Automorphic Forms and Representations Study Group (Current)
We will study the basic theory of automorphic forms and automorphic representations, including the spectral decomposition, Whittaker models, the adelization of modular forms and the theory of Eisenstein series. We will follow the text "Automorphic Forms and Representations" by Daniel Bump. There will be weekly meetings which will include presentation and discussion of the material and exercises. The main prerequisites are algebraic number theory and basic algebra, analysis and topology.
Organizers:
Prof. Gautam Chinta
Ajith Nair
Ajmain Yamin
Spring 2024 schedule
In Spring 2024, we are meeting every Wednesday starting January 31. All meetings will be hybrid. Please send an email to: anair@gradcenter.cuny.edu for the Zoom link.
January 31: Basic representation theory (speaker: Nathaniel Kingsbury)
Nathaniel gave a recap of section 2.4 Basic Representation theory and a preview of the topics coming up in the section.
February 7: Basic representation theory (speaker: Nathaniel Kingsbury)
Nathaniel proved Theorem 2.4.2 which states that under some technical hypothesis, the dimension of the subspace of K-fixed vectors of an irreducible unitary representation of G is at most one, where G=GL(n,R) and K=O(n) or G=GL(n,R)+ and K=SO(n). He also proved Theorem 2.4.3 and few corollaries.
February 14: Basic representation theory (speaker: Nathaniel Kingsbury)
Nathaniel defined the notion of K-finite vectors and proved Propositions 2.4.4 and 2.4.5.
February 21: Basic representation theory (speaker: Nathaniel Kingsbury)
Nathaniel defined the notion of (g,K)-module for GL(n,R) and Ajmain presented the solution for exercise 2.4.1.
February 28: Irreducible (g,K)-modules for GL(2,R) (speaker: Ajith Nair)
Ajith proved Prop. 2.5.1 and presented Prop. 2.5.1 and Theorems 2.5.1 and 2.5.2 without proofs. He also defined the notion of K-types.
March 6: Irreducible (g,K)-modules for GL(2,R) (speaker: Ajith Nair)
Ajith described the construction of the Hilbert space representation of G=GL(2,R)+ whose space of K-finite vectors contain explicit realizations of the irreducible admissible (g,K)-modules of G. He stated theorem 2.5.3 that describes these (g,K)-modules.
March 13: Irreducible (g,K)-modules for GL(2,R) (speaker: Ajith Nair)
After a brief summary of the material covered in the section so far, Ajith stated Prop. 2.5.5 which gives the connection between irreducible representations of compact Lie groups G and H, where H is an index two subgroup of G.
March 20: Irreducible (g,K)-modules for GL(2,R) (speaker: Ajith Nair)
Ajith presented Theorem 2.5.5 which classifies the irreducible (g,K)-modules of GL(2,R).
March 27: Unitaricity and Intertwining Integrals (speaker: Ajmain Yamin)
Ajmain proved Theorem 2.6.6 which shows that infinitesimal equivalence of irreducible unitary representations of GL(2,R)+ implies equivalence. He also proved Theorems 2.6.2 and 2.6.3 which describe the values of lambda,mu and epsilon for which the corresponding principal series representation of GL(2,R)+ has a unitary representative.
April 3: Unitaricity and Intertwining integrals (speaker: Ajmain Yamin)
Ajmain proved Theorem 2.6.4 which asserts the unitaricity of complementary series. The proof uses the technique of intertwining integrals.
April 17: Unitaricity and Intertwining integrals (speaker: Ajmain Yamin)
Ajmain showed that the only finite-dimensional irreducible unitary representations of GL(2,R)+ are the one dimensional characters. He also discussed the proofs of unitaricity of the discrete series and limits of discrete series representations.
May 1: Representations and the Spectral problem (speaker: Ajith Nair)
Ajith presented the relationship between the spectral problem, as outlined in section 2.1, and the representation theory of GL(2,R).
May 8: Whittaker Models and a theorem of Harish-Chandra (speaker: Nathaniel Kingsbury)
Fall 2023 schedule
We will continue reading and presenting on Chapter 2 of Bump. This semester, we will meet every alternate Friday, starting from September 1st. All meetings will be hybrid. Please send an email to: anair@gradcenter.cuny.edu for the Zoom link.
