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![Arxiv:Math/9911264V1 [Math.NT] 1 Nov 1999](https://docslib.b-cdn.net/cover/0514/arxiv-math-9911264v1-math-nt-1-nov-1999-100514.webp)
Arxiv:Math/9911264V1 [Math.NT] 1 Nov 1999
Annals of Mathematics, 150 (1999), 807–866 On explicit lifts of cusp forms from GLm to classical groups By David Ginzburg, Stephen Rallis, and David Soudry* Introduction In this paper, we begin the study of poles of partial L-functions LS(σ ⊗ τ,s), where σ ⊗ τ is an irreducible, automorphic, cuspidal, generic (i.e. with nontrivial Whittaker coefficient) representation of GA × GLm(A). G is a split classical group and A is the adele ring of a number field F . We also consider Sp2n(A) × GLm(A), where ∼ denotes the metaplectic cover. Examining LS(σ ⊗ τ,s) through the corresponding Rankin-Selberg, or Shimura-typef integrals ([G-PS-R], [G-R-S3], [G], [So]), we find that the global integral contains, in its integrand, a certain normalized Eisenstein series which is responsible for the poles. For example, if G = SO2k+1, then the Eisenstein series is on the adele points of split SO2m, induced from the Siegel parabolic s−1/2 subgroup and τ ⊗ | det ·| . If G = Sp2k (this is a convenient abuse of notation), then the Eisenstein series is on Sp2m(A), induced from the Siegel parabolic subgroup and τ ⊗ | det ·|s−1/2. Thef constant term (along the “Siegel radical”) of such a normalized Eisenstein series involves one of the L-functions LS(τ, Λ2, 2s − 1) or LS(τ, Sym2, 2s − 1). So, up to problems of normalization of intertwining operators, the only pole we expect, for Re(s) > 1/2, is at s = 1 and then τ should be self-dual. (See [J-S1], [B-G].) Thus, let us assume that τ is self-dual. -
![Arxiv:1903.00779V1 [Math.SP] 2 Mar 2019 O Clrsh¨Dne Prtr ( Schr¨Odinger Operators Scalar for Rn O P29177](https://docslib.b-cdn.net/cover/5707/arxiv-1903-00779v1-math-sp-2-mar-2019-o-clrsh%C2%A8dne-prtr-schr%C2%A8odinger-operators-scalar-for-rn-o-p29177-615707.webp)
Arxiv:1903.00779V1 [Math.SP] 2 Mar 2019 O Clrsh¨Dne Prtr ( Schr¨Odinger Operators Scalar for Rn O P29177
THE INVERSE APPROACH TO DIRAC-TYPE SYSTEMS BASED ON THE A-FUNCTION CONCEPT FRITZ GESZTESY AND ALEXANDER SAKHNOVICH Abstract. The principal objective in this paper is a new inverse approach to general Dirac-type systems of the form ′ y (x,z)= i(zJ + JV (x))y(x,z) (x ≥ 0), ⊤ where y = (y1,...,ym) and (for m1, m2 ∈ N) Im1 0m1×m2 0m1 v J = , V = ∗ , m1 + m2 =: m, 0m2×m1 −Im2 v 0m2 m ×m for v ∈ C1([0, ∞)) 1 2 , modeled after B. Simon’s 1999 inverse approach to half-line Schr¨odinger operators. In particular, we derive the A-equation associated to this Dirac-type system in the (z-independent) form x ∂ ∂ ∗ A(x,ℓ)= A(x,ℓ)+ A(x − t, ℓ)A(0,ℓ) A(t, ℓ) dt (x ≥ 0,ℓ ≥ 0). ∂ℓ ∂x ˆ0 Given the fundamental positivity condition ST > 0 in (1.14) (cf. (1.13) for details), we prove that this integro-differential equation for A( · , · ) is uniquely solvable for initial conditions 1 m ×m A( · , 0) = A( · ) ∈ C ([0, ∞)) 2 1 , m ×m and the corresponding potential coefficient v ∈ C1([0, ∞)) 1 2 can be recovered from A( · , · ) via ∗ v(ℓ) = −iA(0,ℓ) (ℓ ≥ 0). Contents 1. Introduction 1 2. Preliminaries 6 3. The A-Function for Dirac-Type Systems: Part I 7 4. The A-Function for Dirac-Type Systems: Part II 9 5. The A-Equation for Dirac-Type Systems 13 6. The Inverse Approach 15 arXiv:1903.00779v1 [math.SP] 2 Mar 2019 Appendix A. Various Results in Support of Step 2 in the Proof of Theorem 5.1 19 References 29 1. -
A New Approach to Inverse Spectral Theory, II. General Real Potentials
