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1. 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

2. 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

3. CJM Online first

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
Category:11B05

4. 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
Category:11M06

5. CJM Online first

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

6. 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

7. 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
Category:11F70

8. 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

9. CJM Online first

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

10. 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

11. CJM Online first

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

12. 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

13. 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

14. 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 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

15. 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

16. 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

17. 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 Atkin--Lehner eigenvalues. The proofs involve the notion of quaternionic $S$-ideal classes and the distribution of Atkin--Lehner signs among newforms.

Keywords:modular forms, congruences, quaternion algebras
Categories:11F33, 11R52

18. CJM Online first

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
Category:11T06

19. 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

20. 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

21. CJM 2016 (vol 69 pp. 826)

Lei, Antonio; Loeffler, David; Zerbes, Sarah Livia
On the Asymptotic Growth of Bloch-Kato-Shafarevich-Tate Groups of Modular Forms over Cyclotomic Extensions
We study the asymptotic behaviour of the Bloch--Kato--Shafarevich--Tate group of a modular form $f$ over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ under the assumption that $f$ is non-ordinary 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, Shafarevich-Tate group, Bloch-Kato Selmer group, modular form, non-ordinary prime, p-adic Hodge theory
Categories:11R18, 11F11, 11R23, 11F85

22. 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

23. CJM 2016 (vol 69 pp. 186)

Pan, Shu-Yen
$L$-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/Harris-Kudla-Sweet 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

24. CJM 2016 (vol 69 pp. 890)

Xu, Bin
On Mœglin's Parametrization of Arthur Packets for p-adic Quasisplit $Sp(N)$ and $SO(N)$
We give a survey on Moeglin's construction of representations in the Arthur packets for $p$-adic quasisplit symplectic and orthogonal groups. The emphasis is on comparing Moeglin's parametrization of elements in the Arthur packets with that of Arthur.

Keywords:symplectic and orthogonal group, Arthur packet, endoscopy
Categories:22E50, 11F70

25. CJM 2016 (vol 68 pp. 1382)

Zydor, Michał
La Variante infinitésimale de la formule des traces de Jacquet-Rallis pour les groupes unitaires
We establish an infinitesimal version of the Jacquet-Rallis trace formula for unitary groups. Our formula is obtained by integrating a truncated kernel à 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 semi-simple classes $\mathfrak{o}$, $J_{\mathfrak{o}}$ is a relative orbital integral of Jacquet-Rallis. For classes $\mathfrak{o}$ called relatively regular semi-simple, we express $J_{\mathfrak{o}}$ in terms of relative orbital integrals regularised by means of zêta functions.

Keywords:formule des traces relative
Categories:11F70, 11F72
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