September 1: A recap of where we are (speaker: Ajmain Yamin)
September 15: Discreteness of the spectrum (speaker: Ajith Nair)
Ajith proved Prop. 2.3.2 which asserts that any nonzero closed invariant subspace H of L^2(Gamma\G,Chi) has a Hilbert space direct sum decomposition into subspaces H_k and the Laplace-Beltrami operator acting on the space H_k has a smooth nonzero eigenvector.
September 29: Discreteness of the spectrum (speaker: Ajith Nair)
Ajith presented Theorem 2.3.3 which states that L^2(Gamma\G,Chi) decomposes into irreducible G-invariant subspaces under the right regular representation.
October 13: Discreteness of the spectrum (speaker: Ajith Nair)
Ajith presented theorem 2.3.4 and the corollary after that which states that L^2(Gamma\H,Chi,k) decomposes into eigenspaces of the weight k Laplacian operator.
October 27: Discreteness of the spectrum (speaker: Ajmain Yamin)
Ajmain showed how to prove the discreteness of the spectrum of the Laplacian on a compact quotient of the hyperbolic plane using the technique of Green's function.
November 10: Basic representation theory (speaker: Nathaniel Kingsbury)
Nathaniel defined the notion of smooth vectors (in a Hilbert space H) for a representation (pi, H) of G=GL(n,R) or its connected component. He showed that the subspace H_infinity of smooth vectors is dense in H, is invariant under the action of G, and is a Lie algebra representation for the Lie algebra g=gl(n,R).
December 8: Basic representation theory (speaker: Nathaniel Kingsbury)
Spring 2023 schedule
We will start reading and presenting on Chapter 2 of Bump which concerns Automorphic forms and representations of GL(2,R). All meetings will be hybrid. Please send an email to: anair@gradcenter.cuny.edu for the Zoom link.
February 1: 1) Summary of chapter 1 (speaker: Ajith Nair)
2) Section 2.1 (Maass forms and the Spectral Problem) (speaker: Ajmain Yamin)
February 8: Maass forms and the spectral problem (speaker: Ajmain Yamin and Ajith Nair)
Ajmain defined the raising and lowering operators and the weight k Laplacian acting on smooth functions on upper half plane. He then presented the proofs of Lemma 2.1.1 and Prop. 2.1.2. Ajith defined operators defined on dense subspaces of a Hilbert space, self-adjoint operators and derived Green's formula.
February 15: Maass forms and the spectral problem (speaker: Ajith Nair)
Ajith presented Prop. 2.1.1, lemma 2.1.3 and Prop. 2.1.4, showing that weight k Laplacian on the space L^2(Gamma\H, Chi, k) is a symmetric operator and stated the version of the spectral problem for the Laplacian on this space.
February 22: Maass forms and the spectral problem (speaker: Ajmain Yamin)
Ajmain presented about Haar measure on topological groups, Riesz representation theorem on positive Borel measures on a space X and positive linear functionals on C_c(X) and presented the proof of Prop. 2.1.5.
March 1: Maass forms and the spectral problem (speaker: Ajith Nair)
Ajith defined the right regular representation of G=GL(2,R)+ on L^2(Gamma\G,Chi) and presented the proof of isomorphism between L^2(Gamma\G,Chi,k) and L^2(Gamma\H, Chi,k).
March 8: Exercises of section 2.1 (speaker: NA)
March 15: Basic Lie theory (speaker: Ajith Nair)
Ajith briefly discussed Lie algebras and Lie groups and showed that the differential of a representation of G=GL(n,R)+ is a representation of the Lie algebra of nxn real matrices, gl(n,R). He also proved Prop. 2.2.2.
March 22: Basic Lie theory (speaker: Ajmain Yamin)
Ajmain explained the construction of a Lie algebra of a Lie group and the representation of the elements of universal enveloping algebra U(g) as left-invariant differential operators on G=GL(n,R)+. He then proved Prop. 2.2.4 which states that central elements in U(g) are also right-invariant as differential operators. Lastly, he proved Theorem 2.2.1.
March 29: Basic Lie theory (speaker: Ajmain Yamin)
Ajmain defined the complexification of a real Lie algebra and proved Prop. 2.2.5 which shows that the raising and lowering operators and the Laplace-Beltrami operator defined in the group setting in section 2.1 can be seen as elements in the complexified universal enveloping algebra of gl(2,R).