Annals of Mathematics, 152 (2000), 593–643 A new approach to inverse spectral theory, II. General real potentials and the connection to the spectral measure∗ By Fritz Gesztesy and Barry Simon Abstract We continue the study of the A-amplitude associated to a half-line d2 2 Schr¨odinger operator, 2 + q in L ((0, b)), b . A is related to the Weyl- − dx ≤∞ Titchmarsh m-function via m( κ2)= κ a A(α)e 2ακ dα+O(e (2a ε)κ) for − − − 0 − − − all ε> 0. We discuss five issues here. First,R we extend the theory to general q in L1((0, a)) for all a, including q’s which are limit circle at infinity. Second, we prove the following relation between the A-amplitude and the spectral measure 1 ρ: A(α) = 2 ∞ λ− 2 sin(2α√λ) dρ(λ) (since the integral is divergent, this − −∞ formula has toR be properly interpreted). Third, we provide a Laplace trans- form representation for m without error term in the case b < . Fourth, we ∞ discuss m-functions associated to other boundary conditions than the Dirichlet boundary conditions associated to the principal Weyl-Titchmarsh m-function. Finally, we discuss some examples where one can compute A exactly. 1. Introduction In this paper we will consider Schr¨odinger operators d2 (1.1) + q −dx2 in L2((0, b)) for 0 <b< or b = and real-valued locally integrable q. arXiv:math/9809182v2 [math.SP] 1 Sep 2000 ∞ ∞ There are essentially four distinct cases. ∗This material is based upon work supported by the National Science Foundation under Grant No. -
Generic Transfer from Gsp(4) to GL(4)
Compositio Math. 142 (2006) 541–550 doi:10.1112/S0010437X06001904 Generic transfer from GSp(4) to GL(4) Mahdi Asgari and Freydoon Shahidi Abstract We establish the Langlands functoriality conjecture for the transfer from the generic spec- trum of GSp(4) to GL(4) and give a criterion for the cuspidality of its image. We apply this to prove results toward the generalized Ramanujan conjecture for generic representations of GSp(4). 1. Introduction Let k be a number field and let G denote the group GSp(4, Ak). The (connected component of the) L-group of G is GSp(4, C), which has a natural embedding into GL(4, C). Langlands functoriality predicts that associated to this embedding there should be a transfer of automorphic representations of G to those of GL(4, Ak)(see[Art04]). Langlands’ theory of Eisenstein series reduces the proof of this to unitary cuspidal automorphic representations. We establish functoriality for the generic spectrum of GSp(4, Ak). More precisely (cf. Theorem 2.4), we prove the following. Let π be a unitary cuspidal representation of GSp(4, Ak), which we assume to be globally generic. Then π has a unique transfer to an automorphic representation Π of GL(4, Ak). 2 The transfer is generic (globally and locally) and satisfies ωΠ = ωπ and Π Π⊗ωπ. Here, ωπ and ωΠ denote the central characters of π and Π, respectively. Moreover, we give a cuspidality criterion for Π and prove that, when Π is not cuspidal, it is an isobaric sum of two unitary cuspidal representations of GL(2, Ak)(cf.Proposition2.2). -
On a Correspondence Between Cuspidal
JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 12, Number 3, Pages 849{907 S 0894-0347(99)00300-8 Article electronically published on April 26, 1999 ON A CORRESPONDENCE BETWEEN CUSPIDAL REPRESENTATIONS OF GL2n AND Sp2n f DAVID GINZBURG, STEPHEN RALLIS, AND DAVID SOUDRY Introduction Let K be a number field and A its ring of adeles. Let η be an irreducible, automorphic, cuspidal representation of GLm(A). Assume that η is self-dual. By Langlands conjectures, we expect that η is the functorial lift of an irreducible, automorphic, cuspidal representation σ of G(A), where G is an appropriate classical group defined over K. We know by [J.S.1] that the partial (as well as the complete) L-function LS(η η, s) has a (simple) pole at s = 1. Thus, since LS(η η, s)= LS(η, Sym2,s)LS⊗(η, Λ2,s), either LS(η, Λ2,s)orLS(η, Sym2,s) has a pole⊗ at s =1. If m =2n, then one expects that in the first case G =SO2n+1, and in the second S 2 case G =SO2n.Ifm=2n+ 1, we know that L (η, Λ ,s)isentire(see[J.S.2]), S 2 and hence L (η, Sym ,s) has a pole at s = 1. Here, one expects that G =Sp2n. In [G.R.S.1], we started a program to prove the existence of σ as above. See also [G.R.S.2], [G.R.S.3]. We proposed to prove this existence by explicit construc- tion. We constructed a space Vσ(η) of automorphic forms on G(A), invariant under right translations. -
