Bulletin of Iranian Mathematical Society
Volume:43 Issue: 4, 2017

Let G be a reductive p-adic groupý. ýWe consider the general questioný ýof whether the reducibility of an induced representation can be detected in aý ý``co-rank oneý" ýsituationý. ýFor smooth complex representations induced from supercuspidalý ýrepresentationsý, ýwe show that a sufficient condition is the existence of a subquotientý ýthat does not appear as a subrepresentationý. ýAn important example is the Langland's quotientý. ýIn additioný, ýwe study the same general question for continuous principal series on p-adic Banach spacesý. ýAlthough we do not give an answer in this caseý, ýwe describeý ýa related filtration on the corresponding Iwasawa modulesý.

ýThis paper gives several examples of the basic functionsý ýintroduced in recent years by Ng\^oý. ýThese are mainlyý ýconjectures based on computer experimentý.

This paper extends to the pro-p Iwahori subgroup of GL(2) over an unramified finite extension of Qp the presentation of the Iwasawa algebra obtained earlier by the author for the congruence subgroup of level one of SL(2ý,ýZp)ý. ýIt then describes a natural base change map between the Iwasawa algebras or more correctlyý, ýas it turns outý, ýbetween the global distribution algebras on the associated rigid-analytic spacesý. ýIn a forthcoming paper this will be applied to p-adic representation theoryý.

ýIn this paperý, ýwe propose some Diffie-Hellman type key exchange protocols using isogenies of elliptic curvesý. ýThe first method which uses the endomorphism ring of an ordinary elliptic curve Eý, ýis a straightforward generalization of elliptic curve Diffie-Hellman key exchangeý. ýThe method uses commutativity of the endomorphism ring End(E)ý. ýThen using dual isogeniesý, ýwe propose a second methodý. ýThis case uses the endomorphism ring of an elliptic curve Eý, ýwhich can be ordinary or supersingularý. ýWe extend this method using isogenies between two elliptic curves E and E′ý. ýOur methods have the security level of that of [Dý. ýJao and Lý. ýDe Feoý, ýTowards quantum-resistant cryptosystems from supersingular elliptic curve isogeniesý, Jý. ýMathý. ýCryptol. 8 (2014)ý, ýnoý. ý3ý, ý209--247]ý, ýwith the advantage of transmitting less information between two partiesý.

We study the automorphic theta representation Θ(r)2n on the r-fold cover of the symplectic group Sp2ný. ýThis representation is obtained from the residues of Eisenstein series on this groupý.
ýIf r is oddý, ýn≤r

ýWe study R-groups for p-adic inner forms of quasi-split special unitary groupsý. ýWe prove Arthur's conjectureý, ýthe isomorphism between the Knapp-Stein R-group and the Langlands-Arthur R-groupý, ýfor quasi-split special unitary groups and their inner formsý. ýFurthermoreý, ýwe investigate the invariance of the Knapp-Stein R-group within L-packets and between inner formsý. ýThis 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 formsý.

Let F be a non-Archimedean locally compact fieldý. ýLet σ and τ be finite-dimensional representations of the Weil-Deligne group of Fý. ýWe give strong upper and lower bounds for the Artin and Swan exponents of σ⊗τ in terms of those of σ and τý. ýWe give a different lower bound in terms of σ⊗σˇ and τ⊗τˇý. ýUsing the Langlands correspondenceý, ýwe obtain the bounds for Rankin-Selberg exponentsý.

In the spirit of the Langlands proposal on Beyond Endoscopy ýwe discuss the explicit relation between the Langlands functorial transfers and automorphic L-functionsý. ýIt is well-known that the poles of the L-functions have deep impact to the Langlands functorialityý. ýOur discussion alsoý ýincludes the meaning of the central value of the tensor product L
-functions in terms of the Langlands functorialityý. ýThisý ýleads to the theory of the twisted automorphic descents for cuspidal automorphic representations of general classical groupsý.

In this paper we study the residual spectrum of the quasi-split unitary group G=U(n,n) defined over a number field Fý, ýcoming from the Borel subgroupsý, ýL2dis(G(F)∖G(A))Tý. ýDue to lack of information on the local resultsý, ýthat isý, ýthe image of the local intertwining operators of the principal seriesý, ýour results are incompleteý. ýHoweverý, ýwe describe a conjecture on the residual spectrum and prove a certain special case by using the Knapp-Stein R-group of the unitary groupý.

We illustrate a method of computing Langlands-Shahidi local coefficients via the theory of types and coversý.
ýThe purpose of this paper is to illustrate a method of computing the Langlands-Shahidi local coefficients using the theory of types and covers.