April 5 and 12: No meeting on account of Spring Break
April 19: Discreteness of the spectrum (speaker: Ajith Nair)
Ajith proved the Spectral theorem for compact self-adjoint operators.
April 26: Discreteness of the spectrum (speaker: Ajith Nair)
Ajith proved Lemma 2.3.1 and Prop. 2.3.1 about Hilbert Schmidt integral operators and stated Prop. 2.3.2.
May 3: Discreteness of the spectrum (speaker: Ajmain Yamin)
Fall 2022 schedule
VENUE: Math Thesis room (4214.03)
31st August: Brief recap and overview of Rankin-Selberg method (speaker: Ajmain Yamin)
Ajmain provided a summary of the material covered so far (sections 1.1-1.5).
7th September: Overview of Rankin-Selberg method (speaker: Ajith Nair)
14th September: Section 1.6 (Rankin-Selberg method) exercises
21st September: Section 1.7 Hecke characters and Hilbert modular forms (speaker: Ajmain Yamin)
Ajmain will start presenting section 1.7.
28th September: Hecke characters and Hilbert modular forms (speaker: Ajmain Yamin)
Ajmain presented the proof of theorem 1.7.2 about the properties of the Hecke L-function of a totally real field of narrow class number 1.
12th October: Hilbert modular forms (speaker: Ajith Nair)
Ajith will start presenting on Hilbert modular forms.
26th October: Hilbert modular forms (speaker: Ajith Nair)
Ajith will continue presenting on Hilbert modular forms.
2nd November: Exercises on Hilbert modular forms
9th November: Artin L-functions and Langlands functoriality (speaker: Nathaniel Kingsbury)
16th November: Artin L-functions and Langlands functoriality (speaker: Nathaniel Kingsbury)
23rd November: Maass forms (speaker: Ajith Nair)
30th November: Maass forms (speaker: Ajith Nair and Ajmain Yamin)
7th December: Maass forms (speaker: Ajmain Yamin)
9th January, 2023: Maass forms (speaker: Ajmain Yamin)
Ajmain presented the proof of construction of a Maass form for the subgroup Gamma_0(D) associated to a Hecke character.
19th January, 2023: Base change (speaker: Ajith Nair)
Ajith described the method of proof of Theorem 1.7.3 about base change and presented the proof of Lemma 1.10.1.
25th January, 2023: Base Change (speaker: Ajith Nair)
We discussed certain Eisenstein series and the proof of Theorem 1.7.3.
Spring 2022 schedule
VENUE: Math Thesis room (4214.03)
18th February: Twisted L-functions (speaker: Ajmain Yamin)
We will discuss some exercises from section 1.4 which introduce the notions of twisted modular forms. Ajmain will then start presenting on section 1.5 - Twisted L-functions.
25th February: Twisted L-functions (speaker: Ajmain Yamin)
Ajmain will start presenting section 1.5 on twisted L-functions.
4th March: Twisted L-functions (speaker: Ajmain Yamin)
Last week, we started section 1.5 with the topics- converse theorem and Mellin transform. Ajmain will continue presenting from section 1.5 on twisted L-functions.
11th March: Twisted L-functions (speaker: Ajmain Yamin)
Last week, Ajmain proved Prop. 1.5.1. This week, he will talk about twisted modular forms and twisted L-functions.
18th March: Twisted L-functions (speaker: Ajmain Yamin)
Last week, we discussed exercise 1.4.6 and twisted modular forms. This week, Ajmain will continue with twisted modular forms and twisted L-functions.
25th March: Twisted L-functions (speaker: Ajmain Yamin)
In the previous meeting and in this week's meeting Ajmain proved the functional equation for twisted L-functions i.e L-functions twisted by a primitive Dirichlet character.
31st March: Weil's converse theorem (speaker: Ajith Nair)
This week, Ajith will present Weil's converse theorem which proves the existence of a cusp form given sufficiently many functional equations.
8th April: Introduction to Multiple Dirichlet series (speaker: Prof. Gautam Chinta)
Multiple Dirichlet series is an active area of research with connections to Riemann hypothesis and random matrix theory. Prerequisites are basics of Dirichlet L-functions such as their analytic continuation and functional equations.
