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1. CJM Online first
Abelian Surfaces with an Automorphism and Quaternionic Multiplication We construct one dimensional families of Abelian surfaces with
quaternionic multiplication
which also have an automorphism of order three or four. Using Barth's
description of the moduli space of $(2,4)$-polarized Abelian surfaces,
we find the Shimura curve parametrizing these Abelian surfaces in a
specific case.
We explicitly relate these surfaces to the Jacobians of genus two
curves studied by Hashimoto and Murabayashi.
We also describe a (Humbert) surface in Barth's moduli space which
parametrizes Abelian surfaces with real multiplication by
$\mathbf{Z}[\sqrt{2}]$.
Keywords:abelian surfaces, moduli, shimura curves Categories:14K10, 11G10, 14K20 |
2. CJM Online first
Metaplectic Tensor Products for Automorphic Representation of $\widetilde{GL}(r)$ Let $M=\operatorname{GL}_{r_1}\times\cdots\times\operatorname{GL}_{r_k}\subseteq\operatorname{GL}_r$ be a Levi
subgroup of $\operatorname{GL}_r$, where $r=r_1+\cdots+r_k$, and $\widetilde{M}$ its metaplectic preimage
in the $n$-fold metaplectic cover $\widetilde{\operatorname{GL}}_r$ of $\operatorname{GL}_r$. For automorphic
representations $\pi_1,\dots,\pi_k$ of $\widetilde{\operatorname{GL}}_{r_1}(\mathbb{A}),\dots,\widetilde{\operatorname{GL}}_{r_k}(\mathbb{A})$,
we construct (under a certain
technical assumption, which is always satisfied when $n=2$) an
automorphic representation $\pi$
of $\widetilde{M}(\mathbb{A})$ which can be considered as the ``tensor product'' of the
representations $\pi_1,\dots,\pi_k$. This is
the global analogue of the metaplectic tensor product
defined by P. Mezo in the sense that locally at each place $v$,
$\pi_v$ is equivalent to the local metaplectic tensor product of
$\pi_{1,v},\dots,\pi_{k,v}$ defined by Mezo. Then we show that if all
of $\pi_i$ are cuspidal (resp. square-integrable modulo center), then
the metaplectic tensor product is cuspidal (resp. square-integrable
modulo center). We also show that (both
locally and globally) the metaplectic tensor product behaves in the
expected way under the action of a Weyl group element, and show the
compatibility with parabolic inductions.
Keywords:automorphic forms, representations of covering groups Category:11F70 |
3. CJM Online first
Growth of Selmer groups of CM Abelian varieties Let $p$ be an odd prime. We study the variation of the $p$-rank of
the Selmer group of Abelian varieties with complex multiplication in
certain towers of number fields.
Keywords:Selmer group, Abelian variety with complex multiplication, $\mathbb{Z}_p$-extension, $p$-Hilbert class tower Categories:11G15, 11G10, 11R23, 11R34 |
4. CJM Online first
Sommes friables d'exponentielles et applications An integer is said to be $y$-friable if its greatest prime factor is less than $y$.
In this paper, we obtain estimates for exponential sums
over $y$-friable numbers up to $x$ which are non-trivial
when $y \geq \exp\{c \sqrt{\log x} \log \log x\}$. As a consequence,
we obtain an asymptotic formula for the
number of $y$-friable solutions to the equation $a+b=c$ which is valid
unconditionnally under the same assumption.
We use a contour integration argument based on the saddle point
method, as developped in the context of friable numbers by Hildebrand
and Tenenbaum,
and used by Lagarias, Soundararajan and Harper to study exponential and character sums over friable numbers.
Keywords:thÃ©orie analytique des nombres, entiers friables, mÃ©thode du col Categories:12N25, 11L07 |
5. CJM Online first
Overconvergent Families of Siegel-Hilbert Modular Forms We construct one-parameter families of overconvergent Siegel-Hilbert
modular forms. This result has applications to construction of
Galois representations for automorphic forms of non-cohomological
weights.
