2017
43
0
0
433
Filtrations of smooth principal series and Iwasawa modules
2
2
Let $G$ be a reductive $p$adic group. We consider the general question of whether the reducibility of an induced representation can be detected in a ``corank one" situation. For smooth complex representations induced from supercuspidal representations, we show that a sufficient condition is the existence of a subquotient that does not appear as a subrepresentation. An important example is the Langlands' quotient. In addition, we study the same general question for continuous principal series on $p$adic Banach spaces. Although we do not give an answer in this case, we describe a related filtration on the corresponding Iwasawa modules.
3

3
16


W.
Alsibiani
United States of America
walsibiani@live.com


D.
Ban
United States of America
dban@siu.edu
Parabolically induced representations
Iwasawa modules
$p$adic groups
Symmetric powers and the Satake transform
2
2
This paper gives several examples of the basic functions introduced in recent years by Ng^o. These are mainly conjectures based on computer experiment.
3

17
54


B.
Casselman
 
wacasselman@gmail.com
Satake transform
basic functions
Globally analytic $p$adic representations of the pro$p$Iwahori subgroup of $GL(2)$ and base change, I : Iwasawa algebras and a base change map
2
2
This paper extends to the pro$p$ Iwahori subgroup of $GL(2)$ over an unramified finite extension of $mathbb{Q}_p$ the presentation of the Iwasawa algebra obtained earlier by the author for the congruence subgroup of level one of $SL(2, mathbb{Z}_p)$. It then describes a natural base change map between the Iwasawa algebras or more correctly, as it turns out, between the global distribution algebras on the associated rigidanalytic spaces. In a forthcoming paper this will be applied to $p$adic representation theory.
3

55
76


L.
Clozel
 
clozel@math.upsud.fr
Iwasawa theory
automorphic representations
rigid analytic geometry
DiffieHellman type key exchange protocols based on isogenies
2
2
In this paper, we propose some DiffieHellman type key exchange protocols using isogenies of elliptic curves. The first method which uses the endomorphism ring of an ordinary elliptic curve $ E $, is a straightforward generalization of elliptic curve DiffieHellman key exchange. The method uses commutativity of the endomorphism ring $ End(E) $. Then using dual isogenies, we propose a second method. This case uses the endomorphism ring of an elliptic curve $ E $, which can be ordinary or supersingular. We extend this method using isogenies between two elliptic curves $ E $ and $ E' $. Our methods have the security level of that of [D. Jao and L. De Feo, Towards quantumresistant cryptosystems from supersingular elliptic curve isogenies, J. Math. Cryptol. 8 (2014), no. 3, 209247], with the advantage of transmitting less information between two parties.
3

77
88


H.
Daghigh
Iran
hassan@kashanu.ac.ir


R.
Khodakaramian Gilan
Iran
rkhodakaramian@123grad.kashanu.ac.ir


F.
Seifi Shahpar
Iran
fatemeh.seifishahpar@gmail.com
Supersingular elliptic curves
isogeny
cryptography
key exchange
Theta functions on covers of symplectic groups
2
2
We study the automorphic theta representation $Theta_{2n}^{(r)}$ on the $r$fold cover of the symplectic group $Sp_{2n}$. This representation is obtained from the residues of Eisenstein series on this group.
If $r$ is odd,
$nle r <2n$, then under a natural hypothesis on the theta representations, we show that
$Theta_{2n}^{(r)}$ may be used to construct a globally generic representation
$sigma_{2nr+1}^{(2r)}$ on the $2r$fold cover of $Sp_{2nr+1}$. Moreover, when $r=n$ the
Whittaker functions of this representation attached to factorizable data are factorizable, and the unramified local factors may be computed in terms of $n$th order Gauss sums. If $n=3$ we prove these results, which in that case pertain to the sixfold cover of $Sp_4$, unconditionally. We expect that in fact the representation constructed here, $sigma_{2nr+1}^{(2r)}$, is precisely $Theta_{2nr+1}^{(2r)}$; that is, we conjecture relations between theta representations on different covering groups.
3

89
116


S.
Friedberg
 
solfriedberg@gmail.com


D.
Ginzburg
 
Symplectic group
metaplectic cover
theta representation
descent integral
unipotent orbit
generic representation
Whittaker function
Behavior of $R$groups for $p$adic inner forms of quasisplit special unitary groups
2
2
We study $R$groups for $p$adic inner forms of quasisplit special unitary groups. We prove Arthur's conjecture, the isomorphism between the KnappStein $R$group and the LanglandsArthur $R$group, for quasisplit special unitary groups and their inner forms. Furthermore, we investigate the invariance of the KnappStein $R$group within $L$packets and between inner forms. This work is applied to transferring known results in the secondnamed author's earlier work for quasisplit special unitary groups to their nonquasisplit inner forms.
3