15th April (virtual): Weil's converse theorem (speaker: Ajith Nair)
Ajith will continue the proof of converse theorem.
Zoom link: https://gc-cuny-edu.zoom.us/j/81538553627?pwd=ZmQ2bWxicGR4Q1o1RlJTR2RvdlRZQT09
Time: 12:30 pm
22nd April (virtual): Multiple Dirichlet series (speaker: Prof. Gautam Chinta)
Zoom link: https://gc-cuny-edu.zoom.us/j/85178388125?pwd=NitZYjRJeldLVysyT3k3bmJuZm1Udz09
Time: 10:00 am
29th April : Weil's converse theorem (speaker: Ajith Nair)
Ajith will continue the proof of converse theorem.
6th May: Multiple Dirichlet series (speaker: Prof. Gautam Chinta)
Zoom link: https://gc-cuny-edu.zoom.us/j/81117913261?pwd=Zi9BWlFtclgvbGdTQUF4TzM2dlJzdz09
Time: 10:00 am
13th May: Weil's converse theorem (speaker: Ajith Nair)
Ajith will finish proof of Weil's converse theorem.
Fall 2021 schedule
In Fall 2021, we discussed sections 1.1-1.4 from Chapter 1 of Bump's Automorphic forms and Representations. This included Dirichlet L-functions, Modular group, Modular forms for SL(2,Z) and Hecke operators.
Automorphic Forms and L-functions Student Seminar (previously)
This is an easygoing study/discussion group to go over some books on automorphic forms and L-functions. We started by reading D. Bump's Automorphic Forms and Representations, published by Cambridge University Press. A learning seminar for Spring and Fall 2018 was primarily run by Lawrence Vu whose seminar webpage is here.
If you want to participate or be on the mailing list, please send an email to the organizers. The meetings will be held on (unless otherwise indicated below) Fridays, 2pm-3pm in Rm 4422 at CUNY, The Graduate Center.
Current Organizers:
Geoffrey Akers: gakers"at"gradcenter.cuny.edu
Ricardo Gonzalez: rgonzalez"at"gradcenter.cuny.edu
Bin Guan: bguan"at"gradcenter.cuny.edu
Fikreab Solomon: fadmasu"at"gradcenter.cuny.edu
Owen Sweeney: osweeney"at"gradcenter.cuny.edu
Lawrence Vu: avu"at"gradcenter.cuny.edu
References:
Bump, Daniel. Automorphic forms and representations. Vol. 55. Cambridge University Press, 1998
Bruinier, J.H., van der Geer, G., Harder, G. and Zagier, D. The 1-2-3 of modular forms, 2008
Diamond, Fred, and Jerry Shurman. A first course in modular forms. Vol. 228. Springer Science & Business Media, 2006
Gelbart, Stephen S. Automorphic forms on adele groups. No. 83. Princeton University Press, 1975
Getz, J. R. ,Hahn H. An Introduction to Automorphic Representations with a view toward Trace Formulae 2018/2019
Goldfeld, Dorian. Automorphic forms and L-functions for the group GL (n, R). Vol. 99. Cambridge University Press, 2006
Jacquet, H. and Langlands, R.P., 2006. Automorphic forms on GL (2) (Vol. 114). Springer.
Milne J. S. Algebraic groups (v2.00), 2015. Available at www.jmilne.org/math/.
Serre, Jean-Pierre. A course in arithmetic. Vol. 7. Springer Science & Business Media, 2012
Spring 2019 Schedule
May 3rd, 2019
Owen Sweeney
The converse of Herbrand's Theorem, 2-3pm
An Euler System's proof.
March 15th, 2019
Owen Sweeney
Vandiver's conjecture, 2-3pm
An alternative formulation of Vandiver's conjecture in terms of Cyclotomic Units.
March 1st, 2019
Fikreab Solomon
Subgroup growth zeta functions and Hecke algebras, 11:45am-1:00pm, Rm 3307
For more information, see https://fsw01.bcc.cuny.edu/cormac.osullivan/Research/number_theory.html
February 27th, 2019
Owen Sweeney
Consequences of Stickelberger's Theorem, 615-715pm
I will present a nice reciprocity law that follows from Stickelberger's Theorem and an application to the first case of Fermat's Last Theorem.