Keywords:p-adic automorphic form, rigid analytic geometry Categories:11F46, 14G22 |
6. CJM Online first
Faithfulness of Actions on Riemann-Roch Spaces Given a faithful action of a finite group $G$ on an algebraic
curve~$X$ of genus $g_X\geq 2$, we give explicit criteria for
the induced action of~$G$ on the Riemann-Roch space~$H^0(X,\mathcal{O}_X(D))$
to be faithful, where $D$ is a $G$-invariant divisor on $X$ of
degree at least~$2g_X-2$. This leads to a concise answer to the
question when the action of~$G$ on the space~$H^0(X, \Omega_X^{\otimes
m})$ of global holomorphic polydifferentials of order $m$ is
faithful. If $X$ is hyperelliptic, we furthermore provide an
explicit basis of~$H^0(X, \Omega_X^{\otimes m})$. Finally, we
give applications in deformation theory and in coding theory
and we discuss the analogous problem for the action of~$G$ on
the first homology $H_1(X, \mathbb{Z}/m\mathbb{Z})$ if $X$ is a Riemann surface.
Keywords:faithful action, Riemann-Roch space, polydifferential, hyperelliptic curve, equivariant deformation theory, Goppa code, homology Categories:14H30, 30F30, 14L30, 14D15, 11R32 |
7. CJM 2014 (vol 66 pp. 993)
Expression d'un facteur epsilon de paire par une formule intÃ©grale Let $E/F$ be a quadratic extension of $p$-adic fields and
let $d$, $m$ be nonnegative integers of distinct parities. Fix
admissible irreducible tempered representations $\pi$ and $\sigma$ of
$GL_d(E)$ and $GL_m(E)$ respectively. We assume that $\pi$ and
$\sigma$ are conjugate-dual. That is to say $\pi\simeq \pi^{\vee,c}$
and $\sigma\simeq \sigma^{\vee,c}$ where $c$ is the non trivial
$F$-automorphism of $E$. This implies, we can extend $\pi$ to an
unitary representation $\tilde{\pi}$ of a nonconnected group
$GL_d(E)\rtimes \{1,\theta\}$. Define $\tilde{\sigma}$ the same
way. We state and prove an integral formula for
$\epsilon(1/2,\pi\times \sigma,\psi_E)$ involving the characters of
$\tilde{\pi}$ and $\tilde{\sigma}$. This formula is related to the
local Gan-Gross-Prasad conjecture for unitary groups.
Keywords:epsilon factor, twisted groups Categories:22E50, 11F85 |
8. CJM Online first
On a sumset conjecture of ErdÅs ErdÅs conjectured that for any set $A\subseteq \mathbb{N}$
with positive
lower asymptotic density, there are infinite sets $B,C\subseteq
\mathbb{N}$
such that $B+C\subseteq A$. We verify ErdÅs' conjecture in
the case that $A$ has Banach density exceeding $\frac{1}{2}$.
As a consequence, we prove that, for $A\subseteq \mathbb{N}$
with
positive Banach density (a much weaker assumption than positive
lower density), we can find infinite $B,C\subseteq \mathbb{N}$
such
that $B+C$ is contained in the union of $A$ and a translate of
$A$. Both of the aforementioned
results are generalized to arbitrary countable
amenable groups. We also provide a positive solution to ErdÅs'
conjecture for subsets of the natural numbers that are pseudorandom.