117
141


K.
Choiy
United States of America
kchoiy@gmail.com


D.
Goldberg
United States of America
davidg77@gmail.com
Invariance of KnappStein $R$groups
tempered spectrum
Arthur's conjecture on $R$groups
$p$adic inner forms of special unitary groups
Strong exponent bounds for the local RankinSelberg convolution
2
2
Let $F$ be a nonArchimedean locally compact field. Let $sigma$ and $tau$ be finitedimensional representations of the WeilDeligne group of $F$. We give strong upper and lower bounds for the Artin and Swan exponents of $sigmaotimestau$ in terms of those of $sigma$ and $tau$. We give a different lower bound in terms of $sigmaotimeschecksigma$ and $tauotimeschecktau$. Using the Langlands correspondence, we obtain the bounds for RankinSelberg exponents.
3

143
167


Colin J.
Bushnell
United Kingdom
colin.bushnell@kcl.ac.uk


G.
Henniart
France
guy.henniart@math.upsud.fr
ocal Langlands correspondence
WeilDeligne groups and representations
tensor products
Artin exponent
Swan exponent
RankinSelberg exponent
On tensor product $L$functions and Langlands functoriality
2
2
In the spirit of the Langlands proposal on Beyond Endoscopy we discuss the explicit relation between the Langlands functorial transfers and automorphic $L$functions. It is wellknown that the poles of the $L$functions have deep impact to the Langlands functoriality. Our discussion also includes the meaning of the central value of the tensor product $L$functions in terms of the Langlands functoriality. This leads to the theory of the twisted automorphic descents for cuspidal automorphic representations of general classical groups.
3

169
189


D.
Jiang
United States of America
dhj55119@yahoo.com
Automorphic representations
$L$functions
Langlands functoriality
endoscopy
automorphic descent
The residual spectrum of $U(n,n)$; contribution from Borel subgroups
2
2
In this paper we study the residual spectrum of the quasisplit unitary group $G=U(n,n)$ defined over a number field $F$, coming from the Borel subgroups, $L_{dis}^2(G(F)backslash G(Bbb A))_T$. Due to lack of information on the local results, that is, the image of the local intertwining operators of the principal series, our results are incomplete. However, we describe a conjecture on the residual spectrum and prove a certain special case by using the KnappStein $R$group of the unitary group.
3

191
219


H.H.
Kim
Canada
henrykim@math.toronto.edu
Automorphic representation
spectral decomposition
residual spectrum
Computing local coefficients via types and covers: the example of $SL(2)$
2
2
We illustrate a method of computing LanglandsShahidi local coefficients via the theory of types and covers. The purpose of this paper is to illustrate a method of computing the LanglandsShahidi local coefficients using the theory of types and covers.
3

221
234


M.
Krishnamurthy
United States of America
muthukrishnamurthy@uiowa.edu


P.
Kutzko
United States of America
philipkutzko@uiowa.edu
Local factors
supercuspidal representations
types and covers
On the analytic properties of intertwining operators I: global normalizing factors
2
2
We provide a uniform estimate for the $L^1$norm (over any interval of bounded length) of the logarithmic derivatives of global normalizing factors associated to intertwining operators for the following reductive groups over number fields: inner forms of $operatorname{GL}(n)$; quasisplit classical groups and their similitude groups; the exceptional group $G_2$. This estimate is a key ingredient in the analysis of the spectral side of Arthur's trace formula. In particular, it is applicable to the limit multiplicity problem studied by the authors in earlier papers.
3

235
277


T.
Finis
Germany
finis@math.unileipzig.de


E.
Lapid
 
erez.m.lapid@gmail.com
Intertwining operators
$L$functions
automorphic forms
Caractérisation des paramètres d'Arthur, une remarque
2
2
In The endoscopic classification of representations, J. Arthur has proved the Langlands' classification for discrete series of padic classical groups. This uses endoscopy and twisted endoscopy. In this very short note, we remark that the normalization $rmgrave{a}$ la LanglandsShahidi of the intertwining operators, allows to avoid endoscopy. This is based on the intertwining relation which is a very important point of this book.
3

279
289


C.
Moeglin
France
colette.moeglin@imjprg.fr
Langlands' classification
discrete series
classical padic groups
intertwining operators
Distinguished positive regular representations
2
2
Let $G$ be a tamely ramified reductive $p$adic group. We study distinction of a class of irreducible admissible representations of $G$ by the group of fixed points $H$ of an involution of $G$. The representations correspond to $G$conjugacy classes of pairs $(T,phi)$, where $T$ is a tamely ramified maximal torus of $G$ and $phi$ is a quasicharacter of $T$ whose restriction to the maximal pro$p$subgroup satisfies a regularity condition. Under mild restrictions on the residual characteristic of $F$, we derive necessary conditions for $H$distinction of a representation corresponding to $(T,phi)$, expressed in terms of properties of $T$ and $phi$ relative to the involution. We prove that if an $H$distinguished representation arises from a pair $(T,phi)$ such that $T$ is stable under the involution and compact modulo $(Tcap H)Z$ (here, $Z$ is the centre of $G$), then the representation is $H$relatively supercuspidal.
3