Fikreab Solomon
Zeta functions of groups and multivariable generalizations, 715-745pm
Bin Guan
Non-vanishing central values of triple product L-functions via the relative trace formula, 715-745pm
February 22nd, 2019
Owen Sweeney
Stickelberger's Theorem on Annihilators of Class Groups, 130-230pm
I will present a relatively short proof of of Stickelberger's Theorem which gives explicit group ring elements annihilating the class group of abelian number fields.
Fall 2018 Schedule
See for most recent schedule: https://lawrencevu.bitbucket.io/seminar_fall_2018.html
Spring 2018 Schedule
See for most recent schedule: https://lawrencevu.bitbucket.io/seminar.html
March 2nd, 2018
Lawrence Vu
Analysis on Locally Compact Group, 2-3pm
A short introduction to Haar measures on locally compact group
February 23rd, 2018
Lawrence Vu
Algebraic Groups V, part II, 3-4pm
Adelic points on algebraic groups and strong approximation
February 16th, 2018
No seminar
February 9th, 2018
Lawrence Vu
Algebraic Groups V, 3-4pm (note time change)
Adelic points on algebraic groups and strong approximation
February 2nd, 2018
Lawrence Vu
Algebraic Groups IV, 2-3pm
Let's talk about Lie algebra and classification of reductive groups via root system.
Winter 2018 Schedule
January 26th, 2018
Prof. Dodziuk's birthday conference at the Graduate Center: www.kimsobel.com/Jozef/index.html
January 19th, 2018
Lawrence Vu
Algebraic Groups III, 2-3pm
We continue our discussion with the notion of unipotent radical, semisimple, torus, reductive group, Borel subgroups, Lie algebra and classification of reductive groups via root system.
January 12th, 2018
Lawrence Vu
Algebraic Groups II, 2-3pm
We continue with various notion related to affine algebraic groups such as algebraic group homomorphisms (so we get a category to work with), algebraic subgroup, quotient, extension of scalars and Weil restrictions of scalars.
January 5th, 2018
Lawrence Vu
Algebraic Groups I, 2-3pm
In this talk, I shall define and give examples of algebraic groups; focussing on linear affine algebraic group.
Fall 2017 Schedule
November 9th 2017
Bin Guan
Spherical Representations, 12:00-1:30pm
November 2nd 2017
Bin Guan
Whittaker Models and the Jacquet Functor, 12:00-1:30pm
October 26th 2017
Bin Guan
The Hecke Algebra and Distributions, 12:00-1:30pm
October 19th 2017
Bin Guan
The Weil Representation of GL(2) Over a Finite Field, 12:00-1:30pm
October 12th 2017
No seminar
October 5th 2017
Bin Guan
Smooth and Admissible Representations, 12:00-1:30pm
September 28th, 2017
Bin Guan
Representations of GL(2) Over a Finite Field, 11:45am-1:45pm
There are parallels between the representation theory of GL(n) over a finite field, a local field, or the adele ring of a global field. Because many of the most important concepts persist at each level, we will begin with the representation theory of GL(2) over a finite field. This time we will focus on the irreducible representations constructed by induction, i.e. the principal series and the Steinberg representations.
September 21st 2017
No seminar (No classes scheduled at the Graduate Center)
September 14th, 2017
Interesting Functions for Arithmetic Groups, 11:45am-1:15pm
September 7th, 2017
Bin Guan
An Overview of Langlands Conjectures for GL(n), 12:00-1:30pm
In its most basic form, the Langlands conjecture is a nonabelian generalization of (abelian) class field theory. We will describe the "Galois side" and the "automorphic side", and see an overview of the conjectures for GL(n), both the local and global case. Also we will see how to relate the Langlands conjectures for GL(1) with class field theory.
August 31st 2017
No seminar
Summer 2017 Schedule
July 27th, August 3rd 2017
MSRI Workshop (https://www.msri.org/summer_schools/792)
July 20th, 2017
No seminar
July 13th, 2017
Galois Representations of the Absolute Galois Group of Number Fields, 2-4pm
July 6th, 2017
Bin Guan
The Global Langlands Conjectures (continued), 2-4pm
June 29th, 2017
Bin Guan
The Global Langlands Conjectures, 2-4pm
June 22th, 2017
Bin Guan
Eisenstein Series and Intertwining Integrals (continued), 2-4pm
We will discuss the Fourier expansions of the Eisenstein series to see how these relate the Whittaker functions and intertwining integrals. Then the analytic continuation and functional equations of "normalized" Eisenstein series will be introduced. We will also prove that the Eisenstein series are orthogonal to the cusp forms.