Converse Theorem
A strengthening of the GL(2) converse theorem Andrew R. Booker and M. Krishnamurthy dedicated to Ilya Piatetski-Shapiro (1929{2009) Abstract We generalize the method of [Bo03] to prove a version of the converse theorem of Jacquet- Langlands with relaxed conditions on the twists by ramified id`eleclass characters. Contents 1 Introduction 1 1.1 Notation . 3 2 The classical case 5 3 Additive and multiplicative twists 8 3.1 Multiplicative to additive twists . 8 3.2 Additive to multiplicative twists . 9 4 Some generalities of the GL2 theory 12 4.1 New vectors at the finite places . 12 4.2 Archimedean Whittaker functions . 13 4.3 Bounds for the archimedean parameters . 16 4.4 Fourier coefficients and Dirichlet series . 16 5 Proof of Theorem 1.1 19 5.1 Reduction to additively twisted L-functions . 19 5.2 Automorphy relations from unramified twists . 20 5.3 Producing additive twists . 21 5.4 Taylor expansions . 22 5.5 Proof of Lemma 5.2 . 27 Appendix A. Binomial coefficients 38 1. Introduction The \converse theorem" in the theory of automorphic forms has a long history, beginning with the work of Hecke [He36] and a paper of Weil [We67] relating the automorphy relations satisfied by classical holomorphic modular forms f to analytic properties of the twisted L-functions L(s; f × χ) for Dirichlet characters χ. Soon after, the classical theory was recast in the modern setting of automorphic representations by Jacquet and Langlands [JL70], who generalized Weil's result to GL2 representations π over a global field, characterizing them in terms of their twists L(s; π ⊗ !) by id`eleclass characters !. -
Notices of the American Mathematical Society
OF THE 1994 AMS Election Special Section page 7 4 7 Fields Medals and Nevanlinna Prize Awarded at ICM-94 page 763 SEPTEMBER 1994, VOLUME 41, NUMBER 7 Providence, Rhode Island, USA ISSN 0002-9920 Calendar of AMS Meetings and Conferences This calendar lists all meetings and conferences approved prior to the date this issue insofar as is possible. Instructions for submission of abstracts can be found in the went to press. The summer and annual meetings are joint meetings with the Mathe· January 1994 issue of the Notices on page 43. Abstracts of papers to be presented at matical Association of America. the meeting must be received at the headquarters of the Society in Providence, Rhode Abstracts of papers presented at a meeting of the Society are published in the Island, on or before the deadline given below for the meeting. Note that the deadline for journal Abstracts of papers presented to the American Mathematical Society in the abstracts for consideration for presentation at special sessions is usually three weeks issue corresponding to that of the Notices which contains the program of the meeting, earlier than that specified below. Meetings Abstract Program Meeting# Date Place Deadline Issue 895 t October 28-29, 1994 Stillwater, Oklahoma Expired October 896 t November 11-13, 1994 Richmond, Virginia Expired October 897 * January 4-7, 1995 (101st Annual Meeting) San Francisco, California October 3 January 898 * March 4-5, 1995 Hartford, Connecticut December 1 March 899 * March 17-18, 1995 Orlando, Florida December 1 March 900 * March 24-25, -
![Arxiv:1412.3500V2 [Math.NT] 29 Oct 2015 Γ En Smrhc(Hn3) Ti Ojcueo Aqe Hti Sh It That Jacquet of Conjecture a Is It Let ([Hen93])](https://docslib.b-cdn.net/cover/7638/arxiv-1412-3500v2-math-nt-29-oct-2015-en-smrhc-hn3-ti-ojcueo-aqe-hti-sh-it-that-jacquet-of-conjecture-a-is-it-let-hen93-2787638.webp)
Arxiv:1412.3500V2 [Math.NT] 29 Oct 2015 Γ En Smrhc(Hn3) Ti Ojcueo Aqe Hti Sh It That Jacquet of Conjecture a Is It Let ([Hen93])
