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1. CJM Online first

Gurevich, Nadya; Segal, Avner
Poles of the Standard $\mathcal{L}$-function of $G_2$ and the Rallis-Schiffmann 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 Rallis-Schiffmann lift for the commuting pair $(\widetilde{SL_2}, G_2)$ in $\widetilde{Sp_{14}}$. The image consists of non-tempered representations. The main tool is the recent construction, by the second author, of a family of Rankin-Selberg integrals representing the standard $\mathcal{L}$-function.

Keywords:automorphic representation, exceptional theta-lift, Siegel-Weil identity
Categories:11F70, 11F27, 11F66

2. CJM Online first

Betina, Adel
Ramification of the Eigencurve at classical RM points
J.Bellaïche and M.Dimitrov have shown that the $p$-adic eigencurve is smooth but not étale 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 Atkin-Lehner involution of the completed local ring of the eigencurve at these points and an universal ring representing a pseudo-deformation 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 pseudo-deformations of reducible Galois representations.

Keywords:weight one RM modular form, eigencurve, pseudo-deformation, deformation of reducible representation
Categories:11F80, 11F33, 11R23

3. CJM Online first

Knightly, Andrew; Reno, Caroline
Weighted distribution of low-lying 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 low-lying 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, L-function
Categories:11M41, 11F11, 11M26

4. CJM Online first

Chapdelaine, Hugo; Kuċera, Radan
Annihilators of the ideal class group of a cyclic extension of an imaginary quadratic field
The aim of this paper is to study the group of elliptic units of a cyclic extension $L$ of an imaginary quadratic field $K$ such that the degree $[L:K]$ is a power of an odd prime $p$. We construct an explicit root of the usual top generator of this group and we use it to obtain an annihilation result of the $p$-Sylow subgroup of the ideal class group of $L$.

Keywords:annihilator, class group, elliptic unit
Categories:11R20, 11R27, 11R29

5. 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 idèle class group, and ray class groups in a cocycle-theoretic 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

6. CJM Online first

Cahn, Jordan; Jones, Rafe; Spear, Jacob
Powers in orbits of rational functions: cases of an arithmetic dynamical Mordell-Lang 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 Lattès 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 Möbius-conjugate 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 Mordell-Lang 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 variables-separated curve, Lattès map
Categories:37P05, 11G05, 37P15

7. CJM Online first

Bary-Soroker, 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 $X^3 = Y(Y-1)$ 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 Hardy-Littlewood 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

8. CJM Online first

Hanzer, Marcela; Savin, Gordan
Eisenstein series arising from Jordan algebras
We describe poles and the corresponding residual automorphic representations of Eisenstein series attached to maximal parabolic subgroups whose unipotent radicals admit Jordan algebra structure.

Keywords:Eisenstein series, Jordan algebra, Fourier-Jacobi functor
Categories:11F70, 22E50, 22E55

9. CJM Online first

Bosser, Vincent; Gaudron, Éric
Logarithmes des points rationnels des variétés abéliennes
Nous démontrons une généralisation du théorème des périodes de Masser et Wüstholz où la période est remplacée par un logarithme non nul $u$ d'un point rationnel $p$ d'une variété abélienne définie sur un corps de nombres. Nous en déduisons des minorations explicites de la norme de $u$ et de la hauteur de Néron-Tate de $p$ qui dépendent des invariants classiques du problème dont la dimension et la hauteur de Faltings de la variété abélienne. Les démonstrations reposent sur une construction de transcendance du type Gel'fond-Baker de la théorie des formes linéaires de logarithmes dans laquelle se greffent des formules explicites provenant de la théorie des pentes d'Arakelov.

Keywords:periods theorem, abelian variety, logarithm, Gel'fond-Baker method, slope theory, Néron-Tate height, interpolation lemma
Categories:11J86, 11J95, 11G10, 11G50, 14G40

10. CJM 2018 (vol 70 pp. 1373)

Tuxanidy, Aleksandr; Wang, Qiang
A New Proof of the Hansen-Mullen Irreducibility Conjecture
We give a new proof of the Hansen-Mullen 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, Hansen-Mullen conjecture, symmetric function, $q$-symmetric, discrete Fourier transform, finite field

11. 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

12. CJM 2018 (vol 70 pp. 1096)

Müllner, Clemens
The Rudin-Shapiro 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 Rudin-Shapiro sequence and some generalizations. This gives, for any base, a class of explicit normal numbers that can be efficiently generated.

Keywords:Rudin-Shapiro sequence, digital sequence, normality, exponential sum
Categories:11A63, 11B85, 11L03, 11N60, 60F05

13. 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 non-critical slope rational modular forms, the theory has been extended to construct $p$-adic $L$-functions for non-critical 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), p-adic L-function, L-function, modular symbol, overconvergent, cohomology, automorphic cycle, control theorem, L-value, distribution
Categories:11F41, 11F67, 11F85, 11S40, 11M41

14. CJM Online first

Hartl, Urs; Singh, Rajneesh Kumar
Local Shtukas and Divisible Local Anderson Modules
We develop the analog of crystalline Dieudonné 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 Dieudonné modules are replaced by local shtukas. We show that the categories of divisible local Anderson modules and of effective local shtukas are anti-equivalent 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 t-module
Categories:11G09, 13A35, 14L05

15. CJM 2018 (vol 70 pp. 1390)

Xiao, Stanley Yao
Square-free 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 square-free 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:square-free value, decomposable form, Selberg sieve

16. CJM 2018 (vol 70 pp. 1173)

Viada, Evelina
An Explicit Manin-Dem'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 Néron-Tate 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 Manin-Dem'janenko method in products of elliptic curves.

Keywords:height, elliptic curve, explicit Mordell conjecture, explicit Manin-Demjanenko 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 Rankin-Selberg 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

19. CJM 2017 (vol 70 pp. 1319)

Macourt, Simon; Shkredov, Ilya D.; Shparlinski, Igor E.
Multiplicative Energy of Shifted Subgroups and Bounds On Exponential Sums with Trinomials in Finite Fields
We give a new bound on collinear triples in subgroups of prime finite fields and use it to give some new bounds on exponential sums with trinomials.

Keywords:exponential sum, sparse polynomial, trinomial
Categories:11L07, 11T23

20. CJM 2017 (vol 70 pp. 451)

Zhang, Chao
Ekedahl-Oort 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 Ekedahl-Oort stratifications on the special fibers of those integral canonical models when $p\gt 2$. This generalizes Ekedahl-Oort 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 Ekedahl-Oort 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 non-empty. We also determine the closure of each stratum.

Keywords:Shimura variety, F-zip
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 non-analytic groups. We exploit the interplay of group-theoretical and number-theoretical 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, pro-p group, p-adic analytic group, Brauer-Siegel 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 Non-regular 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 self-similar measure $\mu =\sum p_{j}\mu \circ S_{j}^{-1}$, of finite type. In this paper we study the multi-fractal analysis of such measures, extending the theory to measures arising from non-regular 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:multi-fractal 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 Gross-Deligne 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 non-trivial 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 $F|A^{(j)}(M)$ 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
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