Student Workshop on Tate's thesis, August 2426, 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 emails about this yet), please email 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 Lfucntions (the Lfunctions 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 Lfunctions, and with the concept of a number field and local field. Participation in the summer student
seminar on Lfunctions 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
nonarchimedean 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 afterlunch, 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
nonarchimedean local field: the Haar measure, the SchwartzBruhat 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 zetaintegrals.
 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 Lfunctions.
 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 nonarchimedean 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 14 in Chapter 4).
(Athena)
 Adeles and ideles.
Topology; the diagonal embedding of the field,
and the quotient by its image.
Sources: CasselsFrohlich;
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 nonarchimedean cases.
Haar measure.
Schwartz spaces and distributions.
 Archimedean case (XinYu Liu)
 Nonarchimedean case  sources:
Tate's thesis, sections 2.12.2,
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 nonarchimedean cases.
Sources: Tate's thesis, section 2.3.
(Lance)
 Local Lfunctions, local zetaintegrals, 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.
SchwartzBruhat 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 zetaintegrals and the completed Lfunction.
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 zetafunction, Dedekind zetafunctions,
Automorophic forms...
(Justin)