1. CMB Online first
 Gu, Miao; Martin, Gregory George

Factorization tests and algorithms arising from counting modular forms and automorphic representations
A theorem of Gekeler compares the number of nonisomorphic automorphic
representations associated with the space of cusp forms of weight
$k$ on $\Gamma_0(N)$ to a simpler function of $k$ and $N$, showing
that the two are equal whenever $N$ is squarefree. We prove the
converse of this theorem (with one small exception), thus providing
a characterization of squarefree integers. We also establish
a similar characterization of prime numbers in terms of the number
of Hecke newforms of weight $k$ on $\Gamma_0(N)$.
It follows that a hypothetical fast algorithm for computing the
number of such automorphic representations for even a single
weight $k$ would yield a fast test for whether $N$ is squarefree.
We also show how to obtain bounds on the possible square divisors
of a number $N$ that has been found to not be squarefree via
this test, and we show how to probabilistically obtain
the complete factorization of the squarefull part of $N$ from
the number of such automorphic representations for two different
weights. If in addition we have the number of such Hecke newforms
for even a single weight $k$, then we show how to probabilistically
factor $N$ entirely.
All of these computations could be performed quickly in practice,
given the number(s) of automorphic representations and modular
forms as input.
Keywords:modular form, automorphic representation, squarefree number, primality testing, factorization algorithm Categories:11F70, 11N25, 11N60, 11Y05, 11Y16 

2. CMB Online first
 Cook, Brian

Discrete multilinear spherical averages
In this note we give a characterization of $\ell^{p}\times \cdots\times
\ell^{p}\to\ell^q$ boundedness of maximal operators associated
to multilinear convolution averages over spheres in $\mathbb{Z}^n$.
Keywords:discrete maximal function, multilinear average Categories:11L07, 42B25 

3. CMB Online first
 Porritt, Sam

Irreducible polynomials over a finite field with restricted coefficients
We prove a function field analogue of Maynard's celebrated result
about primes with restricted digits. That is, for certain ranges
of parameters $n$ and $q$, we prove an asymptotic formula for
the number of irreducible polynomials of degree $n$ over a finite
field $\mathbb{F}_q$ whose coefficients are restricted to lie
in a given subset of $\mathbb{F}_q$
Keywords:finite field, irreducible polynomial, restricted coefficients Category:11T55 

4. CMB Online first
 Meisner, Patrick

One Level Density for Cubic Galois Number Fields
Katz and Sarnak predicted that the one level density of the zeros
of a family of $L$functions would fall into one of five categories.
In this paper, we show that the one level density for $L$functions
attached to cubic Galois number fields falls into the category
associated with unitary matrices.
Keywords:Lfunction, one level density Categories:11M06, 11M26, 11M50 

5. CMB Online first
 Asgarli, Shamil

Sharp Bertini theorem for plane curves over finite fields
We prove that if $C$ is a reflexive smooth plane curve of degree
$d$ defined over a finite field $\mathbb{F}_q$ with $d\leq q+1$, then
there is an $\mathbb{F}_q$line $L$ that intersects $C$ transversely.
We also prove the same result for nonreflexive curves of degree
$p+1$ and $2p+1$ where $q=p^{r}$.
Keywords:Bertini theorem, transversality, finite field Categories:14H50, 11G20, 14N05 

6. CMB 2018 (vol 61 pp. 878)
7. CMB Online first
 Nguyen, Khoa Dang

The HermiteJoubert Problem and a Conjecture of BrassilReichstein
show that Hermite's theorem fails for every
integer $n$ of the form $3^{k_1}+3^{k_2}+3^{k_3}$
with integers $k_1\gt k_2\gt k_3\geq 0$. This confirms
a conjecture of Brassil and Reichstein. We also
obtain new results for the relative
HermiteJoubert problem over a finitely generated
field of characteristic $0$.
Keywords:HermiteJoubert problem, BrassilReichstein conjecture, diophantine equation Categories:11D72, 11G05 

