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1. CMB Online first

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 criteriaCategories:14G05, 11R58

2. 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

3. 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 equationCategories:11D72, 11G05

4. CMB Online first

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 programCategories:11F66, 11F55

5. 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 densityCategories:11M06, 11M26, 11M50

6. CMB Online first

Maier, Helmut; Rassias, Michael Th.
 On the size of an expression in the Nyman-Beurling-BÃ¡ez-Duarte criterion for the Riemann Hypothesis A crucial role in the Nyman-Beurling-BÃ¡ez-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-BÃ¡ez-Duarte criterionCategories:30C15, 11M26

7. CMB Online first

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 dynamicsCategories:37P20, 11S20

8. CMB Online first

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 varietyCategories:14K10, 20G40, 11G18

9. CMB Online first

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 theoryCategories:11F85, 11F67, 11G40, 14G35

10. 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 spaceCategories:46H05, 46J10, 11J83

11. CMB Online first

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 functionCategories:11F12, 35Q15, 32L10

12. 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 jacobiansCategories:14J15, 11F67

13. 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 measureCategories:11R09, 11R06

14. 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 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 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:non-vanishing of L-series, linear independence of logarithms of algebraic numbersCategories:11J86, 11J72

15. 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 progressionCategories:11L07, 11B83

16. 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 martingalesCategories:42C40, 42C15, 43A70, 11S85

17. 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 methodCategories:11P32, 11P05, 11P55

18. 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 sumsCategories:11S80, 11G15, 11R18

19. 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 progressionsCategory:11A51

20. 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 seriesCategories:11A55, 13J05

21. 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 PÃ³lya-Vinogradov and Burgess. These estimates are applied to obtain results on recurrence mod $p$ by special elements. Keywords:character sums, Bohr sets, finite fieldsCategories:11L40, 11T24, 11T23

22. CMB 2015 (vol 58 pp. 858)

Williams, Kenneth S.
 Ternary Quadratic Forms and Eta Quotients Let $\eta(z)$ $(z \in \mathbb{C},\;\operatorname{Im}(z)\gt 0)$ denote the Dedekind eta function. We use a recent product-to-sum formula in conjunction with conditions for the non-representability of integers by certain ternary quadratic forms to give explicitly 10 eta quotients $f(z):=\eta^{a(m_1)}(m_1 z)\cdots \eta^{{a(m_r)}}(m_r z)=\sum_{n=1}^{\infty}c(n)e^{2\pi i nz},\quad z \in \mathbb{C},\;\operatorname{Im}(z)\gt 0,$ such that the Fourier coefficients $c(n)$ vanish for all positive integers $n$ in each of infinitely many non-overlapping arithmetic progressions. For example, it is shown that for $f(z)=\eta^4(z)\eta^{9}(4z)\eta^{-2}(8z)$ we have $c(n)=0$ for all $n$ in each of the arithmetic progressions $\{16k+14\}_{k \geq 0}$, $\{64k+56\}_{k \geq 0}$, $\{256k+224\}_{k \geq 0}$, $\{1024k+896\}_{k \geq 0}$, $\ldots$. Keywords:Dedekind eta function, eta quotient, ternary quadratic forms, vanishing of Fourier coefficients, product-to-sum formulaCategories:11F20, 11E20, 11E25

23. CMB 2015 (vol 59 pp. 119)

Hu, Pei-Chu; Li, Bao Qin
 A Simple Proof and Strengthening of a Uniqueness Theorem for L-functions We give a simple proof and strengthening of a uniqueness theorem for functions in the extended Selberg class. Keywords:meromorphic function, Dirichlet series, L-function, zero, order, uniquenessCategories:30B50, 11M41

24. CMB 2015 (vol 58 pp. 548)

Lü, Guangshi; Sankaranarayanan, Ayyadurai
 Higher Moments of Fourier Coefficients of Cusp Forms Let $S_{k}(\Gamma)$ be the space of holomorphic cusp forms of even integral weight $k$ for the full modular group $SL(2, \mathbb{Z})$. Let $\lambda_f(n)$, $\lambda_g(n)$, $\lambda_h(n)$ be the $n$th normalized Fourier coefficients of three distinct holomorphic primitive cusp forms $f(z) \in S_{k_1}(\Gamma), g(z) \in S_{k_2}(\Gamma), h(z) \in S_{k_3}(\Gamma)$ respectively. In this paper we study the cancellations of sums related to arithmetic functions, such as $\lambda_f(n)^4\lambda_g(n)^2$, $\lambda_g(n)^6$, $\lambda_g(n)^2\lambda_h(n)^4$, and $\lambda_g(n^3)^2$ twisted by the arithmetic function $\lambda_f(n)$. Keywords:Fourier coefficients of automorphic forms, Dirichlet series, triple product $L$-function, Perron's formulaCategories:11F30, 11F66

25. CMB 2015 (vol 58 pp. 730)

Efrat, Ido; Matzri, Eliyahu
 Vanishing of Massey Products and Brauer Groups Let $p$ be a prime number and $F$ a field containing a root of unity of order $p$. We relate recent results on vanishing of triple Massey products in the mod-$p$ Galois cohomology of $F$, due to Hopkins, Wickelgren, MinÃ¡Ä, and TÃ¢n, to classical results in the theory of central simple algebras. For global fields, we prove a stronger form of the vanishing property. Keywords:Galois cohomology, Brauer groups, triple Massey products, global fieldsCategories:16K50, 11R34, 12G05, 12E30
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