June 15th, 2017
Bin Guan
Eisenstein Series and Intertwining Integrals, 2-3pm
In this talk I will explain how to generalize the definition of the Eisenstein series for SL(2,Z) defined on the upper half plane. We will see the definition of the Eisenstein series for GL(2,A), and see how to consider it as a function of one complex variable by defining a flat section of the principal series representations.
Spring 2017 Schedule
May 12th, 2017
Hecke's Correspondence for Siegel Modular Forms, 1:00pm-2:30pm
We will discuss Siegel modular forms which are generalizations of elliptic modular forms to higher genus and their properties. We then see K. Imai's generalization of Hecke's correspondence to that between Siegel modular cusp forms of genus two and two variable Dirichlet series with functional equations.
May 5th, 2017
Lawrence Vu
Adelization of Classical Automorphic Forms (continued), 1:00pm-2:30pm
In this talk, I will explain how one obtains an automorphic representation from classical modular forms or Maass forms by the process called adelization which works for general automorphic forms on GL(n, F_\infty) but for simplicity, we will limit to the case F = Q and n = 2.
April 28th, 2017
Hecke's Theory of Holomorphic Modular forms, 1:00pm-2:30pm
We will discuss Hecke’s main correspondence theorem between Dirichlet series with functional equations and modular forms of some weight and apply it to show how finding the number of Dirichlet series with a given signature (λ, k, γ) is equivalent to determining the dimension of M_0(λ, k, γ), the space of entire automorphic forms with respect to G(λ) of same signature.
April 21st, 2017
No seminar
April 14th, 2017
No seminar(Spring break)
April 7th, 2017
(Postponed)
March 31st, 2017
Lawrence Vu
Adelization of Classical Automorphic Forms, 1:00pm-2:30pm
In this talk, I will explain how one obtains an automorphic representation from classical modular forms or Maass forms by the process called adelization which works for general automorphic forms on GL(n, F_\infty) but for simplicity, we will limit to the case F = Q and n = 2.
March 24th, 2017
Whittaker Models and Automorphic Forms, part II, 1:00pm-2:30pm(note time change)
We will discuss the Whittaker function approach of constructing the standard L-function of an automorphic cuspidal representation and also local functional equation of a local zeta integral and global functional equation of a partial L-function.
March 17th, 2017
Whittaker Models and Automorphic Forms, part I, 1:30pm-3:00pm
We will discuss how the uniqueness of Whittaker models(local multiplicity one theorem) leads to the proof of the multiplicity one theorem and also to the functional equations of the standard L-function of an automorphic cuspidal representation of GL(2).
March 10th, 2017
No seminar (GC Math Fest 2017)
March 3rd, 2017
Bin Guan
Automorphic Representations of GL(n) (continued), 12:30pm-2pm
We give the definition of the space of automorphic forms and the definition of automorphic representations. And we will describe precisely the tensor product theorem and multiplicity one theorem.
February 24th, 2017
Bin Guan
Automorphic Representations of GL(n), 12:30pm-2pm
We begin with the definition of GL(n) as an affine algebraic group. Then we use the strong approximation theorem to relate the functions on the upper half plane with those on the adelic GL(2). And we will define the square-integrable space that GL(n,A) acts on, and define the cuspidal condition on GL(n).
February 17th, 2017
Classical Automorphic Forms and Representations, 12pm-1:30pm
We will define automorphic forms as elements of the space of functions A(Γ\G, χ, ω) and relate this notion of an automorphic form to the classical notions of modular forms and Maass forms.
February 10th, 2017
Bin Guan
An Introduction to Tate's Thesis, 12pm-1:30pm
We begin with a brief introduction of valuations, adeles and ideles. Then I'll give some definitions which would be in the description of Tate's thesis, such as the characters, measures, and zeta functions, both in local and global case. If time permits, we will see the sketch of the proof of the functional equation of the global zeta functions.
