Equidistribution in Number Theory

(MA 841)

Upon Akshay Venkatesh winning the fields medal I decided to dedicate this term to various aspects of equidistribution results in number theory and their relations to L-functions. I am aiming to cover basic results like Linnik's and Duke's theorems, as well as certain aspects of subconvexity. As we move along we may briefly touch several other aspects like quantum unique ergodicity (QUE) or equidistribution in other settings (e.g. over function fields).

There is an abundance of material on this topic. We will not follow a single book or an article (although the first book and the surv ey articles following that will be the main reference), however here are a bunch of helpful papers/books. I will update these references as we move along.

Suggested references

Notes for the course (realtime, thanks to Alex Best!).

Time/Location: Tuesday and Thursday 9:30 am - 10:45 am, CGS 521

Office Hours: If you have questions etc. please send an email to set a time to meet.

References for the lectures

(MA 841)

Upon Akshay Venkatesh winning the fields medal I decided to dedicate this term to various aspects of equidistribution results in number theory and their relations to L-functions. I am aiming to cover basic results like Linnik's and Duke's theorems, as well as certain aspects of subconvexity. As we move along we may briefly touch several other aspects like quantum unique ergodicity (QUE) or equidistribution in other settings (e.g. over function fields).

There is an abundance of material on this topic. We will not follow a single book or an article (although the first book and the surv ey articles following that will be the main reference), however here are a bunch of helpful papers/books. I will update these references as we move along.

Suggested references

- Equidistribution in Number Theory (book)
- A treatment of Linnik, by Ellenberg, Michel, and Venkatesh
- A short book by Sarnak (Some applications of modular forms)
- Topics in classical automorphic forms (by Iwaniec)
- Analyitc Number Theory (Iwaniec-Kowalski)

- Survey articles by Duke (same article as in the book above)and Michel-Venkatesh
- Duke's original article

- Baiscs of equidistribution, by Pete L. Clark
- Uniform Distribution of Sequences, by Kuipers and Niederreiter

- An nice treatment of a related setting, by Garrett
- Duke, Rudnick, and Sarnak's article on equidistribtuion in more general settings

- Venkatesh's article on subconvexity and equidistribution

- Quanta interview for Venkatesh's Fields medal

- Sarnak's article on QUE
- Sarnak and Iwaniec on analytic theory of L-functions (GAFA article)
- Efrat's article on PGL(2) over function fields

Time/Location: Tuesday and Thursday 9:30 am - 10:45 am, CGS 521

Office Hours: If you have questions etc. please send an email to set a time to meet.

References for the lectures

- Lecture 1: Introduction, Linnik's and Duke's Theorems

- Duke's article in the first reference above (book), Duke's original article, Quanta article on Venkatesh.
- Lecture 2: Basic Diophantine approximation, Dirichlet, Liouville, and Kronecker's theorems
- There are so many references, I honestly forgot what I used for this lecture.
- Lecture
3: Basics of uniform distribution, examples of uniformly (and not
uniformly) distributed sequences, uniform distribution of \{n\alpha\},
Weyl's criterion

- Granville and Rudnick's article in the first reference above (book), Pete L. Clark's notes, Elkies' notes, Chapter 1 of Uniform Distribution of Sequences
- Lecture
4: Proof of Weyl's criterion, more examples of uniformly distributed
sequences, Vinogradov's theorem on the distribution of {p_n\alpha},
Koksma's theorem, Mahler's 3/2 problem, equidistribution of various
sequences.

- Lecture 5: Weyl differencing, Weyl's theorem on \alpha such that \{a_n\alpha\} is equidistributed

- Lecture
6: Van der Corput lemma, a little bit of ergodic theory: Birkhoff
pointwise ergodic theorem, Furstenberg's proof of the equidistribution
of n^2\alpha

- Montgomery's article in this book, Chapter 2 of this book by Montgomery, Lindenstrauss' article in the first reference above (book)
- Lecture 7: Furstenberg's proof of the equidistribution of n^2\alpha, Introduction to Duke's Theorem

- Lindenstrauss' article in the first reference above (book), Michael Hochman's lecture notes, Wikipedia page for Benford's Law , Duke's article in the first reference above (book)

- Lecture 8: Duke's theorem (Strategy of the proof), Digression into half-integral weight modular forms

- Duke's article in the first reference above (book), Sarnak's book
- Lecture
9: More on theta functions and half-integral weight forms, integral and
half-integral weight Ramanujan Conjecture, Hecke bound, Shimura
correspondence

- Duke's article in the first reference above (book), Sarnak's book, Shimura's article
- Lecture 10: Equidistribution of rational points on S^2, Shimura correspondence, Ramanujan conjecture, Poincare series
- Duke's article in the first reference above (book), Sarnak's book, Shimura's article, Iwaniec's book
- Lecture 11: Fourier expansion of Poincare series, Kloosterman sums, Petersson formula
- Sarnak's book, Iwaniec's book

- Lecture 12: Petersson formula, Weil bound, finishing the proof of equidistribution of rational points on S^2
- Duke's article in the first reference above (book), Sarnak's book

- Lecture 13: Back to Duke's theorem, Salie sums, Iwaniec's bound on Fourier coefficients of half-integral weight forms, Siegel's theorem, and finishing the proof of Duke's theorem
- Duke's article in the first reference above (book), Sarnak's book, Iwaniec-Kowalski
- Lecture 14: Duke's theorem on equidistribution of Heegner points, some spectral theory of automorphic forms on the upper haf plane, Eisenstein series
- Duke's article in the first reference above (book), lecture notes from last year's class
- Lecture 15: Waldspurger's theorem and connections of equidistribution to subconvexity estimates. Introduction to Linnik's ergodic method.
- Duke's
article in the first reference above (book), Sarnak's book,
Waldspurger's article on Fourier coefficients of half-integral weight
forms, EMV article

(Reference for lectures 16-21: EMV article)

- Lecture 16: Overview of Linnik's method, action of the class group on the set of solutions

- Lecture 17: The set of integral points on spheres as a principal homogeneous space for class groups.
- Lecture
18: Explicit description of the action of the class group,

- Lecture 19: Linnik's basic lemma, a lower bound on the number of non-backtracking paths
- Lecture 20: Chernoff bound, End of the proof (modulo the two black boxes)
- Lecture 21: The adelic picture
- Lecture
22: Spectral decomposition for a symmetric space for PGL(2) over the
function field of the projective line over a finite field.