Online first
 Bijakowski, Stephane

Partial Hasse invariants, partial degrees, and the canonical subgroup
If the Hasse invariant of a $p$divisible group is small enough,
then one can construct a canonical subgroup inside its $p$torsion.
We prove that, assuming the existence of a subgroup of adequate
height in the $p$torsion with high degree, the expected properties
of the canonical subgroup can be easily proved, especially the
relation between its degree and the Hasse invariant. When one
considers a $p$divisible group with an action of the ring of
integers of a (possibly ramified) finite extension of $\mathbb{Q}_p$,
then much more can be said. We define partial Hasse invariants
(they are natural in the unramified case, and generalize a construction
of Reduzzi and Xiao in the general case), as well as partial
degrees. After studying these functions, we compute the partial
degrees of the canonical subgroup.
Keywords:canonical subgroup, Hasse invariant, $p$divisible group Categories:11F85, 11F46, 11S15 

12. CJM Online first
 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 

13. CJM 2016 (vol 69 pp. 1169)
 Varma, Sandeep

On Residues of Intertwining Operators in Cases with Prehomogeneous Nilradical
Let $\operatorname{P} = \operatorname{M} \operatorname{N}$ be a Levi decomposition of a maximal parabolic
subgroup of a connected
reductive group $\operatorname{G}$ over a $p$adic field $F$. Assume that there
exists $w_0 \in \operatorname{G}(F)$ that normalizes $\operatorname{M}$ and conjugates $\operatorname{P}$
to an opposite parabolic subgroup.
When $\operatorname{N}$ has a Zariski dense $\operatorname{Int} \operatorname{M}$orbit,
F. Shahidi and X. Yu describe a certain distribution $D$ on
$\operatorname{M}(F)$
such that,
for irreducible unitary supercuspidal representations $\pi$ of
$\operatorname{M}(F)$ with
$\pi \cong \pi \circ \operatorname{Int} w_0$,
$\operatorname{Ind}_{\operatorname{P}(F)}^{\operatorname{G}(F)} \pi$ is
irreducible
if and only if $D(f) \neq 0$ for some pseudocoefficient $f$ of
$\pi$. Since
this irreducibility is conjecturally related to $\pi$ arising
via
transfer from certain twisted endoscopic groups of $\operatorname{M}$, it is
of interest
to realize $D$ as endoscopic transfer from a simpler distribution
on a twisted
endoscopic group $\operatorname{H}$ of $\operatorname{M}$. This has been done in many situations
where $\operatorname{N}$ is abelian. Here, we handle the `standard examples'
in cases
where $\operatorname{N}$ is nonabelian but admits a Zariski dense
$\operatorname{Int} \operatorname{M}$orbit.
Keywords:induced representation, intertwining operator, endoscopy Categories:22E50, 11F70 

14. CJM 2016 (vol 69 pp. 186)
 Pan, ShuYen

$L$Functoriality for Local Theta Correspondence of Supercuspidal Representations with Unipotent Reduction
The preservation principle of local theta correspondences of reductive dual pairs over
a $p$adic field predicts the existence of a sequence of irreducible supercuspidal
representations of classical groups.
Adams/HarrisKudlaSweet
have a conjecture
about the Langlands parameters for the sequence of supercuspidal representations.
In this paper we prove modified versions of their conjectures for the case of
supercuspidal representations with unipotent reduction.
Keywords:local theta correspondence, supercuspidal representation, preservation principle, Langlands functoriality Categories:22E50, 11F27, 20C33 

15. CJM 2016 (vol 69 pp. 826)
 Lei, Antonio; Loeffler, David; Zerbes, Sarah Livia

On the Asymptotic Growth of BlochKatoShafarevichTate Groups of Modular Forms over Cyclotomic Extensions
We study the asymptotic behaviour of the BlochKatoShafarevichTate
group of a modular form $f$ over the cyclotomic $\mathbb{Z}_p$extension
of $\mathbb{Q}$ under the assumption that $f$ is nonordinary at $p$.
In particular, we give upper bounds of these groups in terms
of Iwasawa invariants of Selmer groups defined using $p$adic
Hodge Theory. These bounds have the same form as the formulae
of Kobayashi, Kurihara and Sprung for supersingular elliptic
curves.
Keywords:cyclotomic extension, ShafarevichTate group, BlochKato Selmer group, modular form, nonordinary prime, padic Hodge theory Categories:11R18, 11F11, 11R23, 11F85 