8. CMB 2018 (vol 61 pp. 822)
 Pollack, Aaron; Shah, Shrenik

Multivariate RankinSelberg Integrals on $GL_4$ and $GU(2,2)$
Inspired by a construction by Bump, Friedberg, and Ginzburg of
a twovariable integral representation on $\operatorname{GSp}_4$ for the product
of the standard and spin $L$functions, we give two similar multivariate
integral representations. The first is a threevariable RankinSelberg
integral for cusp forms on $\operatorname{PGL}_4$ representing the product
of the $L$functions attached to the three fundamental representations
of the Langlands $L$group $\operatorname{SL}_4(\mathbf{C})$. The second integral,
which is closely related, is a twovariable RankinSelberg integral
for cusp forms on $\operatorname{PGU}(2,2)$ representing the product of the
degree 8 standard $L$function and the degree 6 exterior square
$L$function.
Keywords:automorphic form, Lfunction, RankinSelberg method, unitary group, exterior square, Langlands program Categories:11F66, 11F55 

9. CMB 2018 (vol 61 pp. 531)
 Ingram, Patrick

$p$adic uniformization and the action of Galois on certain affine correspondences
Given two monic polynomials $f$ and $g$ with coefficients in
a number field $K$, and some $\alpha\in K$, we examine the action
of the absolute Galois group $\operatorname{Gal}(\overline{K}/K)$ on the directed
graph of iterated preimages of $\alpha$ under the correspondence
$g(y)=f(x)$, assuming that $\deg(f)\gt \deg(g)$ and that $\gcd(\deg(f),
\deg(g))=1$. If a prime of $K$ exists at which $f$ and $g$ have
integral coefficients, and at which $\alpha$ is not integral,
we show that this directed graph of preimages consists of finitely
many $\operatorname{Gal}(\overline{K}/K)$orbits. We obtain this result by
establishing a $p$adic uniformization of such correspondences,
tenuously related to BÃ¶ttcher's uniformization of polynomial
dynamical systems over $\mathbb{CC}$, although the construction of a
BÃ¶ttcher coordinate for complex holomorphic correspondences
remains unresolved.
Keywords:arithmetic dynamics Categories:37P20, 11S20 

10. CMB Online first
 Lee, Hao

Irregular Weight one points with $D_{4}$ Image
Darmon, Lauder and Rotger conjectured that the relative tangent space of the eigencurve at a classical, ordinary, irregular weight one point is of dimension two. This space can be identified with the space of normalized overconvergent generalized eigenforms, whose Fourier coefficients can be conjecturally described explicitly in terms of $p$adic logarithms of algebraic numbers. This article presents the proof of this conjecture in the case where the weight one point is the intersection of two Hida families of Hecke theta series.
Keywords:weight one points, irregular, dihedral image, generalized eigenform, eigencurve, tangent space, Categories:11F33, 11F80 

11. CMB 2018 (vol 61 pp. 622)
 Maier, Helmut; Rassias, Michael Th.

On the size of an expression in the NymanBeurlingBÃ¡ezDuarte criterion for the Riemann Hypothesis
A crucial role in the NymanBeurlingBÃ¡ezDuarte approach to
the Riemann Hypothesis is played by the distance
\[
d_N^2:=\inf_{A_N}\frac{1}{2\pi}\int_{\infty}^\infty
\left1\zeta A_N
\left(\frac{1}{2}+it
\right)
\right^2\frac{dt}{\frac{1}{4}+t^2}\:,
\]
where the infimum is over all Dirichlet polynomials
$$A_N(s)=\sum_{n=1}^{N}\frac{a_n}{n^s}$$
of length $N$.
In this paper we investigate $d_N^2$ under the assumption that
the Riemann zeta function has four nontrivial zeros off the
critical line.
Keywords:Riemann hypothesis, Riemann zeta function, NymanBeurlingBÃ¡ezDuarte criterion Categories:30C15, 11M26 

12. CMB 2017 (vol 61 pp. 572)
 Koskivirta, JeanStefan

Normalization of closed EkedahlOort strata
We apply our theory of partial flag spaces developed
with W. Goldring
to study a grouptheoretical generalization of the canonical
filtration of a truncated BarsottiTate group of level 1. As
an application, we determine explicitly the normalization of
the Zariski closures of EkedahlOort strata of Shimura varieties
of Hodgetype as certain closed coarse strata in the associated
partial flag spaces.
Keywords:EkedahlOort stratification, Shimura variety Categories:14K10, 20G40, 11G18 

