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:
Equidistribution of Heegner Points and Fourier Coefficient Growth of Automorphic Forms
Spectral Reciprocity, Moments and Nonvanishing of Automorphic Lfunctions
Constant Term Formula and Eisenstein Series of Reductive Groups
Please email me if you would like to join the mailing list for this seminar.
Announcement
 There will be a special seminar by Prof. Gunnells from UMass Amherst (currently scheduled on Tue, 19 Feb, 4.306.00).
General References
This list will be expanded soon.
 [G] D. Goldfeld, Automorphic Forms and Lfunctions 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 Lfunctions II , Invent. Math 1994.
 [I2] H. Iwaniec, Topics in Classical Automorphic Forms , Graduate Studies in Mathematics, 1997. (esp. Chp. 45, 1113)
 [I3] H. Iwaniec, Spectral Methods of Automorphic Forms , Graduate Studies in Mathematics, 2002. (esp. Chp. 89)
 [IK] H. Iwaniec, E. Kowalski Analytic Number Theory , Colloquium Publications Vol. 53, 2004. (esp. Chp. 1516)
 [L1] X. Li, Arithmetic Trace Formula and Kloostermania
References for Equidistribution of Heegner points
This list will be expanded soon.
 [D1] W. Duke, Hyperbolic distribution problems and halfintegral 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 halfintegral 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 RankinSelberg Lfunctions and equidistributions of Heegner points , Ann. of Math, 2004
References for Spectral Reciprocity
This list will be expanded soon.
Schedule
The schedule is tentative and subject to change as the seminar moves along.

Date 
Speaker 
Topic 
References 
Lecture 1 
January 29 
Asbjorn Nordentoft 
Background & Review 
[D1], [I3] 
Lecture 2 
February 5 
Asbjorn Nordentoft 
Bounds for Fourier Coefficients of Halfweight Maass forms 
[I1], [I2], [Sar] 
Lecture 3 
February 12 
Asbjorn Nordentoft 
Distribution of Heegner points on Shimura curves associated to Definite Quaternion Algebras 
[M] 
Lecture 4 
February 19 
Prof. Paul Gunnells (UMass Amherst) 
Introduction to Modular Symbols 

Lecture 5 
February 26 
Kevin Kwan 
Background info of Spectral Reciprocity 

Lecture 6 
March 5 
Kevin Kwan 


Lecture 7 
March 12 
Kevin Kwan 


Lecture 8 
March 19 (Spring Break) 



Lecture 9 
March 26 



Lecture 10 
April 2 



Lecture 11 
April 9 



Lecture 12 
April 16 



Lecture 13 
April 23 



Lecture 14 
April 30 



Lecture 15 
7 May 



Lecture 16 
14 May 



Abstracts
 Lectures 13
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 Lfunctions. Furthermore I will discuss certain applications of the equidistribution results to other subconvexity problems, which came up in my own work.
The plan is as follows, but it might change slightly.
 Talk 1: Introduction and background; Kuznetsov trace formula and the theta correspondance in the nonholomorphic case.
 Talk 2: Bounds for Fourier coefficients of halfintegral Maass forms (following Iwaniec)
 Talk 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 nonvanishing and subconvexity bounds of Lfunctions. The red thread will be the triad (1). Fourier coefficients of halfintegral weight automorphic forms; (2). Weyl sums for /(traces over) Heegner points; (3). Central values of Lfunctions
 Lecture 4
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).
Last updated: Feb 12, 2019.
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