16. CJM 2016 (vol 69 pp. 579)
 Lee, Jungyun; Lee, Yoonjin

Regulators of an Infinite Family of the Simplest Quartic Function Fields
We explicitly find regulators of an infinite family $\{L_m\}$
of the simplest quartic function fields
with a parameter $m$ in a polynomial ring $\mathbb{F}_q [t]$, where
$\mathbb{F}_q$
is the finite field of order $q$
with odd characteristic. In fact, this infinite family of the
simplest quartic function fields are
subfields of maximal real subfields of cyclotomic function fields,
where they have the same conductors.
We obtain a lower bound on the class numbers of the family $\{L_m\}$
and some result on the divisibility
of the divisor class numbers of cyclotomic function fields which
contain $\{L_m\}$ as their subfields.
Furthermore, we find an explicit criterion for the characterization
of splitting types of all the primes
of the rational function field $\mathbb{F}_q (t)$ in $\{L_m\}$.
Keywords:regulator, function field, quartic extension, class number Categories:11R29, 11R58 

17. CJM 2016 (vol 69 pp. 890)
18. CJM 2016 (vol 68 pp. 1382)
 Zydor, Michał

La Variante infinitÃ©simale de la formule des traces de JacquetRallis pour les groupes unitaires
We establish an infinitesimal version of the
JacquetRallis trace formula for unitary groups.
Our formula is obtained by integrating a
truncated kernel Ã la Arthur.
It has a geometric side which is a
sum of distributions $J_{\mathfrak{o}}$ indexed by classes of
elements
of the Lie algebra of $U(n+1)$ stable by $U(n)$conjugation
as well as the "spectral side"
consisting of the Fourier transforms
of the aforementioned distributions.
We prove that the distributions $J_{\mathfrak{o}}$
are invariant and depend only on the choice of
the Haar measure on $U(n)(\mathbb{A})$.
For regular semisimple classes $\mathfrak{o}$, $J_{\mathfrak{o}}$
is
a relative orbital integral of JacquetRallis.
For classes $\mathfrak{o}$ called relatively regular semisimple,
we express $J_{\mathfrak{o}}$
in terms of relative orbital integrals regularised by means of
zÃªta functions.
Keywords:formule des traces relative Categories:11F70, 11F72 

19. CJM 2016 (vol 68 pp. 1362)
 Papikian, Mihran; Rabinoff, Joseph

Optimal Quotients of Jacobians with Toric Reduction and Component Groups
Let $J$ be a Jacobian variety with toric reduction
over a local field $K$.
Let $J \to E$ be an optimal quotient defined over $K$, where
$E$ is an elliptic curve.
We give examples in which the functorially induced map $\Phi_J
\to \Phi_E$
on component groups of the NÃ©ron models is not surjective.
This answers a question of Ribet and Takahashi.
We also give various criteria under which $\Phi_J \to \Phi_E$
is surjective, and discuss
when these criteria hold for the Jacobians of modular curves.
Keywords:Jacobians with toric reduction, component groups, modular curves Categories:11G18, 14G22, 14G20 

20. CJM 2016 (vol 69 pp. 532)
 Ganguly, Arijit; Ghosh, Anish

Dirichlet's Theorem in Function Fields
We study metric Diophantine approximation for function fields
specifically the problem of improving Dirichlet's theorem in
Diophantine
approximation.
Keywords:Dirichlet's theorem, Diophantine approximation, positive characteristic Categories:11J83, 11K60, 37D40, 37A17, 22E40 

