For interested students, we put
here some suggestions by the mini-course speakers for preparatory reading.
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 GL
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 -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).
Tentative title of the lecture
series: -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.
Background reading for
students. Three papers would be (1) Ribet's paper "A
modular construction of unramified p-extensions of Q(mu_ (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).
Tentative title: 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. |