GAMMA FACTORS OF PAIRS AND A LOCAL CONVERSE THEOREM IN FAMILIES GILBERT MOSS Abstract. We prove a GL(n) × GL(n − 1) local converse theorem for ℓ-adic families of smooth representations of GLn(F ) where F is a finite extension of Qp and ℓ =6 p. Along the way, we extend the theory of Rankin-Selberg integrals, first introduced in [JPSS83], to the setting of families, continuing previous work of the author [Mos]. 1. Introduction Let F be a finite extension of Qp. A local converse theorem is a result along the ′ following lines: given V1 and V2 representations of GLn(F ), if γ(V1 × V ,X,ψ) = ′ ′ γ(V2×V ,X,ψ) for all representations V of GLn−1(F ), then V1 and V2 are the same. There exists such a converse theorem for complex representations: if V1, V2, and ′ V are irreducible admissible generic representations of GLn(F ) over C, “the same” means isomorphic ([Hen93]). It is a conjecture of Jacquet that it should suffice to ′ n let V vary over representations of GL n (F ), or in other words a GL(n)×GL(⌊ ⌋) ⌊ 2 ⌋ 2 converse theorem should hold. In this paper we construct γ(V ×V ′,X,ψ) and prove a GL(n)×GL(n−1) local converse theorem in the setting of ℓ-adic families. We deal with admissible generic families that are not typically irreducible, so “the same” will mean that V1 and V2 have the same supercuspidal support. Over families, there arises a new dimension to the local converse problem: determining the smallest coefficient ring over which the twisting representations V ′ can be taken while still having the theorem hold. -
Functorial Products for GL2× GL3 and the Symmetric Cube For
Annals of Mathematics, 155 (2002), 837–893 Functorial products for GL2× GL3 and the symmetric cube for GL2 By Henry H. Kim and Freydoon Shahidi* Dedicated to Robert P. Langlands Introduction In this paper we prove two new cases of Langlands functoriality. The first is a functorial product for cusp forms on GL GL as automorphic forms 2 × 3 on GL6, from which we obtain our second case, the long awaited functorial symmetric cube map for cusp forms on GL2. We prove these by applying a recent version of converse theorems of Cogdell and Piatetski-Shapiro to analytic properties of certain L-functions obtained from the method of Eisenstein series (Langlands-Shahidi method). As a consequence we prove the bound 5/34 for Hecke eigenvalues of Maass forms over any number field and at every place, finite or infinite, breaking the crucial bound 1/6 (see below and Sections 7 and 8) towards Ramanujan-Petersson and Selberg conjectures for GL2. As noted below, many other applications follow. To be precise, let π1 and π2 be two automorphic cuspidal representations of GL2(AF ) and GL3(AF ), respectively, where AF is the ring of ad`eles of a number field F . Write π1 = ⊗vπ1v and π2 = ⊗vπ2v. For each v, finite or otherwise, let π1v ⊠ π2v be the irreducible admissible representation of GL6(Fv), attached to (π1v,π2v) through the local Langlands correspondence by Harris-Taylor [HT], Henniart [He], and Langlands [La4]. We point out that, if ϕiv, i = 1, 2, are the two- and the three-dimensional representations of Deligne-Weil group, arXiv:math/0409607v1 [math.NT] 30 Sep 2004 parametrizing πiv, respectively, then π1v⊠π2v is attached to the six-dimensional representation ϕ ϕ . -
Automorphic Forms and L-Functions
Program of the Israel Science Foundation Workshop on Automorphic Forms and L-functions May 15-19, 2006, Rehovot and Tel-Aviv MONDAY, May 15 (Weizmann Institute, Schmidt Lecture Hall) 09:00 – 10:00 Registration of participants 10:00 – 12:30 Morning session on the trace formula James Arthur (University of Toronto) 12:30 LUNCH 14:30 – 15:30 Marie-France Vigneras (Institut Mathematiques de Jussieu) Irreducibility and cuspidality of the Steinberg representation modulo p 16:00 – 17:00 Prof. Chaim Leib Pekeris Memorial Lecture by Peter Sarnak (Princeton University) Equidistribution and Primes (Dolfi and Lola Ebner Auditorium) TUESDAY, May 16 (Tel-Aviv University, Melamed Auditorium) 08:30 – 9:30 Bus from San-Martin to Tel-Aviv University 10:00 – 12:30 Morning session on L-functions Daniel Bump (Stanford University) James Cogdell (Ohio State University) 12:30 LUNCH 14:00 – 15:00 Akshay Venkatesh (Courant Institute) A spherical simple trace formula, and Weyl’s law for cusp forms 15:30 – 16:30 Wee Teck Gan (University of California, San Diego) The regularized Siegel-Weil formula for exceptional groups 17:00 – 18:00 Jiu-Kang Yu (Purdue University) Construction of tame types WEDNESDAY, May 17 (Weizmann Institute, Schmidt Lecture Hall) 10:00 – 12:30 Morning session on theta correspondence Roger Howe (Yale University) Stephen Kudla (University of Maryland) 13:00 – 22:00 A tour and dinner in Jerusalem (bus leaves from San-Martin) THURSDAY, May 18 (Weizmann Institute, Schmidt Lecture Hall) 10:00 – 11:00 Birgit Speh (Cornell University) The restriction -
