TITLE/ABSTRACT Brakocevic: Anticyclotomic p-adic L-function of central critical
Rankin-Selberg L-value.
Januszewski: p-adic Rankin-Selberg convolutions. Abstract: Fix a number field k, a finite place p, and two (suitably nice) automorphic representations of GL(n) and GL(n-1) respectively.
We will discuss how to construct a p-adic L-function interpolating the
critical values of the associated Rankin-Selberg
L-function. In motivic terms this L-function
corresponds to the tensor product of the motives (conjecturally) associated to
these two automorphic representations. This
construction works for arbitrary n and thus gives a higher dimensional
generalization of Mazur's construction of the p-adic L-function associated to
an elliptic curve over Q (resp. a cusp form). Recommended reading, I would
suggest Mazur and Swinnerton-Dyer's "Arithmetic of Weil curves", Inventiones Mathematicae 25,
1974. Kakde: Noncommutative main conjecture of Iwasawa theory for totally real number fields. Abstract: I will first give a formulation of the main conjecture of Iwasawa theory for totally real number fields. Then we
will see how a wonderful strategy of David Burns and Kazuya Kato together
with our computations of K_1 groups of Iwasawa
algebra reduces the proof of the main conjecture in the non-commutative case
to the commutative case and certain congruences
between Deligne-Ribet, Cassou-Nogues
p-adic L-functions. Finally we will see that commutative case can be deduced
from the result of Wiles on classical Iwasawa main
conjecture and the congruences are proven using the
q-expansion principle of Deligne-Ribet. Pilloni: Geometric overconvergent modular forms. Abstract : In the first lecture we will give a geometric definition of overconvergent modular forms of any p-adic weight for the
modular curve. We can then re-obtain Coleman's families without using the
Eisenstein series. In the second lecture we will
generalize the construction of
the first lecture to PEL Shimura varieties to obtain finite slope families of
modular forms over the total weight space. Prasanna: p-adic L-functions and the Griffiths group. In the first lecture, I will
give an introduction to some problems on algebraic cycles. In particular, I
will discuss the Griffiths group (of cycles homologically equivalent to zero
mod those algebraically equivalent to zero) and its role in the formulation
of the Bloch-Beilinson conjecture. In the second
lecture I will discuss some applications of p-adic L-functions to these
questions. Sharifi: Galois cohomology, Iwasawa
theory, and p-adic L-functions. Abstract: Galois cohomology is a powerful tool in
the study of the arithmetic of local and global fields. In particular, it is needed to define
Selmer groups, the structure of which are expected to be connected with
p-adic L-functions. In Iwasawa theory, it is a ubiquitous tool, yet much about
the actual groups that one studies remains mysterious. Moreover, work of the speaker has
exposed some connections between operations in Galois cohomology
and certain p-adic L-functions that were not previously expected. Assuming a basic knowledge of Galois cohomology and class field theory, the speaker will
elucidate something of the structure of cohomology
groups of interest, dualities among them, different ways to think about
operations on them, and their applications to Iwasawa
theory, even of the noncommutative sort. Connections will be drawn to classical
Iwasawa theory and its long-since proven main
conjecture, which relates the structure of certain Galois groups to Kubota-Leopoldt p-adic L-functions. Finally, we will explore the more
mysterious connection between cup products on cohomology
groups and p-adic L-functions of certain modular eigenforms, both on a basic level and in a Hida-theoretic and Iwasawa-theoretic
context. Tilouine: Introduction to companion forms. Abstract: In my first two talks, I'll explain the definition and the use of BGG
complex to prove Gross theorem on companion forms by solving a differential
equation. In the third talk, I'll explain how this may generalize to certain
bigger groups. 1) BGG complex and companion
forms for $GL_2(\Q)$ 2) same, part II 3) The case of $GSp_4(\Q)$ Vatsal: Algebracity of L functions for GL_{2}. I will give 3 talks , at the end
of which I will explain how to prove classical algebraicity
results of Shimura on the special values of L functions for modular forms on
GL_2. The talks should be relatively elementary, assuming no more than basic
facts about modular forms and modular curves. A basic reference for all the facts we
need is the survey article by Diamond and Im,
"modular forms and modular curves" (1995). Weinstein: Local Langlands and the tower of modular curves. Abstract: A cuspidal
eigenform, as we know from Deligne,
determines a Galois representation.
A cuspidal eigenform
also determines (indeed is the same thing as) an automorphic representation of GL(2). Galois representations and automorphic representations both have "local
components". By a very nice
theorem of Deligne-Carayol, one sort of local
component determines the other.
(Before this it was not known that a modular elliptic curve had the
same conductor as the level of the corresponding cusp form!) We will revisit this result and
interpret it in the language of the geometry of the tower of modular curves
X(p^n).
In doing so we will encounter the local Langlands
correspondence for GL(2), which we will summarize. In the final half hour we will present
some work in progress concerning the resolution of singularities of the tower
of modular curves. There will be
pictures. (We will keep this talk as
accessible as possible for graduate students.) |