Keywords:sumsets of integers, asymptotic density, amenable groups, nonstandard analysis Categories:11B05, 11B13, 11P70, 28D15, 37A45 |
9. CJM Online first
Mahler Measures as Linear Combinations of $L$-values of Multiple Modular Forms We study the Mahler measures of certain families of Laurent
polynomials in two and three variables. Each of the known Mahler
measure formulas for these families involves $L$-values of at most one
newform and/or at most one quadratic character. In this paper, we
show, either rigorously or numerically, that the Mahler measures of
some polynomials are related to $L$-values of multiple newforms and
quadratic characters simultaneously. The results suggest that the
number of modular $L$-values appearing in the formulas significantly
depends on the shape of the algebraic value of the parameter chosen
for each polynomial. As a consequence, we also obtain new formulas
relating special values of hypergeometric series evaluated at
algebraic numbers to special values of $L$-functions.
Keywords:Mahler measures, Eisenstein-Kronecker series, $L$-functions, hypergeometric series Categories:11F67, 33C20 |
10. CJM Online first
On Littlewood Polynomials with Prescribed Number of Zeros Inside the Unit Disk We investigate the numbers of complex zeros of Littlewood polynomials
$p(z)$ (polynomials with coefficients $\{-1, 1\}$) inside or
on the unit circle $|z|=1$, denoted by $N(p)$ and $U(p)$, respectively.
Two types of Littlewood polynomials are considered: Littlewood
polynomials with one sign change in the sequence of coefficients
and Littlewood polynomials with one negative coefficient. We
obtain explicit formulas for $N(p)$, $U(p)$ for polynomials $p(z)$
of these types. We show that, if $n+1$ is a prime number, then
for each integer $k$, $0 \leq k \leq n-1$, there exists a Littlewood
polynomial $p(z)$ of degree $n$ with $N(p)=k$ and $U(p)=0$. Furthermore,
we describe some cases when the ratios $N(p)/n$ and $U(p)/n$
have limits as $n \to \infty$ and find the corresponding limit
values.
Keywords:Littlewood polynomials, zeros, complex roots Categories:11R06, 11R09, 11C08 |
11. CJM 2014 (vol 67 pp. 198)
Tate Cycles on Abelian Varieties with Complex Multiplication We consider Tate cycles on an Abelian variety $A$ defined over
a sufficiently large number field $K$ and having complex
multiplication. We show that
there is an effective bound $C = C(A,K)$ so that
to check whether a given cohomology class is a Tate class on
$A$, it suffices to check the action of
Frobenius elements at primes $v$ of norm $ \leq C$.
We also show that for a set of primes $v$ of $K$ of density
$1$, the space of Tate cycles on the special fibre $A_v$ of the
NÃ©ron model of $A$ is isomorphic to the space of Tate cycles
on $A$ itself.
Keywords:Abelian varieties, complex multiplication, Tate cycles Categories:11G10, 14K22 |
12. CJM 2014 (vol 66 pp. 1078)
Values of Twisted Tensor $L$-functions of Automorphic Forms Over Imaginary Quadratic Fields Let $K$ be a complex quadratic extension of $\mathbb{Q}$ and let $\mathbb{A}_K$
denote the adeles of $K$.
We find special values at all of the critical points of twisted
tensor $L$-functions attached to cohomological cuspforms on $GL_2(\mathbb{A}_K)$,
and establish Galois equivariance of the values.
To investigate the values, we determine the archimedean factors
of a class of integral representations of these $L$-functions,
thus proving a conjecture due to Ghate. We also investigate
analytic properties of these $L$-functions, such as their functional
equations.
Keywords:twisted tensor $L$-function, cuspform, hypergeometric series Categories:11F67, 11F37 |
13. CJM Online first
A Skolem-Mahler-Lech Theorem for Iterated Automorphisms of $K$-algebras This paper proves a commutative algebraic extension
of a generalized Skolem-Mahler-Lech theorem due to the first
author.
Let $A$ be a finitely generated commutative $K$-algebra
over a field of characteristic $0$, and let $\sigma$ be
a $K$-algebra automorphism of $A$.
Given ideals $I$ and $J$ of $A$, we show that
the set $S$ of integers $m$ such that
$\sigma^m(I) \supseteq J$ is a finite union of
complete doubly infinite arithmetic progressions in $m$, up to
the addition of a finite set.
