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1. CJM Online first

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

2. 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

3. 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

4. CJM Online first

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

5. CJM Online first

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

6. CJM Online first

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

7. CJM Online first

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

8. 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

9. 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

10. CJM Online first

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

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

12. CJM Online first

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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. CJM 2016 (vol 68 pp. 1362)

Papikian, Mihran; Rabinoff, Joseph
Optimal Quotients of Jacobians with Toric Reduction and Component Groups
Let $J$ be a Jacobian variety with toric reduction over a local field $K$. Let $J \to E$ be an optimal quotient defined over $K$, where $E$ is an elliptic curve. We give examples in which the functorially induced map $\Phi_J \to \Phi_E$ on component groups of the Néron models is not surjective. This answers a question of Ribet and Takahashi. We also give various criteria under which $\Phi_J \to \Phi_E$ is surjective, and discuss when these criteria hold for the Jacobians of modular curves.

Keywords:Jacobians with toric reduction, component groups, modular curves
Categories:11G18, 14G22, 14G20

20. CJM 2016 (vol 69 pp. 532)

Ganguly, Arijit; Ghosh, Anish
Dirichlet's Theorem in Function Fields
We study metric Diophantine approximation for function fields specifically the problem of improving Dirichlet's theorem in Diophantine approximation.

Keywords:Dirichlet's theorem, Diophantine approximation, positive characteristic
Categories:11J83, 11K60, 37D40, 37A17, 22E40

21. CJM 2016 (vol 68 pp. 1120)

Stange, Katherine E.
Integral Points on Elliptic Curves and Explicit Valuations of Division Polynomials
Assuming Lang's conjectured lower bound on the heights of non-torsion points on an elliptic curve, we show that there exists an absolute constant $C$ such that for any elliptic curve $E/\mathbb{Q}$ and non-torsion point $P \in E(\mathbb{Q})$, there is at most one integral multiple $[n]P$ such that $n \gt C$. The proof is a modification of a proof of Ingram giving an unconditional but not uniform bound. The new ingredient is a collection of explicit formulae for the sequence $v(\Psi_n)$ of valuations of the division polynomials. For $P$ of non-singular reduction, such sequences are already well described in most cases, but for $P$ of singular reduction, we are led to define a new class of sequences called \emph{elliptic troublemaker sequences}, which measure the failure of the Néron local height to be quadratic. As a corollary in the spirit of a conjecture of Lang and Hall, we obtain a uniform upper bound on $\widehat{h}(P)/h(E)$ for integer points having two large integral multiples.

Keywords:elliptic divisibility sequence, Lang's conjecture, height functions
Categories:11G05, 11G07, 11D25, 11B37, 11B39, 11Y55, 11G50, 11H52

22. CJM 2016 (vol 69 pp. 807)

Günther, Christian; Schmidt, Kai-Uwe
$L^q$ Norms of Fekete and Related Polynomials
A Littlewood polynomial is a polynomial in $\mathbb{C}[z]$ having all of its coefficients in $\{-1,1\}$. There are various old unsolved problems, mostly due to Littlewood and Erdős, that ask for Littlewood polynomials that provide a good approximation to a function that is constant on the complex unit circle, and in particular have small $L^q$ norm on the complex unit circle. We consider the Fekete polynomials \[ f_p(z)=\sum_{j=1}^{p-1}(j\,|\,p)\,z^j, \] where $p$ is an odd prime and $(\,\cdot\,|\,p)$ is the Legendre symbol (so that $z^{-1}f_p(z)$ is a Littlewood polynomial). We give explicit and recursive formulas for the limit of the ratio of $L^q$ and $L^2$ norm of $f_p$ when $q$ is an even positive integer and $p\to\infty$. To our knowledge, these are the first results that give these limiting values for specific sequences of nontrivial Littlewood polynomials and infinitely many $q$. Similar results are given for polynomials obtained by cyclically permuting the coefficients of Fekete polynomials and for Littlewood polynomials whose coefficients are obtained from additive characters of finite fields. These results vastly generalise earlier results on the $L^4$ norm of these polynomials.

Keywords:character polynomial, Fekete polynomial, $L^q$ norm, Littlewood polynomial
Categories:11B83, 42A05, 30C10

23. CJM 2016 (vol 69 pp. 258)

Brandes, Julia; Parsell, Scott T.
Simultaneous Additive Equations: Repeated and Differing Degrees
We obtain bounds for the number of variables required to establish Hasse principles, both for existence of solutions and for asymptotic formulæ, for systems of additive equations containing forms of differing degree but also multiple forms of like degree. Apart from the very general estimates of Schmidt and Browning--Heath-Brown, which give weak results when specialized to the diagonal situation, this is the first result on such "hybrid" systems. We also obtain specialised results for systems of quadratic and cubic forms, where we are able to take advantage of some of the stronger methods available in that setting. In particular, we achieve essentially square root cancellation for systems consisting of one cubic and $r$ quadratic equations.

Keywords:equations in many variables, counting solutions of Diophantine equations, applications of the Hardy-Littlewood method
Categories:11D72, 11D45, 11P55

24. CJM 2016 (vol 68 pp. 961)

Greenberg, Matthew; Seveso, Marco
$p$-adic Families of Cohomological Modular Forms for Indefinite Quaternion Algebras and the Jacquet-Langlands Correspondence
We use the method of Ash and Stevens to prove the existence of small slope $p$-adic families of cohomological modular forms for an indefinite quaternion algebra $B$. We prove that the Jacquet-Langlands correspondence relating modular forms on $\textbf{GL}_2/\mathbb{Q}$ and cohomomological modular forms for $B$ is compatible with the formation of $p$-adic families. This result is an analogue of a theorem of Chenevier concerning definite quaternion algebras.

Keywords:modular forms, p-adic families, Jacquet-Langlands correspondence, Shimura curves, eigencurves
Categories:11F11, 11F67, 11F85

25. CJM 2016 (vol 69 pp. 1143)

Sikirić, Mathieu Dutour
The seven Dimensional Perfect Delaunay Polytopes and Delaunay Simplices
For a lattice $L$ of $\mathbb{RR}^n$, a sphere $S(c,r)$ of center $c$ and radius $r$ is called empty if for any $v\in L$ we have $\Vert v - c\Vert \geq r$. Then the set $S(c,r)\cap L$ is the vertex set of a {\em Delaunay polytope} $P=\operatorname{conv}(S(c,r)\cap L)$. A Delaunay polytope is called {\em perfect} if any affine transformation $\phi$ such that $\phi(P)$ is a Delaunay polytope is necessarily an isometry of the space composed with an homothety. Perfect Delaunay polytopes are remarkable structure that exist only if $n=1$ or $n\geq 6$ and they have shown up recently in covering maxima studies. Here we give a general algorithm for their enumeration that relies on the Erdahl cone. We apply this algorithm in dimension $7$ which allow us to find that there are only two perfect Delaunay polytopes: $3_{21}$ which is a Delaunay polytope in the root lattice $\mathsf{E}_7$ and the Erdahl Rybnikov polytope. We then use this classification in order to get the list of all types Delaunay simplices in dimension $7$ and found $11$ types.

Keywords:Delaunay polytope, enumeration, polyhedral methods
Categories:11H06, 11H31
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