We provide a uniform estimate for the L1-norm (over any interval of bounded length) of the logarithmic derivativesý ýof global normalizing factors associated to intertwining operators for the following reductive groups over number fieldsý: ýinner forms of GL(n); quasi-split classical groups and their similitude groups; the exceptional group G2ý. ýThis estimate is a key ingredient in the analysis of the spectral side of Arthur's trace formulaý. ýIn particularý, ýit is applicable to the limit multiplicity problem studied by the authors in earlier papersý.

In The endoscopic classification of representationsýý, ýJý. ýArthur has proved the Langland's classification for discrete series of p-adic classical groupsý. ýThis uses endoscopy and twisted endoscopyý. ýIn this very short noteý, ýwe remark that the normalization ýýýa`ýý la Langlands-Shahidi of the intertwining operatorsý, ýallows to avoid endoscopyý. ýThis is based on the intertwining relation which is a very important point of this booký.

Let G be a tamely ramified reductive p-adicý ýgroupý. ýWe study distinction of a class of irreducible admissible representationsý ýof G by the group of fixed points H of an involutioný ýof Gý. ýThe representations correspond to G-conjugacy classes ofý ýpairs (T,ϕ)ý, ýwhere T is aý ýtamely ramified maximal torus of G and ϕ is a quasicharacterý ýof T whose restriction to the maximal pro-p-subgroupý ýsatisfies a regularity conditioný.
ýUnder mild restrictions on the residual characteristic ofý ýFý, ýwe derive necessary conditions for H-distinction ofý ýa representation corresponding to (T,ϕ)ý, ýexpressed in terms of properties of T and ϕý ýrelative to the involutioný.
ýWe prove that if an H-distinguished representation arises fromý ýa pair (T,ϕ) such that T is stable under the involution andý ýcompact modulo (T∩H)Z (hereý, ýZ is the centre ofýýG)ý, ýthen the representation is H-relatively supercuspidalý.

We show that for all normalized Hecke eigenforms fý ýwith weight one and of CM typeý, ýthe number (f,f) where (⋅ý,ý⋅) denotesý ýthe Petersson inner productý, ýis a linear form in logarithms andý ýhence transcendentalý.

Let G=ResF/Q(GLn) where F is a number fieldý. ýLet SGKf denote an ad\`elic locally symmetric space for some level structure Kf. Let Mμ,C be an algebraic irreducible representation of G(R) and we let M˜μ,C denote the associated sheaf on SGKf. The aim of this paper is to classify the data (F,n,μ) for which cuspidal cohomology of G with μ-coefficientsý, ýdenoted H∙cusp(SGKfý,ýM˜μ,C)ý, ýis nonzero for some Kf. We prove nonvanishing of cuspidal cohomology when F is a totally real field or a totally imaginary quadratic extension of a totally real fieldý, ýand also for a general number field but when μ
is a parallel weightý.

We introduce a higher dimensional Atkin-Lehner theory forý ýSiegel-Parahoric congruence subgroups of GSp(2g)ý. ýOldý ýSiegel forms are induced by geometric correspondences on Siegelý ýmoduli spaces which commute with almost all local Hecke algebrasý. ýWe also introduce an algorithm to get equations for moduli spaces ofý ýSiegel-Parahoric level structuresý, ýonce we have equations for prime levels and square prime levelsý ýover the level one Siegel spaceý. ýThis way we give equations for an infinite tower of Siegel spacesý ýafter Ný. ýElkies who did the genus one caseý.

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 squaresý. ýWe examine several statistics of these point setsý, ýsuch as the electrostatic potentialý, ýRipley's functioný, ýthe variance of the number of points in random spherical capsý, ýand the covering radiusý. ýSome of the results are conditional on the Generalized Riemann Hypothesisý. ý

ýIn this paperý, ýwe find a relation between the proportionality factors which arise from the functional equations of two families of local Rankin-Selberg convolutions forý ýirreducible admissible representations of orthogonal groupsý, ýor unitary groupsý. ýOne family is that of local integrals of the doubling methodý, ýand the other family isý ýthat of local integrals expressed in terms of spherical Bessel modelsý.

We first give bounds for domains where the unitarizabile subquotients can show up in the parabolically induced representations of classical p-adic groupsý. ýRoughlyý, ýthey can show up only if theý ýcentral character of the inducing irreducible cuspidal representation is dominated by theý ýsquare root of the modular character of the minimal parabolic subgroupý. ýFor unitarizable subquotients supported by a fixed parabolic subgroupý, ýor in a specific Bernstein componentý, ýa more precise bound is givený.
ýFor the reductive groups of rank at least twoý, ýthe trivial representation is always isolated in the unitary dual (Dý. ýKazhdan)ý. ýStillý, ýwe may ask if the level of isolation is higher in the case of the automorphic dualsý, ýas it is a case in the rank oneý. ýWe show that the answer is negative to this question for symplectic p-adic groupsý.