Fall 2016 Schedule
November 28th, 2016
Joseph DiCapua
Unitarizable Admissible Representations and Intertwining Integrals, 11am-1pm
First we construct unitary admissible representations of G=GL(2,R)^+ via the principal series from section 2.5. We then prove a necessary condition for an admissible representation of G to be unitarizable, and as a result we show there exist non-unitary representations from the principal series. We discuss intertwining integrals with the goal of showing the unitaricity of the complementary series..
November 21st, 2016
Bin Guan
Irreducible (g,K)-Modules for GL(2,R) (3/3), 11am-1pm
To show the existence in the classification theorem of all the irreducible admissible (g,K)-modules for G=GL(2,R)^+, we will realize each module as an invariant subspace (or quotient of that) in L^2(G), on which G acts by right translation. And we would also see the classification of all the irreducible admissible (g,K)-modules for G=GL(2,R).
November 14th, 2016
Bin Guan
Irreducible (g,K)-Modules for GL(2,R) (2/3), 11am-1pm
The talk will classify all the irreducible admissible (g,K)-modules for G=GL(2,R), and realize each module as an invariant subspace (or quotient of that) in L^2(G), on which G acts by right translation.
November 7th, 2016
Basic Representation Theory II: Classification of (g,K)-modules for GL(2,R), 11am-1pm
Continuing the discussion on admissible representations, (g,K)-modules and infinitesimal equivalence, we will prove a complete explicit list of the irreducible admissible (g,K)-modules when g=gl(2,R) and K=SO(2).
October 24th, 2016
Basic Representation Theory I, 11am-1pm
In the next two talks, we will discuss section 2.4 on representation theory notions we will use for the study of automorphic forms. This will include smooth vectors, K-finite vectors, admissible representations, (g,K)-modules and infinitesimal equivalence. Although the discussion is for G=GL(n,R), if time permits, we will see how we can generalize each time to the context of an arbitrary reductive group.
October 17th, 2016
Bin Guan
Discreteness of the Spectrum, 11am-1pm
The talk will cover Section 2.3 of Bump's Automorphic Forms and Representations. The goal is to prove that the spectrum of the Laplace-Beltrami operator $\Delta$ is discrete when $\Gamma\backslash H$ is compact, using the spectral theorem for compact operators. We will also give several theorems about the decomposition of the right regular representation of GL(2,R) on the space $L^2 (\Gamma \backslash G, \chi)$.
September 26th, 2016
Bin Guan
Representations of the Lie Algebra sl(2,C), 11am-1pm
I'll introduce some basic notions in the representation theory of groups, algebras, and Lie algebras. Irreducible and indecomposable representations, Schur's lemma, induced representations (and more) will show up. As an example, I'll state the classification of irreducible representations of the Lie algebra sl(2,C).
September 19th, 2016
Basic Lie Theory II(continued from last), 11am-1pm
The talk will be on Section 2.2 of Bump's Automorphic Forms and Representations. The goal is to discuss the Lie theory notions we need to reinterpret the raising operator, the lowering operator and the Laplace-Beltrami operator as elements in the universal enveloping algebra of the Lie algebra gl(2,R) of GL(2,R).
September 12th, 2016
Basic Lie Theory I, 12pm-1:30pm
The talk will be on Section 2.2 of Bump's Automorphic Forms and Representations. The goal is to discuss the Lie theory notions we need to reinterpret the raising operator, the lowering operator and the Laplace-Beltrami operator as elements in the universal enveloping algebra of the Lie algebra gl(2,R) of GL(2,R).
Summer 2016 Schedule
August 19th, 2016
Bin Guan
Hilbert Modular Forms and Base Change, 12pm-2pm
I'll continue the second part of Section 1.7. We start with some properties of real quadratic extensions over Q (Exercise 3,5,6 in Section 1.1). Then we show the "base change" identity in GL(1) case, that is the base change lift of a Dirichlet character relative to a quadratic extension is a particular Hecke character of the extension field. Before studying base change in GL(2) case, we define the Hilbert modular forms and their L-functions. The goal is to understand the idea to prove the existence of a base change lift of a Hecke eigenform.
August 12th, 2016
Zachary McGuirk
The Selberg Trace Formula in the Compact Setting from a Geometric Viewpoint (continued), 12pm-2pm
Some basic results regarding the structure and geometry of hyperbolic space will be presented, along with some algebraic properties of the group of isometries for hyperbolic n-space. This will be followed by an introduction of point-pair invariants and the Eichler Pre-Trace Formula. The talk will be concluded with the statement of the Selberg Trace Formula in the compact setting.