291
311


F.
Murnaghan
Canada
fiona@math.toronto.edu
Distinguished
representation
relatively supercuspidal
On the transcendence of certain Petersson inner products
2
2
We show that for all normalized Hecke eigenforms $f$ with weight one and of CM type, the number $(f,f)$ where $(cdot, cdot )$ denotes the Petersson inner product, is a linear form in logarithms and hence transcendental.
3

313
316


M. Ram
Murty
Canada
murty@mast.queensu.ca


V. Kumar
Murty
Canada
murty@math.toronto.edu
CM modular forms
Petersson inner product
transcendence
Endoscopy and the cohomology of $GL(n)$
2
2
Let $G = {rm Res}_{F/mathbb{Q}}(GL_n)$ where $F$ is a number field. Let $S^G_{K_f}$ denote an ad`elic locally symmetric space for some level structure $K_f.$ Let ${mathcal M}_{mu,{mathbb C}}$ be an algebraic irreducible representation of $G({mathbb R})$ and we let $widetilde{mathcal{M}}_{mu,{mathbb C}}$ denote the associated sheaf on $S^G_{K_f}.$ The aim of this paper is to classify the data $(F,n,mu)$ for which cuspidal cohomology of $G$ with $mu$coefficients, denoted $H^{bullet}_{rm cusp}(S^G_{K_f}, widetilde{mathcal{M}}_{mu,{mathbb C}})$, is nonzero for some $K_f.$ We prove nonvanishing of cuspidal cohomology when $F$ is a totally real field or a totally imaginary quadratic extension of a totally real field, and also for a general number field but when $mu$ is a parallel weight.
3

317
335


C.
Bhagwat
India
cbhagwat@iiserpune.ac.in


A.
Raghuram
India
raghuram@iiserpune.ac.in
Locally symmetric spaces
cuspidal cohomology
On AtkinLehner correspondences on Siegel spaces
2
2
We introduce a higher dimensional AtkinLehner theory for SiegelParahoric congruence subgroups of $GSp(2g)$. Old Siegel forms are induced by geometric correspondences on Siegel moduli spaces which commute with almost all local Hecke algebras. We also introduce an algorithm to get equations for moduli spaces of SiegelParahoric level structures, once we have equations for prime levels and square prime levels over the level one Siegel space. This way we give equations for an infinite tower of Siegel spaces after N. Elkies who did the genus one case.
3

337
359


A.
Rastegar
Iran
rastegar1352@gmail.com
AtkinLehner theory
Siegel moduli space
old and new Siegel modular forms
Spatial statistics for lattice points on the sphere I: Individual results
2
2
We study the spatial distribution of point sets on the sphere obtained from the representation of a large integer as a sum of three integer squares. We examine several statistics of these point sets, such as the electrostatic potential, Ripley's function, the variance of the number of points in random spherical caps, and the covering radius. Some of the results are conditional on the Generalized Riemann Hypothesis.
3

361
386


J.
Bourgain
United States of America


Z.
Rudnick
 


P.
Sarnak
 
sarnakpeter@gmail.com
Sums of three squares
spatial statistics
Ripley's functions
On local gamma factors for orthogonal groups and unitary groups
2
2
In this paper, we find a relation between the proportionality factors which arise from the functional equations of two families of local RankinSelberg convolutions for irreducible admissible representations of orthogonal groups, or unitary groups. One family is that of local integrals of the doubling method, and the other family is that of local integrals expressed in terms of spherical Bessel models.
3

387
403


S.
Rallis
 


D.
Soudry
 
soudryhadas@gmail.com
Gamma factors
RankinSelberg convolutions
intertwining operators
Some bounds on unitary duals of classical groups  nonarchimeden case
2
2
We first give bounds for domains where the unitarizabile subquotients can show up in the parabolically induced representations of classical $p$adic groups. Roughly, they can show up only if the central character of the inducing irreducible cuspidal representation is dominated by the square root of the modular character of the minimal parabolic subgroup. For unitarizable subquotients supported by a fixed parabolic subgroup, or in a specific Bernstein component, a more precise bound is given. For the reductive groups of rank at least two, the trivial representation is always isolated in the unitary dual (D. Kazhdan). Still, we may ask if the level of isolation is higher in the case of the automorphic duals, as it is a case in the rank one. We show that the answer is negative to this question for symplectic $p$adic groups.
3

405
433


M.
Tadić
Croatia
tadic@math.hr
Irreducible unitary representations
parabolic induction
unitary dual
automorphic dual