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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 non-isomorphic 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

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:L-function, 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 non-reflexive 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)

Sun, Chia-Liang
Weak Approximation for Points with Coordinates in Rank-one Subgroups of Global Function Fields
For every affine variety over a global function field, we show that the set of its points with coordinates in an arbitrary rank-one multiplicative subgroup of this function field satisfies the required property of weak approximation for finite sets of places of this function field avoiding arbitrarily given finitely many places.

Keywords:weak approximation, global function fields, local-global criteria
Categories:14G05, 11R58

7. CMB Online first

Nguyen, Khoa Dang
The Hermite-Joubert Problem and a Conjecture of Brassil-Reichstein
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 Hermite-Joubert problem over a finitely generated field of characteristic $0$.

Keywords:Hermite-Joubert problem, Brassil-Reichstein conjecture, diophantine equation
Categories:11D72, 11G05

8. CMB 2018 (vol 61 pp. 822)

Pollack, Aaron; Shah, Shrenik
Multivariate Rankin-Selberg Integrals on $GL_4$ and $GU(2,2)$
Inspired by a construction by Bump, Friedberg, and Ginzburg of a two-variable 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 three-variable Rankin-Selberg 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 two-variable Rankin-Selberg 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, L-function, Rankin-Selberg 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 Böttcher's uniformization of polynomial dynamical systems over $\mathbb{CC}$, although the construction of a Böttcher 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 Nyman-Beurling-Báez-Duarte criterion for the Riemann Hypothesis
A crucial role in the Nyman-Beurling-Báez-Duarte approach to the Riemann Hypothesis is played by the distance \[ d_N^2:=\inf_{A_N}\frac{1}{2\pi}\int_{-\infty}^\infty \left|1-\zeta A_N \left(\frac{1}{2}+it \right) \right|^2\frac{dt}{\frac{1}{4}+t^2}\:, \] 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 non-trivial zeros off the critical line.

Keywords:Riemann hypothesis, Riemann zeta function, Nyman-Beurling-Báez-Duarte criterion
Categories:30C15, 11M26

12. CMB 2017 (vol 61 pp. 572)

Koskivirta, Jean-Stefan
Normalization of closed Ekedahl-Oort strata
We apply our theory of partial flag spaces developed with W. Goldring to study a group-theoretical generalization of the canonical filtration of a truncated Barsotti-Tate group of level 1. As an application, we determine explicitly the normalization of the Zariski closures of Ekedahl-Oort strata of Shimura varieties of Hodge-type as certain closed coarse strata in the associated partial flag spaces.

Keywords:Ekedahl-Oort stratification, Shimura variety
Categories:14K10, 20G40, 11G18

13. CMB 2017 (vol 61 pp. 608)

Loeffler, David
A note on $p$-adic Rankin-Selberg $L$-functions
We prove an interpolation formula for the values of certain $p$-adic Rankin-Selberg $L$-functions associated to non-ordinary modular forms.

Keywords:$p$-adic $L$-function, Iwasawa theory
Categories:11F85, 11F67, 11G40, 14G35

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; Al-Shbeil, 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 half-plane 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 Atkin-Lehner 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 Atkin-Lehner 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 Erdös regarding the non-vanishing 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 Erdös'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 Erdös's conjecture for prime $q$.

Keywords:non-vanishing of L-series, linear independence of logarithms of algebraic numbers
Categories:11J86, 11J72

19. CMB 2016 (vol 59 pp. 592)

Liu, H. Q.
The Dirichlet Divisor Problem of Arithmetic Progressions
We design an elementary method to study the problem, getting an asymptotic formula which is better than Hooley's and Heath-Brown's results for certain cases.

Keywords:Dirichlet divisor problem, arithmetic progression
Categories:11L07, 11B83

20. CMB 2016 (vol 59 pp. 528)

Jahan, Qaiser
Characterization of Low-pass Filters on Local Fields of Positive Characteristic
In this article, we give necessary and sufficient conditions on a function to be a low-pass 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, low-pass 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 non-zero 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), 785-794) 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, Waring-Goldbach 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, Ihara-Anderson 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 $p|n$, we have $p-a|n-a$. This can be seen as a generalization of Carmichael numbers, which are integers $n$ such that $p-1|n-1$ for every $p|n$.

Keywords:Carmichael number, pseudoprime, Korselt's Criterion, primes in arithmetic progressions

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 square-free 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, non-squares 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 Pólya-Vinogradov 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
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