Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X43Issue 4 (Special Issue)20170830Filtrations of smooth principal series and Iwasawa modules3161153ENW. AlsibianiD. BanJournal Article20170513Let $G$ be a reductive $p$-adic group. We consider the general question of whether the reducibility of an induced representation can be detected in a ``co-rank 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.<br /> Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X43Issue 4 (Special Issue)20170830Symmetric powers and the Satake transform17541154ENB. CasselmanJournal Article20170513This paper gives several examples of the basic functions introduced in recent years by Ng^o. These are mainly conjectures based on computer experiment.<br /> Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X43Issue 4 (Special Issue)20170830Globally analytic $p$-adic representations of the pro--$p$--Iwahori subgroup of $GL(2)$ and base change, I : Iwasawa algebras and a base change map55761155ENL. ClozelJournal Article20170513This 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 rigid-analytic spaces. In a forthcoming paper this will be applied to $p$-adic representation theory.Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X43Issue 4 (Special Issue)20170830Diffie-Hellman type key exchange protocols based on isogenies77881156ENH. DaghighR. Khodakaramian GilanF. Seifi ShahparJournal Article20170513In this paper, we propose some Diffie-Hellman 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 Diffie-Hellman 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 quantum-resistant cryptosystems from supersingular elliptic curve isogenies, J. Math. Cryptol. 8 (2014), no. 3, 209--247], with the advantage of transmitting less information between two parties.
Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X43Issue 4 (Special Issue)20170830Theta functions on covers of symplectic groups891161157ENS. FriedbergD. GinzburgJournal Article20171001We 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
$sigma_{2n-r+1}^{(2r)}$ on the $2r$-fold cover of $Sp_{2n-r+1}$. 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 six-fold cover of $Sp_4$, unconditionally. We expect that in fact the representation constructed here, $sigma_{2n-r+1}^{(2r)}$, is precisely $Theta_{2n-r+1}^{(2r)}$; that is, we conjecture relations between theta representations on different covering groups.
Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X43Issue 4 (Special Issue)20170830Behavior of $R$-groups for $p$-adic inner forms of quasi-split special unitary groups1171411158ENK. ChoiyD. GoldbergJournal Article20170513We study $R$-groups for $p$-adic inner forms of quasi-split special unitary groups. We prove Arthur's conjecture, the isomorphism between the Knapp-Stein $R$-group and the Langlands-Arthur $R$-group, for quasi-split special unitary groups and their inner forms. Furthermore, we investigate the invariance of the Knapp-Stein $R$-group within $L$-packets and between inner forms. This work is applied to transferring known results in the second-named author's earlier work for quasi-split special unitary groups to their non-quasi-split inner forms.Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X43Issue 4 (Special Issue)20170830Strong exponent bounds for the local Rankin-Selberg convolution1431671159ENColin J. BushnellG. HenniartJournal Article20170515Let $F$ be a non-Archimedean locally compact field. Let $sigma$ and $tau$ be finite-dimensional representations of the Weil-Deligne 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 Rankin-Selberg exponents.Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X43Issue 4 (Special Issue)20170830On tensor product $L$-functions and Langlands functoriality1691891160END. JiangJournal Article20170515In 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 well-known 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.<br /> Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X43Issue 4 (Special Issue)20170830The residual spectrum of $U(n,n)$; contribution from Borel subgroups1912191161ENH.H. KimJournal Article20170515In this paper we study the residual spectrum of the quasi-split 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 Knapp-Stein $R$-group of the unitary group.Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X43Issue 4 (Special Issue)20170830Computing local coefficients via types and covers: the example of $SL(2)$2212341162ENM. KrishnamurthyP. KutzkoJournal Article20170515We illustrate a method of computing Langlands-Shahidi local coefficients via the theory of types and covers.<br /> The purpose of this paper is to illustrate a method of computing the Langlands-Shahidi local coefficients using the theory of types and covers.Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X43Issue 4 (Special Issue)20170830On the analytic properties of intertwining operators I: global normalizing factors2352771163ENT. FinisE. LapidJournal Article20171001We 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 $operatorname{GL}(n)$; quasi-split 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.Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X43Issue 4 (Special Issue)20170830Caractérisation des paramètres d'Arthur, une remarque2792891164ENC. MoeglinJournal Article20170515In The endoscopic classification of representations, J. Arthur has proved the Langlands' classification for discrete series of p-adic classical groups. This uses endoscopy and twisted endoscopy. In this very short note, we remark that the normalization $rmgrave{a}$ la Langlands-Shahidi of the intertwining operators, allows to avoid endoscopy. This is based on the intertwining relation which is a very important point of this book.Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X43Issue 4 (Special Issue)20170830Distinguished positive regular representations2913111165ENF. MurnaghanJournal Article20170515Let $G$ 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 <br /> 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.<br /> <br /> 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.<br /> <br /> 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<br /> $G$), then the representation is $H$-relatively supercuspidal.Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X43Issue 4 (Special Issue)20170830On the transcendence of certain Petersson inner products3133161166ENM. Ram MurtyV. Kumar MurtyJournal Article20170515We 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.Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X43Issue 4 (Special Issue)20170830Endoscopy and the cohomology of $GL(n)$3173351167ENC. BhagwatA. RaghuramJournal Article20170515Let $G = {rm Res}_{F/mathbb{Q}}(GL_n)$ 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.Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X43Issue 4 (Special Issue)20170801On Atkin-Lehner correspondences on Siegel spaces3373591168ENA. RastegarJournal Article20170515We introduce a higher dimensional Atkin-Lehner theory for Siegel-Parahoric 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 Siegel-Parahoric 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. Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X43Issue 4 (Special Issue)20170830Spatial statistics for lattice points on the sphere I: Individual results3613861169ENJ. BourgainZ. RudnickP. SarnakJournal Article20171001We 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. Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X43Issue 4 (Special Issue)20170830On local gamma factors for orthogonal groups and unitary groups3874031170ENS. RallisD. SoudryJournal Article20171001In this paper, we find a relation between the proportionality factors which arise from the functional equations of two families of local Rankin-Selberg 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. Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X43Issue 4 (Special Issue)20170830Some bounds on unitary duals of classical groups - non-archimeden case4054331171ENM. TadićJournal Article20170515We 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. <br /> <br /> 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.