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 GL2.
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.)