Number theory learning seminar 20112012
The seminar will meet at the beginning Fridays 2:15  4:15 in Room 380W . Starting on Feb 15th, we will shift the seminar to Wednesdays 1:303:30 in Room 383N . In the spring quarter, which begins on Apr 4th, we will meet Thursday 122 in Room 383N .
The topic for 20112012 is padic Lfunctions and padic modular forms;
in past years we covered the Mordell conjecture (2010), modularity lifting (2009),
and the proof of the Weil conjectures (2008).
Here are list of references which may be relevant for this year's seminar:
(1) "The rationality of StarkHeegner points over genus fields of real quadratic fields" by Bertolini and Darmon
(2) "Hida families and rational points on elliptic curves" by Bertolini and Darmon
(3) "padic Lfunctions and padic periods of modular forms" by Greenberg and Stevens
(4) "On a Conjecture of Mazur, Tate, Teitelbaum" by Greenberg and Stevens
(5) Hida's papers, if people suggest specific ones, I will add them
References for CM theory:
(1) Background on Abelian Varieties from Mordell seminar by Brian Conrad
(2) Lecture Notes on Neron Models from Mordell seminar by Sam Lichtenstein
(3) Lecture Notes on semistable reduction from Mordell seminar by Brian Conrad
(4) (draft) Book on CM Liftings by Chai, Conrad, and Oort. Especially relevant are Chapters 1 and 2, and Appendix A.
The speaking slots are tentative, not set in stone: please do contact Brian, Samit or Akshay and express your desire to speak. The tentative schedule for the seminar can be found below after the notes for past talks.
Notes  use at your own risk.
These are informal notes written by each speaker. They may change
from time to time as we edit them.
Fall quarter 
1 
Sept 30 
Akshay + Samit 
Overview of StarkHeegner points and BertoliniDarmon Theorem 

2 
Oct 7,14 
Brandon 
Modular Symbols, Shimura periods, and Lfunctions 
pdf

3 
Oct 21 
Rebecca 
padic Lfunctions for Dirichlet characters 
pdf

4 
Oct 28 
Cameron 
padic Modular forms and Hida families (partial notes) 
pdf

5 
Nov 11 
Mike 
padic Lfunction of a modular form 
pdf

6 
Dec 6 
Payman 
Twovariable padic Lfunction 
pdf

Winter Quarter 
1 
Feb 15 
Conrad 
Intro to CM: Endomorphisms 
pdf

2 
Feb 22 
Conrad 
Intro to CM: CM Types, Reflex Fields, and Main Theorem 
pdf

3 
Mar 7, 14 
Arnav 
Endomorphisms: See schedule below for references to [CCO] 
None.

4 
Apr 22 
Mike 
Algebraic Hecke characters 
pdf

5 
Apr 29 
Brandon 
Algebraic Form of the Main Theorem of CM 
pdf

6 
May 10 
Daniel 
ShimuraTaniyama formula 
pdf

7 
May 17 
Jeremy 
Constructions with fractional ideals 
pdf

8 
May 24 
Iurie 
Proof of the adelic Main th'm 
pdf

Detailed Schedule
Fall:
Sept 30: [Akshay + Samit]
Intro lecture (heegner points, modular curves, field of definition, moduli interpretation, GZ, vague idea of SH points, DarmonBertolini theorem on rationality over genus fields, outline of proof  Introduction of BD Annals paper).
Oct 7,14: [Brandon] Lfunctions of modular forms, modular symbols and relation to Lvalues, Shimura periods (Section 1.1 of BD Annals paper and section 1.1 of BD Inv paper, including full proof of Prop 1.3.)
Oct 14,21: [Rebecca] padic measures, weight space, padic Lfunctions of Dirichlet characters.
Oct 28: [Cameron] Hida families, Eisenstein example (constant terms = padic Lfunction of Dirichlet characters), relationship to modular symbols, inverse limit of cohomology of tower of modular curves of ppower level, ordinary operator, Hida's theorem that the ordinary part is finite free Lambdamodule. Hecke operators, Hida Hecke algebra. Hida's control theorem. (Sections 2.12.2 in BD Annals paper, sections 1.21.3 in BD Inventiones paper, and relevant material from GreenbergStevens and Hida's papers.)
Nov 4: [Payman] continuation of last time. Additional topic: Lambdaadic galois representation and specialization to other weights.
Nov 11: [Mike] padic Lfunctions of classical modular forms, interpolation property and construction via modular symbols.
Nov 18: Cancelled.
Nov 25: off, thanksgiving
Dec 2: [Payman] GreenbergStevens perspective on padic modular forms: D = measures on Z_p \times Z_p, modular symbols valued in D, relationship to Hida families, 2variable padic Lfunction, MazurKitagawa padic Lfunction, (Section 3.1 of BD Annals paper and Section 1.4 of Inventiones paper)
Dec 9: No seminar due to an arithmetically interesting algebraic geometry seminar at 3 pm by Max Lieblich.
Winter
Jan 13: [Samit] Five Linvariants: (Darmon, GS, "algebraic", log(q)/ord(q), analytic). Proof that all are equal.
Jan 20: [Cameron] SL_2(Z[1/p])invariant modular symbols valued in Meas(P^1(Q_p)).
Jan 27: [Cameron] StarkHeegner points, proof that the SH point is given by an integral over D. (Sections 1.21.4 of BD Annals paper, section 2.3 of annals paper)
Feb 3: [Samit] Continuation of StarkHeegner points: NOTE Samit will speak for just 1 hour from 2:153:15 due to conflict with algebraic geometry seminar.
Feb 10: [Akshay] Proof of main Theorem in DarmonBertolini Annals paper (basically, section 3.2 to end). State Popa's formula without proof.
Feb 15: [Brian] Overview of CM Theory: Endormorphism Algebras and CM structures NOTE the change of day, the time is now 1:30  3:30 in 383N. Will assume background on abelian varieties as covered here.
Feb 22: [Brian] Overview of CM Theory Part II: Arithmetic aspects and preliminary versions of Main Theorem.
Feb 29: [Brian] Overview of CM Theory Part III: Reflex norm, algebraic Hecke characters, and precise versions of Main Theorem.
Mar 7: [Arnav] Results on endomorphism: Proofs (or highlights thereof) for 1.2.1.2, 1.2.1.3, 1.3.1.1, 1.3.4, 1.4.4.1, 1.6.4, 1.6.2.3, and 2.2.2 (in char. 0) in the CMlifting book.
Mar 14: [Arnav] Endomorphisms continued.
Mar 21: SPRING BREAK
Change in time and room, seminar will be held Thurs 12  2 pm in 383N.
Apr 5: [Mike] Algebraic Hecke characters and $\ell$adic avatars (2.4.12.4.9). Also discuss archimedean avatar.
Apr 12: [Arnav] Endomorphisms finale. Continuation of Mike's talk if necessary.
Apr 19: [Mike] Algebraic Hecke characters finale.
Apr 26: [Brandon] Qpolarizations, reflex torus, and adelic Main Theorem and applications (A.2.2A.2.5.9).
May 3: [Brandon and Daniel] End of adelic Main Theorem and beginning of ShimuraTaniyama formula.
May 10: [Daniel] ShimuraTaniyama formula (2.1.5.1 and here. )
May 17: [Jeremy] Constructions with fractional ideals (A.2.6).
May 24: [Iurie] Completion of the proof of adelic Main Theorem (A.2.7).
May 31: [Akshay] adelic lattices and analytic results (A.2.8, A.2.5.11, and A.4.6.1 if time permits).