Student Seminar on Analytic Aspects of Automorphic Forms (Spring 2019)

Logistics

Time: 4:30 pm - 6:00 pm on Tuesdays
Location: 507 Mathematics

This semester Asbjorn Nordentoft and I are organizing a learning seminar on analytic number theory. The seminars will be divided into several blocks and each focuses on one particular topic. Currently, we intend to include:

Please email me if you would like to join the mailing list for this seminar.

Announcement (New!!!)

• (New!!!) There will be a new series of talks by Mike Woodbury on harmonic Maass forms!
• There will be a guest lecture by Jeanine Van Order (Universitat Bielefeld). Currently, it is scheduled on Tue 16 April, from 4.30-6.00.
• There will be a new series of talks by Sam Mundy on Eisenstein series and Cohomology of Arithmetic Groups!
• There will be a location change for the seminar on Tuesday, 5 March, due to the Minerva Lecture. The talk will be held in Room 622 (Math Building) from 4.30-6.00 pm.
• There will be a guest lecture by Paul Gunnells (UMass Amherst). Currently, it is scheduled on Tue 19 Feb, from 4.30-6.00.

References for Mike's Talk

• [BF] J. H. Bruinier, J. Funke, On two geometric theta lifts , Duke Math. J., vol. 125, no. 1 (2004), 45-90.
• [BFOR] K. Bringmann, A. Folsom, K. Ono, L. Rolen, Harmonic Maass Forms and Mock Modular Forms: Theory and Applications , Colloquium Publications (2017), Vol. 64 (Electronic copy is available via Columbia network)

References for Jeanine's Talk

• [LRS1] W. Luo, Z. Rudnick and P. Sarnak, On Selberg's Eigenvalue Conjecture , GAFA 5 no. 2 (1995), 388-401
• [LRS2] W. Luo, Z. Rudnick and P. Sarnak, On the generalized Ramanujan conjecture for GL(n) , Proc. Sympos. Pure Math, Vol. 66.2 (1999)

References for Eisenstein Series of Reductive Groups and Cohomology of Arithmetic Groups

• [Sp] T.A. Springer, Reductive Groups, Proceedings of Symposia in Pure Mathematics Vol. 33 (1979) (Corvallis), part I, pp. 3-27
• [Kn] A.W. Knapp Lie Groups: Beyond an Introduction , Progress in Mathematics book series PM, Vol. 140 (1996), Springer
• [MoWa] C. Moeglin, J. L. Waldspurger Spectral Decomposition and Eisenstein Series , Cambridge University Press (1995)
• [La] R. P. Langlands Euler Products , Yale Mathematical Monograph, Yale University Press (1971)
• [Sha1] F. Shahidi Intertwining Operators, L-Functions, and Representation Theory , Lecture Notes of the Eleventh KAIST Mathematics Workshop (1996)
• [Sha2] F. Shahidi Eisenstein Series and Automorphic L-Functions , Colloquium Publications Vol. 58 (2010)
• [Fra] J. Franke Harmonic Analysis in Weighted L2-spaces , Annales scientifiques de l'E.N.S. 4eserie, tome 31, no2 (1998), p. 181-279
• [LiSch] J-S Li, J. Schwermer On the Eisenstein cohomology of arithmetic groups , Duke. Math. J., Vol. 123, No. 1 (2004), 141-169.

References for Equidistribution of Heegner points

• [D1] W. Duke, Hyperbolic distribution problems and half-integral weight Maass forms , Invent. Math 1988.
• [KS] S. Katok, P. Sarnak, Heegner points, cycles and Maass forms , Israel J. Math 1993.
• [I1] H. Iwaniec, Fourier coefficients of modular forms of half-integral weight , Invent. Math 1987.
• [Sar] P. Sarnak, Some Applications of Modular Forms , Cambridge University Press, 1990. (esp. Chp. 4)
• [GS] D. Goldfeld, P. Sarnak Sums of Kloosterman Sums , Invent. Math, 1983
• [M] P. Michel The subconvexity problem for Rankin-Selberg L-functions and equidistributions of Heegner points , Ann. of Math, 2004

References for Spectral Reciprocity

• [BK] V. Blomer, R. Khan, Twisted moments of L-functions and spectral reciprocity To appear in Duke Math. J.
• [BLM] V. Blomer, X. Li, S. Miller A spectral reciprocity formula and non-vanishing for L-functions on GL(4) x GL(2) , arxiv
• [L2] X. Li Bounds for GL(3)×GL(2) L-functions and GL(3) L-functions , Ann. of Math, 2011

