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### Automorphic Forms and L-functions Student Seminar

 This is an easygoing study/discussion group to go over some books on automorphic forms and L-functions. We are currently reading D. Bump's Automorphic Forms and Representations, published by Cambridge University Press.     If you want to participate or be on the mailing list, please send an email to the organizers. The meetings will be held on Thursdays, 11:45am-1:45pm in Room 3212 at CUNY, The Graduate Center.Organizers:     Bin Guan: bguan"at"gradcenter.cuny.edu    Fikreab Solomon: fadmasu"at"gradcenter.cuny.eduReferences: Bump, Daniel. Automorphic forms and representations. Vol. 55. Cambridge University Press, 1998Bruinier, J.H., van der Geer, G., Harder, G. and Zagier, D. The 1-2-3 of modular forms,  2008Diamond, Fred, and Jerry Shurman. A first course in modular forms. Vol. 228. Springer Science & Business Media, 2006Gelbart, Stephen S. Automorphic forms on adele groups. No. 83. Princeton University Press, 1975Goldfeld, Dorian. Automorphic forms and L-functions for the group GL (n, R). Vol. 99. Cambridge University Press, 2006Serre, Jean-Pierre. A course in arithmetic. Vol. 7. Springer Science & Business Media, 2012                Fall 2017 ScheduleNovember 9th 2017    Bin Guan    Spherical Representations, 12:00-1:30pmNovember 2nd 2017    Bin Guan    Whittaker Models and the Jacquet Functor, 12:00-1:30pmOctober 26th 2017    Bin Guan    The Hecke Algebra and Distributions, 12:00-1:30pmOctober 19th 2017    Bin Guan    The Weil Representation of GL(2) Over a Finite Field, 12:00-1:30pmOctober 12th 2017    No seminar October 5th 2017    Bin Guan    Smooth and Admissible Representations, 12:00-1:30pmSeptember 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    Fikreab Admasu    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 seminarSummer 2017 ScheduleJuly 27th,  August 3rd 2017    MSRI Workshop (https://www.msri.org/summer_schools/792)   July 20th, 2017    No seminarJuly 13th, 2017    Fikreab Admasu    Galois Representations of the Absolute Galois Group of Number Fields, 2-4pmJuly 6th, 2017    Bin Guan    The Global Langlands Conjectures (continued), 2-4pmJune 29th, 2017    Bin Guan    The Global Langlands Conjectures, 2-4pmJune 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 ScheduleMay 12th, 2017    Fikreab Admasu     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     Fikreab Admasu     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 seminarApril 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    Fikreab Admasu     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    Fikreab Admasu     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    Fikreab Admasu     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 ScheduleNovember 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    Fikreab Admasu    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    Fikreab Admasu    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    Fikreab Admasu    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    Fikreab Admasu    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 ScheduleAugust 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    Fikreab Admasu    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    Fikreab Admasu    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.