Alternatively, this result states that for an affine scheme
$X$ of finite type over $K$,
an automorphism $\sigma \in \operatorname{Aut}_K(X)$, and $Y$ and $Z$
any two closed subschemes of $X$, the set
of integers $m$ with $\sigma^m(Z ) \subseteq Y$ is as above.
The paper presents examples
showing that this result may fail to hold if the affine scheme
$X$ is
not of finite type, or if $X$ is of finite type but the field
$K$ has positive characteristic.
Keywords:automorphisms, endomorphisms, affine space, commutative algebras, Skolem-Mahler-Lech theorem Categories:11D45, 14R10, 11Y55, 11D88 |
14. CJM 2013 (vol 66 pp. 826)
Signed-Selmer Groups over the $\mathbb{Z}_p^2$-extension of an Imaginary Quadratic Field Let $E$ be an elliptic curve over $\mathbb Q$ which has good supersingular
reduction at $p\gt 3$. We construct what we call the $\pm/\pm$-Selmer
groups of $E$ over the $\mathbb Z_p^2$-extension of an imaginary quadratic
field $K$ when the prime $p$ splits completely over $K/\mathbb Q$, and
prove they enjoy a property analogous to Mazur's control theorem.
Furthermore, we propose a conjectural connection between the
$\pm/\pm$-Selmer groups and Loeffler's two-variable $\pm/\pm$-$p$-adic
$L$-functions of elliptic curves.
Keywords:elliptic curves, Iwasawa theory Category:11Gxx |
15. CJM 2013 (vol 66 pp. 1305)
Congruence Relations for Shimura Varieties Associated with $GU(n-1,1)$ We prove the congruence relation for the mod-$p$ reduction of Shimura
varieties associated to a unitary similitude group $GU(n-1,1)$ over
$\mathbb{Q}$, when $p$ is inert and $n$ odd. The case when $n$
is even was obtained by T. Wedhorn and O. B?ltel, as a special case
of a result of B. Moonen, when the $\mu$-ordinary locus of the $p$-isogeny
space is dense. This condition fails in our case. We show that every
supersingular irreducible component of the special fiber of $p\textrm{-}\mathscr{I}sog$
is annihilated by a degree one polynomial in the Frobenius element
$F$, which implies the congruence relation.
Keywords:Shimura varieties, congruence relation Categories:11G18, 14G35, 14K10 |
16. CJM 2013 (vol 67 pp. 214)
Symmetric Genuine Spherical Whittaker Functions on $\overline{GSp_{2n}(F)}$ Let $F$ be a p-adic field of odd residual characteristic. Let
$\overline{GSp_{2n}(F)}$ and $\overline{Sp_{2n}(F)}$ be the metaplectic double covers of the general
symplectic group and the symplectic group attached to the $2n$
dimensional symplectic space over $F$. Let $\sigma$ be a genuine,
possibly reducible, unramified principal series representation of
$\overline{GSp_{2n}(F)}$. In these notes we give an explicit formulas for a spanning
set for the space of Spherical Whittaker functions attached to
$\sigma$. For odd $n$, and generically for even $n$, this spanning set
is a basis. The significant property of this set is that each of its
elements is unchanged under the action of the Weyl group of
$\overline{Sp_{2n}(F)}$.
If $n$ is odd then each element in the set has an equivariant property
that generalizes a uniqueness result of Gelbart, Howe and
Piatetski-Shapiro. Using this symmetric set, we
construct a family of reducible genuine unramified principal series
representations which have more then one generic constituent. This
family contains all the reducible genuine unramified principal series
representations induced from a unitary data and exists only for $n$
even.
Keywords:metaplectic group, Casselman Shalika Formula Category:11F85 |
17. CJM 2013 (vol 65 pp. 1320)
Orbital $L$-functions for the Space of Binary Cubic Forms We introduce the notion of orbital $L$-functions
for the space of binary cubic forms
and investigate their analytic properties.