13. CMB 2017 (vol 61 pp. 608)
14. CMB 2017 (vol 60 pp. 673)
 Abtahi, Fatemeh; Azizi, Mohsen; Rejali, Ali

Character Amenability of the Intersection of Lipschitz Algebras
Let $(X,d)$ be a metric space and $J\subseteq [0,\infty)$ be
nonempty. We study the structure of the arbitrary intersections
of
Lipschitz algebras, and define a special Banach subalgebra of
$\bigcap_{\gamma\in J}\operatorname{Lip}_\gamma X$, denoted by
$\operatorname{ILip}_J X$. Mainly,
we investigate $C$character amenability of $\operatorname{ILip}_J X$, in
particular Lipschitz algebras. We address a gap in the proof
of a
recent result in this field. Then we remove this gap, and obtain
a
necessary and sufficient condition for $C$character amenability
of $\operatorname{ILip}_J X$, specially Lipschitz algebras, under an additional
assumption.
Keywords:amenability, character amenability, Lipschitz algebra, metric space Categories:46H05, 46J10, 11J83 

15. CMB 2017 (vol 61 pp. 376)
 Sebbar, Abdellah; AlShbeil, Isra

Elliptic Zeta Functions and Equivariant Functions
In this paper we establish a close connection between three
notions attached to a modular subgroup. Namely the set of weight
two meromorphic modular forms, the set of equivariant functions
on the upper halfplane commuting with the action of the modular
subgroup and the set of elliptic zeta functions generalizing
the Weierstrass zeta functions. In particular, we show that the
equivariant functions can be parameterized by modular objects
as well as by elliptic objects.
Keywords:modular form, equivariant function, elliptic zeta function Categories:11F12, 35Q15, 32L10 

16. CMB 2017 (vol 60 pp. 329)
 Le Fourn, Samuel

Nonvanishing of Central Values of $L$functions of Newforms in $S_2 (\Gamma_0 (dp^2))$ Twisted by Quadratic Characters
We prove that for $d \in \{ 2,3,5,7,13 \}$ and $K$ a quadratic
(or rational) field of discriminant $D$ and Dirichlet character
$\chi$, if a prime $p$ is large enough compared to $D$, there
is a newform $f \in S_2(\Gamma_0(dp^2))$ with sign $(+1)$ with
respect to the AtkinLehner involution $w_{p^2}$ such that $L(f
\otimes \chi,1) \neq 0$. This result is obtained through an estimate
of a weighted sum of twists of $L$functions which generalises
a result of Ellenberg. It relies on the approximate functional
equation for the $L$functions $L(f \otimes \chi, \cdot)$ and
a Petersson trace formula restricted to AtkinLehner eigenspaces.
An application of this nonvanishing theorem will be given in
terms of existence of rank zero quotients of some twisted jacobians,
which generalises a result of Darmon and Merel.
Keywords:nonvanishing of $L$functions of modular forms, Petersson trace formula, rank zero quotients of jacobians Categories:14J15, 11F67 

17. CMB 2016 (vol 60 pp. 484)
 Dobrowolski, Edward

A Note on Lawton's Theorem
We prove Lawton's conjecture about the upper bound on the measure
of the set on the unit circle on which a complex polynomial with
a bounded number of coefficients takes small values. Namely,
we prove that Lawton's bound holds for polynomials that are not
necessarily monic. We also provide an analogous bound for polynomials
in several variables. Finally, we investigate the dependence
of the bound on the multiplicity of zeros for polynomials in
one variable.
Keywords:polynomial, Mahler measure Categories:11R09, 11R06 

18. CMB 2016 (vol 60 pp. 184)
 Pathak, Siddhi

On a Conjecture of Livingston
In an attempt to resolve a folklore conjecture of ErdÃ¶s regarding
the nonvanishing at $s=1$ of the $L$series
attached to a periodic arithmetical function with period $q$
and values in $\{ 1, 1\} $, Livingston conjectured the $\bar{\mathbb{Q}}$
 linear independence of logarithms of certain algebraic numbers.
In this paper, we disprove Livingston's conjecture for composite
$q \geq 4$, highlighting that a new approach is required to settle
ErdÃ¶s's conjecture. We also prove that the conjecture is
true for prime $q \geq 3$, and indicate that more ingredients
will be needed to settle ErdÃ¶s's conjecture for prime $q$.
Keywords:nonvanishing of Lseries, linear independence of logarithms of algebraic numbers Categories:11J86, 11J72 

19. CMB 2016 (vol 59 pp. 592)
20. CMB 2016 (vol 59 pp. 528)
 Jahan, Qaiser

Characterization of Lowpass Filters on Local Fields of Positive Characteristic
In this article, we give necessary and sufficient conditions
on a function to be a lowpass filter on a local field $K$ of
positive characteristic associated to the scaling function for
multiresolution analysis of $L^2(K)$. We use probability and
martingale methods to provide such a characterization.
Keywords:multiresolution analysis, local field, lowpass filter, scaling function, probability, conditional probability and martingales Categories:42C40, 42C15, 43A70, 11S85 

