Math 207A: How much covolume does tell us about a lattice?

Fall 2015

 Lectures: M-W-F 10:00 - 10:50 APM 5829 Office Hour: Send me an e-mail.
Send me an e-mail, we can meet and discuss math (possibly in a coffee shop).

Course description:

The following is a tentative list of topics to be covered:

• Hyperbolic plane: Gauss-Bonnet in this case, fundamental domain of SL(2,Z), lattice of minimum covolume in SL(2,R).
• Geometry of numbers: space of lattices in Rn, Mahler's compactness criteria, Minkowski's reduction theory.
• Kazhdan-Margulis theorem, existence of a lattice of minimum covolume in semisimple Lie groups without compact factors, Wang's theorem on the finiteness of the number of lattices with covolume at most x, up to conjugation.
• Covolume of SL(n,Z): Siegel's approach.
• Strong approximation, adeles and covolume of SL(n,Q) in SL(n,A).
• Siegel's mass formula and Tamagawa number of an orthogonal group.
• Eskin-Rudnick-Sarnak's approach: counting integer points and Tamagawa number of orthogonal groups.
• Weil's conjecture on Tamagawa number.
• Prasad's volume formula.
• Related open problems and projects.

Prerequisite:

will be kept to a minimum.

Resources:

I will not follow a particular book, but I will post the related books and articles in the course's webpage.

Here are a few related references:

• M. S. Raghunathan, Discrete Subgroups of Lie Groups, Springer, New York, 1972
• D. W. Morris, Introduction to arithmetic groups, go here for an electronic version.
• S. Katok, Fuchsian groups.
• A. Weil, Adeles and algebraic groups.
• C. Maclachan, A. Reid, The arithmetic of hyperbolic 3-manifolds.
• C. L. Siegel, A mean value theorem in geometry of numbers, Annals of math. (2nd ser.) 46, no. 2, (1945), 340-347.
• T. Tamagawa, Adeles, Proc. Symp. Pure Math., 9 113-121, Amer. Math. Soc, Providence, 1966.
• G. Prasad, Volume of S-arithmetic quotients of semisimple groups, Publ. math. IHES 69 (1989) 91-114.

Notes related to lectures and supplementary materials:
• In the first lecture, I said what a lattice in a topological group is, explained that any lattice in R is cyclic, and gave an overview of the course. Then I defined the hyperbolic metric, and explored the basic properties of hyperbolic plane.
• In the second lecture, I showed that Mobius transformations are hyperbolic isometries and understood hyperbolic geodesics.
• In the third lecture, we saw the classification of hyperbolic isometries, and Schottky's ping-pong argument.
• In the forth and the fifth lecture, we computed the area of a hyperbolic triangle, reviewed covering spaces and the group of deck transformations, and mentioned the connection between the Teichmuller space and the character variety. You can see more precise statements in the note.
• In the sixth and seventh lecture, we introduced the symmetric space of SL(n,R).
• In the eighth lecture, we introduced Dirichlet domain, and found a fundamental domain of PSL(2,Z).
• In the ninth and tenth lecture, we talked about Haar measure, found explicit Haar measures of certain groups, found various volume forms of SL(2,R). Here is the exercise on the decomposition of the Haar measure.
• In the eleventh lecture, we talked about reduction theory.
• In the twelfth and thirteenth lectures, we continued the study of reduction theory.
• In the fortheenth and fifteenthlectures, we proved Mahler's compactness criterion, SL(n,Z) is a lattice in SL(n,R), etc.
• In the sixteenth, seventeenth, and eighteenth lectures, we defined Siegel transform, proved the average of the Siegel transform is equal to the Lebesgue integral of the function over Rn, computed the covolume of SLn(Z) inductively, and proved Hlawka's theorem (Siegel's approach) which asserts that for any bounded region A in Rn with area less that ζ(n) one can find a unimodular lattice Δ in Rn that meets A only possibily at the origin.
• In the ninteenth and twentieth lectures, we found an SL(n)-invariant gauge form and compute the covolume of SLn(Z) with respect to the induced Haar measure. Then we defined p-adic numbers, and proved a generalization of Hensel's lemma.
• In the lectures 21-24, we covered basics of algebraic number thoery: ring of integers in a number field is a Dedekind domain, it is a finitely generated free abelain group, defined the discriminant of a number field, proved the class number is finite, defined valuations and their completion, defined ring of adeles, proved strong approximation (in the additive case), showed k is a lattice in the ring of adeles of k and computed its covolume.