21. CJM 2016 (vol 68 pp. 1120)
 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 

22. CJM 2016 (vol 69 pp. 807)
 Günther, Christian; Schmidt, KaiUwe

$L^q$ Norms of Fekete and Related Polynomials
A Littlewood polynomial is a polynomial in $\mathbb{C}[z]$ having all
of its coefficients in $\{1,1\}$. There are various old unsolved
problems, mostly due to Littlewood and ErdÅs, that ask for
Littlewood polynomials that provide a good approximation to a
function that is constant on the complex unit circle, and in
particular have small $L^q$ norm on the complex unit circle.
We consider the Fekete polynomials
\[
f_p(z)=\sum_{j=1}^{p1}(j\,\,p)\,z^j,
\]
where $p$ is an odd prime and $(\,\cdot\,\,p)$ is the Legendre
symbol (so that $z^{1}f_p(z)$ is a Littlewood polynomial). We
give explicit and recursive formulas for the limit of the ratio
of $L^q$ and $L^2$ norm of $f_p$ when $q$ is an even positive
integer and $p\to\infty$. To our knowledge, these are the first
results that give these limiting values for specific sequences
of nontrivial Littlewood polynomials and infinitely many $q$.
Similar results are given for polynomials obtained by cyclically
permuting the coefficients of Fekete polynomials and for Littlewood
polynomials whose coefficients are obtained from additive characters
of finite fields. These results vastly generalise earlier results
on the $L^4$ norm of these polynomials.
Keywords:character polynomial, Fekete polynomial, $L^q$ norm, Littlewood polynomial Categories:11B83, 42A05, 30C10 

23. CJM 2016 (vol 69 pp. 258)
 Brandes, Julia; Parsell, Scott T.

Simultaneous Additive Equations: Repeated and Differing Degrees
We obtain bounds for the number of variables required to establish
Hasse principles, both for existence of solutions and for asymptotic
formulÃ¦, for systems of additive equations containing forms
of differing degree but also multiple forms of like degree.
Apart from the very general estimates of Schmidt and BrowningHeathBrown,
which give weak results when specialized to the diagonal situation,
this is the first result on such "hybrid" systems. We also obtain
specialised results for systems of quadratic and cubic forms,
where we are able to take advantage of some of the stronger methods
available in that setting. In particular, we achieve essentially
square root cancellation for systems consisting of one cubic
and $r$ quadratic equations.
Keywords:equations in many variables, counting solutions of Diophantine equations, applications of the HardyLittlewood method Categories:11D72, 11D45, 11P55 

24. CJM 2016 (vol 68 pp. 961)
25. CJM 2016 (vol 69 pp. 1143)
 Sikirić, Mathieu Dutour

The seven Dimensional Perfect Delaunay Polytopes and Delaunay Simplices
For a lattice $L$ of $\mathbb{RR}^n$, a sphere $S(c,r)$ of center $c$
and radius $r$
is called empty if for any $v\in L$ we have $\Vert v 
c\Vert \geq r$.
Then the set $S(c,r)\cap L$ is the vertex set of a {\em Delaunay
polytope}
$P=\operatorname{conv}(S(c,r)\cap L)$.
A Delaunay polytope is called {\em perfect} if any affine transformation
$\phi$ such that $\phi(P)$ is a Delaunay polytope is necessarily
an isometry
of the space composed with an homothety.
Perfect Delaunay polytopes are remarkable structure that exist
only
if $n=1$ or $n\geq 6$ and they have shown up recently in covering
maxima studies.
Here we give a general algorithm for their enumeration that relies
on
the Erdahl cone.
We apply this algorithm in dimension $7$ which allow us to find
that there are only two perfect Delaunay polytopes: $3_{21}$
which
is a Delaunay polytope in the root lattice $\mathsf{E}_7$ and
the
Erdahl Rybnikov polytope.
We then use this classification in order to get the list of all
types
Delaunay simplices in dimension $7$ and found $11$ types.
Keywords:Delaunay polytope, enumeration, polyhedral methods Categories:11H06, 11H31 
