Student Workshop on Tate's thesis, August 24-26, 2011

Organizational matters:

Location: Math 126 (the lecture room in the lounge)
Time: start at 10am every day
Scroll down for the schedule of talks.
The speakers can get credit for Math 592 (Summer term).
If you'd like to be on the mailing list (and have not received any e-mails about this yet), please e-mail me at "gor at math etc".

The goals and format:

Our goal is, as in the famous thesis of J. Tate, to prove the analytic contunuation and functional equation for Hecke L-fucntions (the L-functions attached to Grossencharacters). Tate's thesis remains our main reference. We plan to have three intensive days of meetings, with roughly four hours of talks (by the student participants) each day. Below is the approximate list (subject to change) of specific topics and speakers.

Prerequisites:

Some familiarity with the classical Dirichlet L-functions, and with the concept of a number field and local field. Participation in the summer student seminar on L-functions would have been an excellent preparation. If you did not participate in this seminar, please familiarize yourself somewhat with the topics covered there. Other prerequisites, such as adeles and ideles, will be covered in the first day of talks.

References:

Tate's thesis, in Cassels and Frohlich.
Notes by S. Kudla, posted here
Notes by P. Walls.
Notes by D. Prasad.
A. Weil, "Fonction zeta et distributions", 1966.
Also, the book "Basic Number Theory" by A. Weil.
Milne's notes on Class Field Theory (this is a reference for the division algebras). Also, Milne's notes on algebraic number theory can be useful.

Suggested schedule (all times except 10am are very approximate, all talks in MATH 126):

Comments on this suggested schedule are welcome!

Wednesday August 24:

10am: Opening remarks.
10:10am -- Athena: local fields and division algebras (the basic definitions, topology).
11:45 am -- Lance: adeles and ideles (topology, the diagonal embedding of the field, compactness of the quotient by the image of this embedding.)
1:30pm -- lunch.
2:30pm Lance: Additive and multiplicative characters of non-archimedean local fields.
NOTE: the talks in the first day can turn out to be slightly shorter than the time allocated here; in this case, we might move Lance's talk to before lunch, and Carmen's talk from day 2 to after-lunch, to leave more time for the possibly longer talks on the following days.

Thursday August 25:

10am: XinYu: Fourier analysis, the archimedean place.
11:45am: Carmen: Fourier analysis on the additive group of a non-archimedean local field: the Haar measure, the Schwartz-Bruhat space and its dual -- definitions and topology. Action of the multiplicative group on these spaces.
1:30pm -- Lunch.
2:30pm Athena: local functional equation: the exact sequence of Schwartz spaces, the dimension theorem, and the proof of local functional equation.

Friday August 26:

10am: Asif: Fourier analysis on adeles and ideles: characters, Schwartz spaces, distributions, global zeta-integrals.
11:45am: Carmen: global functional equation -- eigendistibutions, dimension theorem, Fourier transform and the global functional equation.
1:30pm -- lunch.
2:30pm Justin: connections with other L-functions.
3:30pm Prof. Casselman, concluding remarks.

Approximate list of topics and sources:

(if the topic has been claimed, the tentative speaker's name appears in parentheses after the description).

Local and global fields and division algebras. Local fields: recall the definitions of: valuations, maximal ideal, residue field; topology. Global fields: archimedean and non-archimedean places, completions. [Division algebras: the definitions, compactness modulo centre; the classification of division algebras over a local field (maybe). -- it looks like in the end we will not be doing division algebras...] Sources: any book on number theory for the local fields; for the division algebras, Milne's notes on Class Field Theory, Section 1 in Chapter 4 (if ambitious, Sections 1-4 in Chapter 4). (Athena)
Adeles and ideles. Topology; the diagonal embedding of the field, and the quotient by its image. Sources: Cassels-Frohlich; a vignette by P. Garrett (especially the appendices, for this talk); Notes by D. Prasad, Section 1; Kudla's notes, section 1. Tate's Thesis, Section 3. (Lance)
Fourier analysis on the additive group of the local field (division algebra) -- the archimedean and non-archimedean cases. Haar measure. Schwartz spaces and distributions.
- Archimedean case (XinYu Liu)
- Non-archimedean case -- sources: Tate's thesis, sections 2.1-2.2, Notes by P. Walls, sections 1.2 -- 1.5; 1.12. Kudla's notes, section 3. (Lance and Carmen)
Fourier analysis on the multiplicative group the local field (division algebra) -- the archimedean and non-archimedean cases. Sources: Tate's thesis, section 2.3. (Lance)
Local L-functions, local zeta-integrals, local functional equation. Sources: Tate's thesis, sections 2.4 -2.5, Kudla's notes, Section 3; Notes by P. Walls, sections 1.6 - 1.11. Please note: Kudla's lectures followed the approach to proving the functional equation by A. Weil; P. Walls follows Kudla's article in his notes. It is better if we follow this approach too, rather than Section 2.4 of Tate's thesis. The idea is to use the relationship between Schwartz spaces on k^* and on k to compute the dimension of the space of eigendistributions, and to derive the functional equation from this. (Athena)
Fourier analysis on adeles and ideles: grossencharacters. Schwartz-Bruhat functions and distributions. Sources: Tate's thesis, section 4.1, Kudla's notes, Section 4; Notes by P. Walls, Sections 2.1 -- 2.5 (Asif)
Global zeta-integrals and the completed L-function. Sources: Tate's thesis, section 3, Kudla's notes, Section 4; Notes by P. Walls, Section 2.6. ; Eigendistributions, the dimension theorem, Global functional equation. Sources: Tate's thesis, sections 4.2 -4.4, Kudla's notes, section 4; Notes by P. Walls, Section 2.7 -2.11 (Carmen and Asif?)
Connections with Riemann zeta-function, Dedekind zeta-functions, Automorophic forms... (Justin)