Suggested reading for mini-courses (three lectures by each speaker) before the workshop


For interested students, we put here some suggestions by the mini-course speakers for preparatory reading.



Jacques Tilouine (Introduction to companion modular forms modulo  p):

Use of BGG complex to prove existence of companion modular forms first proven by B. Gross.

-I recommend our Asterisque volume 280 by Mokrane, Polo and myself for the definition of the BGG complex and its basic properties.

-There is a down-to-earth paper by Tilouine on BGG for GL2 submitted for Vasudevan's 60th birthday.



Vinayak Vatsal (Algebraicity and integrality of modular L values):

Books recommended for preparatory reading.

-Shimura: Introduction to the arithmetic theory of automorphic functions,Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers,Tokyo.

-H. Hida: Elementary theory of p-adic L-functions and Eisenstein series, Cambridge University Press, 1993.

-F. Diamond and J. Im: Modular Curves and Modular Forms, In the book Seminar on Fermat's last theorem, Publihsed by the Amer. Math. Soc. in the series C.M.S. Conference Proceedings (Ed: K Murty).



Chris Skinner and Eric Urban

Tentative title of the lecture series: Pull-back formulas, differential operators and construction of p-adic families of holomorphic cusp forms for unitary groups.

-The main reference is Shimura's books (AMS Mathematical Surveys and Monographs V. 82 2000 and CBMS Regional conference series No.93 1991) on automorphic forms on unitary and symplectic groups, there will be also Urban's paper on the eigen-varieties and some other things we will specify later.



Samit Dasgupta (Hida families and Gross--Stark units over totally real fields):

Background reading for students.  Three papers would be

(1) Ribet's paper "A modular construction of unramified p-extensions of Q(mu_p)",

(2) Greenberg-Stevens paper on the MTT conjecture, entitled "On the conjecture of Mazur, Tate, and Teitelbaum",

(3) Wiles paper on "The Iwasawa conjecture over totally real fields."

Most relevant, of course, will be knowledge of what a Hida family is, and what the associated Galois representation looks like.  Presumably this will be important background for all the lectures (esp. mine, Skinner, Urban, Tilouine).  An easiest reference would be Hida's blue book on Eisenstein series and Hida's papers in Inventiones Math. 85 (1986) and Ann. Scient. Ec. Norm. Sup. 4th series 19 (1986).



Romyar Sharifi

Tentative title: Applications of cohomological operations to Iwasawa theory and modular L-values.

Possible reading (and suggestions regarding it):

  (1) McCallum, Sharifi - A cup product in the Galois cohomology of number fields (first six to eight sections),

  (2) Neukirch, Schmidt, Wingberg - Cohomology of number fields (best as a reference for duality, Galois groups with restricted ramification, and Iwasawa theory),

  (3) Washington - Introduction to cyclotomic fields (to get a glimpse of Iwasawa theory),

  (4) Mazur, Tate, Teitelbaum - On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer,

  (5) The papers (esp. Ribet, Wiles, 2 papers of Hida) suggested by Dasgupta,

  (6) M. Ohta - Ordinary p-adic tale cohomology groups attached to towers of elliptic modular curves (just to get a feel of it and his related papers),

  (7) J. Nekovar - Selmer complexes (try the introduction).



More reading suggestions will be forthcoming.