General References

• [G] D. Goldfeld, Automorphic Forms and L-functions for the Group GL(n,R) , Cambridge Studies in Advanced Mathematics, 2006.
• [DI] J. -M. Deshouillers and H.Iwaniec, Kloosterman sums and Fourier coefficients of cusp forms, Invent. Math 1982.
• [DFI] W. Duke, J. Friedlander and H. Iwaniec, Bounds for Automorphic L-functions II , Invent. Math 1994.
• [I2] H. Iwaniec, Topics in Classical Automorphic Forms , Graduate Studies in Mathematics, 1997. (esp. Chp. 4-5, 11-13)
• [I3] H. Iwaniec, Spectral Methods of Automorphic Forms , Graduate Studies in Mathematics, 2002. (esp. Chp. 8-9)
• [IK] H. Iwaniec, E. Kowalski Analytic Number Theory , Colloquium Publications Vol. 53, 2004. (esp. Chp. 15-16)
• [L1] X. Li, Arithmetic Trace Formula and Kloostermania

Schedule

The schedule is tentative and subject to change as the seminar moves along.

Abstracts

• Lectures 1-3 (Asbjorn Nordentoft)
  In the first three talks, I will present different aspects of the analytic theory of Heegner points. The principal goal is to go through Duke's work on equidistribution of Heegner points. The beef of the proof are certain bound on sums of Kloosterman sums due to Iwaniec and we will try to point to the main points in the argument. If time permits I will also discuss P. Michel's work on equidistribution of Heegner points on Shimura curves associated to definite quaternion algebras over Q and the related subconvexity bounds of central values of L-functions. Furthermore I will discuss certain applications of the equidistribution results to other sub-convexity problems, which came up in my own work. The plan is as follows, but it might change slightly.
• (1). Introduction and background; Kuznetsov trace formula and the theta correspondance in the non-holomorphic case.
• (2). Bounds for Fourier coefficients of half-integral Maass forms (following Iwaniec)
• (3). This last talk of the series on the analytic theory of Heegner points will be less technical than the preceding one and should be more relatable for more algebraically orientated people. I will talk about certain work of P. Michel on distribution of Heegner points on Shimura curves associated to definite quaternion algebras. Furthermore I will discuss some applications of equidistribution to non-vanishing and sub-convexity bounds of L-functions. The red thread will be the triad (i). Fourier coefficients of half-integral weight automorphic forms; (ii). Weyl sums for /(traces over) Heegner points; (iii). Central values of L-functions

• Lecture 4 (Paul Gunnells)
  Modular symbols, due to Birch and Manin, provide a very concrete way to compute with classical holomorphic modular forms. In this talk we explain how this works, and also present techniques due to Ash that allow computation with certain automorphic forms on GL(3).

• Lecture 5 (Chung-Hang Kwan)
  One has seen in the first three talks by Asbjorn the remarkable applications of subconvexity bounds of GL(2) and GL(2)×GL(1) L-functions in various equidistribution problems with arithmetic flavour. This is a new series of talks which focuses on the methods behind proving the subconvexity bounds for automorphic L-functions of high-ranks. This is a relatively new area with a lot of exciting development. The first talk in this series will mainly be non-technical and aiming at motivating the subject, providing an overview and setting-up of the problem. For the sake of being self-contained, we shall also review the basics of GL(3)-automorphic forms and standard tools in analytic number theory briefly.

• Lecture 6 (Chung-Hang Kwan)
  This is a continuation of the first talk in this series. We shall explain the ideas in X. Li's important paper on estimating the spectral first moment involving GL(3)×GL(2) and GL(3) L-functions. We also plan to start the discussion of Blomer-Li-Miller's strategies on proving the non-vanishing of GL(4)×GL(2) L-functions.

• Lecture 7 (Chung-Hang Kwan)
  This will be the last talk of the series. The main focus of the talk is to compare the two different approaches of Li and Blomer-Li-Miller in studying the spectral first moment, especially on how the Voronoi summation formulae are applied in respective cases.

• Lecture 8-10 (Sam Mundy)
  Root systems, reductive groups, and parabolic subgroups. In this talk, I'll explain how the root data attached to a reductive groups can be used to better see the structure of that group and its parabolic subgroups.

• Lecture 11 (Jeanine Van Order)
  

• Lecture 12 (Sam Mundy)
  In this talk, I'll outline the basics of the theory of Eisenstein series on reductive groups. Time and interest permitting, I could also explain the calculation of the constant terms of Eisenstein series, a la Langlands-Shahidi.

• Lecture 13-14 (Mike Woodbury)
  I will discuss the theory of so-called harmonic Maass forms as developed by Funke-Bruinier and various applications both to classical modular forms (such as Lehmer's conjecture and Borcherds products) and others (such as Ramanujan's mock modular forms). References: Bruinier and Funke's paper "On two geometric theta lifts", Duke Math. J. (2004)

Last updated: Feb 12, 2019.