We study their functional equations and residue formulas in some detail.
Aside from their intrinsic interest,
the results from this paper are used to
prove the existence of secondary terms in counting
functions for cubic fields.
This is worked out in a companion paper.
Keywords:binary cubic forms, prehomogeneous vector spaces, Shintani zeta functions, $L$-functions, cubic rings and fields Categories:11M41, 11E76 |
18. CJM 2013 (vol 66 pp. 924)
Twists of Shimura Curves Consider a Shimura curve $X^D_0(N)$ over the rational
numbers. We determine criteria for the twist by an Atkin-Lehner
involution to have points over a local field. As a corollary we give a
new proof of the theorem of Jordan-LivnÃ© on $\mathbf{Q}_p$ points
when $p\mid D$ and for the first time give criteria for $\mathbf{Q}_p$
points when $p\mid N$. We also give congruence conditions for roots
modulo $p$ of Hilbert class polynomials.
Keywords:Shimura curves, complex multiplication, modular curves, elliptic curves Categories:11G18, 14G35, 11G15, 11G10 |
19. CJM 2013 (vol 66 pp. 1167)
Galois Representations Over Fields of Moduli and Rational Points on Shimura Curves The purpose of this note is introducing a method for proving the
existence of no rational points on a coarse moduli space $X$ of abelian varieties
over a given number field $K$, in cases where the moduli problem is not fine and
points in $X(K)$ may not be represented by an abelian variety (with additional structure)
admitting a model over the field $K$. This is typically the case when the abelian
varieties that are being classified have even dimension. The main idea, inspired on
the work of Ellenberg and Skinner on the modularity of $\mathbb{Q}$-curves, is that to a
point $P=[A]\in X(K)$ represented by an abelian variety $A/\bar K$ one may still
attach a Galois representation of $\operatorname{Gal}(\bar K/K)$ with values in the quotient
group $\operatorname{GL}(T_\ell(A))/\operatorname{Aut}(A)$, provided
$\operatorname{Aut}(A)$ lies in the centre of $\operatorname{GL}(T_\ell(A))$.
We exemplify our method in the cases where $X$ is a Shimura curve over an imaginary
quadratic field or an Atkin-Lehner quotient over $\mathbb{Q}$.
Keywords:Shimura curves, rational points, Galois representations, Hasse principle, Brauer-Manin obstruction Categories:11G18, 14G35, 14G05 |
20. CJM 2013 (vol 66 pp. 844)
Multidimensional Vinogradov-type Estimates in Function Fields Let $\mathbb{F}_q[t]$ denote the polynomial ring over the finite
field $\mathbb{F}_q$.
We employ Wooley's new efficient congruencing method to prove
certain multidimensional Vinogradov-type estimates in $\mathbb{F}_q[t]$.
These results allow us to apply a variant of the circle method
to obtain asymptotic formulas for a system connected to the problem
about linear spaces lying on hypersurfaces defined over $\mathbb{F}_q[t]$.
Keywords:Vinogradov's mean value theorem, function fields, circle method Categories:11D45, 11P55, 11T55 |
21. CJM 2013 (vol 66 pp. 566)
Transfer of Plancherel Measures for Unitary Supercuspidal Representations between $p$-adic Inner Forms |
Transfer of Plancherel Measures for Unitary Supercuspidal Representations between $p$-adic Inner Forms Let $F$ be a $p$-adic field of characteristic $0$, and let $M$ be an $F$-Levi subgroup of a connected reductive $F$-split group such that $\Pi_{i=1}^{r} SL_{n_i} \subseteq M \subseteq \Pi_{i=1}^{r} GL_{n_i}$ for positive integers $r$ and $n_i$. We prove that the Plancherel measure for any unitary supercuspidal representation of $M(F)$ is identically transferred under the local Jacquet-Langlands type correspondence between $M$ and its $F$-inner forms, assuming a working hypothesis that Plancherel measures are invariant on a certain set. This work extends the result of
MuiÄ and Savin (2000) for Siegel Levi subgroups of the groups $SO_{4n}$ and $Sp_{4n}$ under the local Jacquet-Langlands correspondence. It can be applied to a simply connected simple $F$-group of type $E_6$ or $E_7$, and a connected reductive $F$-group of type $A_{n}$, $B_{n}$, $C_n$ or $D_n$.