21. CMB 2016 (vol 59 pp. 599)
 Liu, Zhixin

Small Prime Solutions to Cubic Diophantine Equations II
Let $a_1, \cdots, a_9$ be nonzero integers and $n$ any integer.
Suppose
that $a_1+\cdots+a_9 \equiv n( \textrm{mod}\,2)$ and $(a_i, a_j)=1$
for $1 \leq i \lt j \leq 9$.
In this paper we prove that
(i) if $a_j$ are not all of the same sign, then the cubic
equation $a_1p_1^3+\cdots +a_9p_9^3=n$ has prime solutions satisfying
$p_j \ll n^{1/3}+\textrm{max}\{a_j\}^{8+\varepsilon};$
(ii) if all $a_j$ are positive and $n \gg \textrm{max}\{a_j\}^{25+\varepsilon}$,
then
$a_1p_1^3+\cdots +a_9p_9^3=n$ is soluble in primes $p_j$.
This results improve our previous results (Canad. Math. Bull.,
56 (2013), 785794)
with the bounds $\textrm{max}\{a_j\}^{14+\varepsilon}$ and
$\textrm{max}\{a_j\}^{43+\varepsilon}$
in place of $\textrm{max}\{a_j\}^{8+\varepsilon}$ and $\textrm{max}\{a_j\}^{25+\varepsilon}$
above, respectively.
Keywords:small prime, WaringGoldbach problem, circle method Categories:11P32, 11P05, 11P55 

22. CMB 2016 (vol 59 pp. 624)
 Otsubo, Noriyuki

Homology of the Fermat Tower and Universal Measures for Jacobi Sums
We give a precise description of the homology group of the Fermat
curve as a cyclic module over a group ring.
As an application, we prove the freeness of the profinite homology
of the Fermat tower.
This allows us to define measures, an equivalent of Anderson's
adelic beta functions,
in a similar manner to Ihara's definition of $\ell$adic universal
power series for Jacobi sums.
We give a simple proof of the interpolation property using a
motivic decomposition of the Fermat curve.
Keywords:Fermat curves, IharaAnderson theory, Jacobi sums Categories:11S80, 11G15, 11R18 

23. CMB 2015 (vol 58 pp. 869)
 Wright, Thomas

Variants of Korselt's Criterion
Under sufficiently strong assumptions about the first term in
an arithmetic progression, we prove that for any integer $a$,
there are infinitely many $n\in \mathbb N$ such that for each
prime factor $pn$, we have $pana$. This can be seen as a
generalization of Carmichael numbers, which are integers $n$
such that $p1n1$ for every $pn$.
Keywords:Carmichael number, pseudoprime, Korselt's Criterion, primes in arithmetic progressions Category:11A51 

24. CMB 2015 (vol 58 pp. 704)
 Benamar, H.; Chandoul, A.; Mkaouar, M.

On the Continued Fraction Expansion of Fixed Period in Finite Fields
The Chowla conjecture
states that,
if $t$ is any given
positive integer, there are infinitely many prime positive
integers $N$ such that $\operatorname{Per} (\sqrt{N})=t$, where
$\operatorname{Per} (\sqrt{N})$
is the period length of the continued fraction expansion for
$\sqrt{N}$.
C. Friesen proved
that, for any $k\in \mathbb{N}$, there are infinitely many
squarefree integers $N$, where the continued fraction expansion
of $\sqrt{N}$ has a fixed period. In this paper, we describe all
polynomials $Q\in \mathbb{F}_q[X] $ for which the continued fraction
expansion of $\sqrt {Q}$ has a fixed period, also we give a
lower
bound of the number of monic, nonsquares polynomials $Q$ such
that $\deg Q= 2d$ and $ Per \sqrt {Q}=t$.
Keywords:continued fractions, polynomials, formal power series Categories:11A55, 13J05 

25. CMB 2015 (vol 58 pp. 774)
 Hanson, Brandon

Character Sums over Bohr Sets
We prove character sum estimates for additive Bohr subsets modulo
a prime.
These estimates are analogous to classical character sum bounds
of
PÃ³lyaVinogradov and Burgess. These estimates are applied to
obtain results on
recurrence mod $p$ by special elements.
Keywords:character sums, Bohr sets, finite fields Categories:11L40, 11T24, 11T23 