Curriculum Vitae
Isaac B. Michael Curriculum Vitae Education 2019 Ph.D. in Mathematics, Baylor University, Waco, TX. Advisor: Fritz Gesztesy Co-Advisor: Lance Littlejohn Thesis: On Birman–Hardy–Rellich-type Inequalities 2015 M.S. in Mathematics, Baylor University, Waco, TX. 4.0 GPA 2013 B.S. in Mathematics, Tarleton State University, Stephenville, TX. 3.3 GPA; 3.9 Institutional GPA 8-12 Math Teaching Certification Research Interests Primary Analysis (Real, Complex, and Functional), Differential Equations (Ordi- nary and Partial), Spectral Theory, Operator Theory Secondary Deterministic Dynamical Systems, Semigroups, Functional Calculus, Calculus of Variations, Lie Theory Publications and Preprints [1] Net Regular Signed Trees, with M. Sepanski, Australasian Journal of Combinatorics 66(2), 192-204 (2016). [2] On Birman’s Sequence of Hardy–Rellich-Type Inequalities, with F. Gesztesy, L. Littlejohn, and R. Wellman, Journal of Differential Equations 264(4), 2761-2801 (2018). [3] Radial and Logarithmic Refinements of Hardy’s Inequality, with F. Gesztesy, L. Littlejohn, and M. M. H. Pang, Algebra i Analiz, 30(3), 55–65 (2018) (Russian), St. Petersburg Math. J., St. 30, 429–436 (2019) (English). Louisiana State University – Department of Mathematics H (254) 366-6861 • B [email protected] Í math.lsu.edu/∼imichael 1/9 [4] On Weighted Hardy-Type Inequalities, with C. Y. Chuah, F. Gesztesy, L. Littlejohn, T. Mei, and M. M. H. Pang, Math. Ineq. & App. 23(2), 625-646 (2020). [5] A Sequence of Weighted Birman–Hardy–Rellich-type Inequalities with Loga- rithmic Refinements, with F. Gesztesy, L. Littlejohn, and M. M. H. Pang (submitted for publication). [6] Optimality of Constants in Weighted Birman–Hardy–Rellich Inequalities with Logarithmic Refinements, with F. -
On $(\Chi, B) $-Factors of Cuspidal Automorphic Representations Of
ON (χ, b)-FACTORS OF CUSPIDAL AUTOMORPHIC REPRESENTATIONS OF UNITARY GROUPS I DIHUA JIANG AND CHENYAN WU Dedicated to Wen-Ching Winnie Li Abstract. Following the idea of [GJS09] for orthogonal groups, we introduce a new family of period integrals for cuspidal auto- morphic representations σ of unitary groups and investigate their relation with the occurrence of a simple global Arthur parameter (χ,b) in the global Arthur parameter ψσ associated to σ, by the endoscopic classification of Arthur ([Art13], [Mok13], [KMSW14]). The argument uses the theory of theta correspondence. This can be viewed as a part of the (χ,b)-theory outlined in [Jia14] and can be regarded as a refinement of the theory of theta correspondences and poles of certain L-functions, which was outlined in [Ral91]. 1. Introduction Let F be a number field and E be a quadratic extension of F . We denote by A = AF the ring of adeles of F , and by AE that of E. Let G be a unitary group associated to an m-dimensional skew-Hermitian vector space X. Consider σ in Acusp(G), the set of equivalence classes of irreducible cuspidal automorphic representations of G that occurs in the discrete spectrum, following the notation of [Art13]. Let χ be an automorphic character of GL1(AE). The tensor product L-functions L(s, σ × χ) has been investigated through the work of Li in [Li92b] arXiv:1409.0767v3 [math.NT] 1 Dec 2014 by using the doubling method of Piatetski-Shapiro and Rallis, and the work of Langlands by calculating constant terms of the Eisenstein series with cuspidal support χ ⊗ σ ([Lan71] and [Sha10]).