Modular Forms and L-functions, Math 8207-8208

A course in modern number theory and harmonic analysis

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( See also: [ vignettes ] ... [ functional analysis ] ... [ intro to modular forms ] ... [ representation theory ] ... [ Lie theory, symmetric spaces ] ... [ buildings notes ] ... [ number theory ] ... [ algebra ] ... [ complex analysis ] ... [ real analysis ] ... [ homological algebra ] )


2019-2020

Phenomena, examples, history

Fall 2019: Vincent Hall 2, 11:15-12:05, MWF


2017-18

Phenomena, examples, history

This introductory Number Theory course will be accessible to first-year and second-year grad students with a modest background, and will proceed by extensive examples throughout, as motivation and explanation for more sophisticated methods and formalism. I will adapt the content and degree of sophistication to the students who show up! :)

Course notes will appear here, based in part on earlier notes as below, with revisions and additions. The general tone and outlook will be similar to past courses.

My book Modern Analysis of Automorphic Forms by Example (also: Cambridge University Press, Cambridge Studies in Advanced Mathematics, vols 173, 174 ). The contract allows CUP to sell physical copies, and (by design) allows me to put this PDF on-line, under the reasonable condition that I ask that people only download copies for personal use (or, conceivably, just link to this page), and that I link to CUP's pages for this book: CUP's page for volume one , CUP's page for volume two

Tentative partial course outline:


2015-16

Phenomena, examples

This introductory Number Theory course will be accessible to first-year and second-year grad students with a modest background, and will proceed by extensive examples throughout, as motivation and explanation for more sophisticated methods and formalism.

I intend (by adapting the content to the population of students that show up!) that this course be interesting not only to students in Number Theory or Automorphic Forms, but also to students whose research areas interact frequently with these subjects, such as Algebraic Geometry, Mathematical Physics, Representation Theory, and Combinatorics, among others.

This course will give the phenomenological background to formalities such as the Langlands program, but I intend to take a broader approach.

Approximate/tentative outline:

[ background on complex analysis ]

  1. Classical GL(1) stories:
  2. Some classical GL(2) stories:
  3. Equidistribution example
  4. Classical L-functions for GL(2)
  5. Waveforms
  6. Pre-trace formulas and spectral theory of automorphic forms/functions
  7. Hilbert-Blumenthal modular forms
  8. Siegel modular forms
  9. Transition to GL2(A) and GLn(A)
  10. Theta lifts/correspondences, Segal-Shale-Weil representations:

Questions? Send me email! :)

(2005-06, 2010-11, and 2013-14 notes lower on the page)


2013-14

This course introduces many phenomena that led to much contemporary research, including the Langlands program and much more. Little prior acquaintance with higher-level prerequisites is assumed. Rather, we will give examples that led to formation of many contemporary concepts and abstractions in number theory, complex analysis, Lie theory, harmonic analysis, representation theory, and algebraic geometry.

Units are listed in reverse chronological order. Notes will be linked-to as we go, somewhat in advance of progress in-class. If you must print notes, please don't do so until just before reading, because many updates will occur.

See also Number theory notes 2011-12 for related discussions

Our course will include much supporting material, beyond the strict topics of the title. Samples of other sources about modular forms themselves are below. Siegel's notes give number-theoretic applications of Hilbert modular forms.

2010-11

Office hours: MWF 1:25-2:15 or by appointment, email anytime

An introduction to number theory, zeta functions and L-functions, and the role of modular and automorphic forms

Notes and exercises (reverse chrono order)


2005-06

Notes (reverse chronological order):
Unless explicitly noted otherwise, everything here, work by Paul Garrett, is licensed under a Creative Commons Attribution 3.0 Unported License. ... [ garrett@math.umn.edu ]

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