August 5th, 2016
Zachary McGuirk
The Selberg Trace Formula in the Compact Setting from a Geometric Viewpoint, 12pm-2pm
Some basic results regarding the structure and geometry of hyperbolic space will be presented, along with some algebraic properties of the group of isometries for hyperbolic n-space. This will be followed by an introduction of point-pair invariants and the Eichler Pre-Trace Formula. The talk will be concluded with the statement of the Selberg Trace Formula in the compact setting.
July 29th, 2016
Ryan Ronan
Maass forms and the spectral problem (continued), 12pm-2pm
We will discuss the second version of the spectral problem as Bump defines it in Section 2.1. After reviewing some basic facts about Haar measures and representation theory, we will define and discuss the Hilbert space $L^2 ( \Gamma \\ GL(2,R)^+, \chi)$ and the spectral problem on this space. This problem is closely connected to the spectral problem discussed in our previous meeting. If time permits, we will discuss some of the exercises at the end of Section 2.1.
July 22nd, 2016
Colette LaPointe
Maass forms and the spectral problem, 12pm-2pm
Section 2.1 in Bump introduces two versions the spectral problem, for which he works out the solutions throughout chapter 2. Version I of the spectral problem, determining the spectrum (i.e. set of eigenvalues of) the symmetric unbounded Laplacian operator \Delta_k on L^2(\Gamma\\HH,\chi,k) -- is described using analytic methods, and some solutions coming from automorphic forms and Eisenstein series will be mentioned. (Version II reformulates the spectral problem in terms of representation theory. This would be another talk so I won't include it.) In working toward presenting version I, we will introduce the Maass differential operators R_k and L_k, and \Delta_k, describe some of their properties, and go over some background on operators on Hilbert spaces.
July 15th, 2016
Maass Forms, 12pm-2pm
After introducing Maass forms of weight zero as in section 1.9, we will show that the non-holomorphic Eisenstein series is a Maass form and define an L-series associated to the forms so we can prove analytic continuation and find a functional equation for them. The construction of a Maass cusp form on \Gamma_0(D) will be best understood after seeing a discussion of Langland's functionality or base change.
July 6th, 2016
Bin Guan
Twisted L-functions & Hecke L-functions, 2pm-4pm
In this talk we'll follow section 1.5 and 1.7. We will see the functional equations of twisted L-functions and try to understand the idea of the converse theorem. Then we follow Hecke's method to get an analogue of the Dirichlet L-functions over any number field, i.e. Hecke L-functions, and their functional equations.
July 1st, 2016
Ryan Ronan
The Rankin-Selberg Method, 12pm-2pm
We will begin with an overview of the Eisenstein series, its analytic continuation, and its functional equation. We will then introduce the Rankin-Selberg Method, a technique used to represent certain L-functions as an integral of an automorphic form against an Eisenstein series. Such an L-function will inherit the functional equation of the Eisenstein series. We will conclude with an application of this method, discovered independently by Rankin (1939) and Selberg (1940).
June 24th, 2016
Hecke Operators, 12pm-2pm
After a brief overview of of congruence modular forms, congruence cusp forms, the Peterson inner product, Hecke operators and the Hecke algebra on the space of modular forms on SL(2,Z), we will show how the commutativity of the Hecke algebra leads to an Euler product for L-functions arising from modular forms. Exercises will be mentioned in passing.
June 16th, 2016
Colette LaPointe
Modular forms, 2pm-4pm
I'll be introducing modular and cusp forms on the modular group SL(2,Z), on congruence subgroups, giving some examples, and showing some dimension properties for spaces of modular and cusp forms. I'll follow Bump's section 1.3, and also use some bits from Fred Diamond's "An Introduction to Modular Forms."
June 9th, 2016
Bin Guan
Dirichlet L-Functions, 2pm-4pm
In this talk we will follow Section 1.1 of Bump's Automorphic Forms and Representations. We will see a brief introduction to Dirichlet L-functions. And when approaching their functional equation, topics as Gauss sums, Poisson summation formula, and theta functions would be introduced. If time permits, we'll show some notation and properties of the modular group.