## Number theory learning seminar 2011-2012

The seminar will meet at the beginning Fridays 2:15 - 4:15 in Room 380-W . Starting on Feb 15th, we will shift the seminar to Wednesdays 1:30-3:30 in Room 383-N . In the spring quarter, which begins on Apr 4th, we will meet Thursday 12-2 in Room 383-N . The topic for 2011-2012 is p-adic L-functions and p-adic 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 Stark-Heegner 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) "p-adic L-functions and p-adic 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 semi-stable 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 Stark-Heegner points and Bertolini-Darmon Theorem 2 Oct 7,14 Brandon Modular Symbols, Shimura periods, and L-functions pdf 3 Oct 21 Rebecca p-adic L-functions for Dirichlet characters pdf 4 Oct 28 Cameron p-adic Modular forms and Hida families (partial notes) pdf 5 Nov 11 Mike p-adic L-function of a modular form pdf 6 Dec 6 Payman Two-variable p-adic L-function 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 Shimura-Taniyama 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 S-H points, Darmon-Bertolini theorem on rationality over genus fields, outline of proof --- Introduction of BD Annals paper).

Oct 7,14: [Brandon] L-functions of modular forms, modular symbols and relation to L-values, 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] p-adic measures, weight space, p-adic L-functions of Dirichlet characters.

Oct 28: [Cameron] Hida families, Eisenstein example (constant terms = p-adic L-function of Dirichlet characters), relationship to modular symbols, inverse limit of cohomology of tower of modular curves of p-power level, ordinary operator, Hida's theorem that the ordinary part is finite free Lambda-module. Hecke operators, Hida Hecke algebra. Hida's control theorem. (Sections 2.1--2.2 in BD Annals paper, sections 1.2-1.3 in BD Inventiones paper, and relevant material from Greenberg-Stevens and Hida's papers.)

Nov 4: [Payman] continuation of last time. Additional topic: Lambda-adic galois representation and specialization to other weights.

Nov 11: [Mike] p-adic L-functions of classical modular forms, interpolation property and construction via modular symbols.

Nov 18: Cancelled.

Nov 25: off, thanksgiving

Dec 2: [Payman] Greenberg-Stevens perspective on p-adic modular forms: D = measures on Z_p \times Z_p, modular symbols valued in D, relationship to Hida families, 2-variable p-adic L-function, Mazur-Kitagawa p-adic L-function, (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 L-invariants: (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] Stark-Heegner points, proof that the SH point is given by an integral over D. (Sections 1.2--1.4 of BD Annals paper, section 2.3 of annals paper)

Feb 3: [Samit] Continuation of Stark-Heegner points: NOTE Samit will speak for just 1 hour from 2:15-3:15 due to conflict with algebraic geometry seminar.

Feb 10: [Akshay] Proof of main Theorem in Darmon-Bertolini 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 383-N. 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 CM-lifting book.

Mar 14: [Arnav] Endomorphisms continued.

Mar 21: SPRING BREAK

Change in time and room, seminar will be held Thurs 12 - 2 pm in 383-N.
Apr 5: [Mike] Algebraic Hecke characters and $\ell$-adic avatars (2.4.1--2.4.9). 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] Q-polarizations, reflex torus, and adelic Main Theorem and applications (A.2.2--A.2.5.9).

May 3: [Brandon and Daniel] End of adelic Main Theorem and beginning of Shimura-Taniyama formula.

May 10: [Daniel] Shimura-Taniyama 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).