Keywords:Plancherel measure, inner form, local to global global argument, cuspidal automorphic representation, Jacquet-Langlands correspondence Categories:22E50, 11F70, 22E55, 22E35 |
22. CJM 2012 (vol 66 pp. 170)
Modular Abelian Varieties Over Number Fields The main result of this paper is a characterization of the abelian
varieties $B/K$ defined over Galois number fields with the
property that the $L$-function $L(B/K;s)$ is a product of
$L$-functions of non-CM newforms over $\mathbb Q$ for congruence
subgroups of the form $\Gamma_1(N)$. The characterization involves the
structure of $\operatorname{End}(B)$, isogenies between the Galois conjugates of
$B$, and a Galois cohomology class attached to $B/K$.
We call the varieties having this property strongly modular.
The last section is devoted to the study of a family of abelian surfaces with quaternionic
multiplication.
As an illustration of the ways in which the general results of the paper can be applied
we prove the strong modularity of some particular abelian surfaces belonging to that family, and
we show how to find nontrivial examples of strongly modular varieties by twisting.
Keywords:Modular abelian varieties, $GL_2$-type varieties, modular forms Categories:11G10, 11G18, 11F11 |
23. CJM 2012 (vol 65 pp. 1201)
Application of the Strong Artin Conjecture to the Class Number Problem We construct unconditionally several families of number fields with
the largest possible class numbers. They are number fields of degree 4
and 5 whose Galois closures have the Galois group $A_4, S_4$ and
$S_5$. We first construct families of number fields with smallest
regulators, and by using the strong Artin conjecture and applying zero
density result of Kowalski-Michel, we choose subfamilies of
$L$-functions which are zero free close to 1.
For these subfamilies, the $L$-functions have the extremal value at
$s=1$, and by the class number formula, we obtain the extreme class
numbers.
Keywords:class number, strong Artin conjecture Categories:11R29, 11M41 |
24. CJM 2012 (vol 65 pp. 544)
Iterated Integrals and Higher Order Invariants We show that higher order invariants of smooth functions can be
written as linear combinations of full invariants times iterated
integrals.
The non-uniqueness of such a presentation is captured in the kernel of
the ensuing map from the tensor product. This kernel is computed
explicitly.
As a consequence, it turns out that higher order invariants are a free
module of the algebra of full invariants.
Keywords:higher order forms, iterated integrals Categories:14F35, 11F12, 55D35, 58A10 |
25. CJM 2012 (vol 65 pp. 403)
On the Dihedral Main Conjectures of Iwasawa Theory for Hilbert Modular Eigenforms We construct a bipartite Euler system in the sense of Howard for Hilbert modular eigenforms of parallel
weight two over totally real fields, generalizing works of Bertolini-Darmon, Longo, Nekovar, Pollack-Weston
and others. The construction has direct applications to Iwasawa main conjectures. For instance, it implies
in many cases one divisibility of the associated dihedral or anticyclotomic main conjecture, at the same
time reducing the other divisibility to a certain nonvanishing criterion for the associated $p$-adic $L$-functions.
It also has applications to cyclotomic main conjectures for Hilbert modular forms over CM fields via the technique
of Skinner and Urban.
Keywords:Iwasawa theory, Hilbert modular forms, abelian varieties Categories:11G10, 11G18, 11G40 |