LECTURE    SCHEDULE   MINI-COURSES  AFTERNOON    TALKS                                                                                                                               Arithmetic Geometry                                         NCBS Bangalore, March 23-29, 2008                                                                               PROGRAMME         Talks will begin on Monday, March 24 and will last through Saturday, March 29.     March 23 (Sunday) is arrival day, and March 30 (Sunday) is departure day.            The lecture schedule is here.  NB: Wednesday afternoon is free.      There will be a dance performance by Kalakshetra Repertory Company on Tuesday evening,      and a conference dinner on Thursday evening.              There will be three mini-courses in the mornings, as follows:     I. Speakers: John Coates and Ralph Greenberg               Title: p-adic L-functions               Abstract: The four lectures will discuss the construction of p-adic        L-functions attached to elliptic curves and modular forms without        assuming any prior knowledge.        Greenberg's two lectures will discuss the construction        given by Mazur and Swinnerton-Dyer and some subsequent developments.               Coates's two lectures will discuss the one and two variable        p-adic L-functions attached to elliptic curves with complex        multiplication, and possibly a little of the first known results (due to        Kakde) on the existence of non-abelian p-adic L-functions.       2. Speakers: Laurent Berger, Kiran Kedlaya, Jean-Pierre Wintenbeger                 Title: p-adic local Langlands and the Fontaine-Mazur conjecture         Abstract:  here     3. Speakers: Philippe Michel (unable to attend) and Ritabrata Munshi         Title: Analytic number theory for modular forms and applications           Abstract: In these instructional lectures we discuss several basic         aspects of the analytic theory of modular forms and of their associated         L-functions along with a discussion of the existing techniques: these         include the various manifestations of the spectral gap property, the         non-vanishing problem and the subconvexity problems for central values         of L-functions.                 In the second part of these lectures, we discuss several applications         and interpretations of some of the results explained above.         In particular, we explain some consequences of these results to the         study on the distribution properties of "quadratic cycles" on various         arithmetic quotients related to modular forms (for instance Heegner         points on a modular surface or closed geodesics on it).         There will be several afternoon talks. Titles and abstracts:         Speaker: Massimo Bertolini     Title: Generalised Heegner cycles and p-adic Rankin L-series     Abstract: In our talk, we consider the image by the p-adic Abel-Jacobi                        map  of certain higher dimensional algebraic cycles,                      generalising the Heegner cycles on Kuga-Sato varieties.                      We explain that this image is related to values outside the range of                      classical interpolation of a p-adic L-function, which interpolates the                      special values of the Rankin L-series attached to a cuspidal eigenform                      and to binary theta-series of varying weight. We will also mention                     applications to the construction of points on CM elliptic curves.                      These results are part of an ongoing collaboration with Henri Darmon                      and Kartik Prasanna.        Speaker: Chandan Dalawat     Title: The ramification filtration in local Kummerian extensions     Abstract: We shall discuss how the ramification filtration on the maximal                      abelian extension of exponent p of a p-adic field containing a                      primitive p-th root of 1 is orthogonal, with respect to the Kummer pairing,                      to the filtration by higher units on the multiplicative group modulo p-th                      powers.  We hope to show how this result is cognate to Stickelberger's                      congruence for the absolute discriminant of a number field.        Speaker: Fred Diamond     Title: Serre's conjecture and mod p Langlands correspondences     Abstract: Serre's conjecture, now a theorem of Khare and Wintenberger,                    states that every odd, irreducible 2-dimensional mod p Galois                    representation arises from a modular form.   This result can                    be viewed as a global mod p Langlands correspondence; moreover                    Serre's refinement predicting the weight and level of the form                    suggests compatibility with a hypothetical local correspondence.                    This talk will be a survey describing what's known for GL_2 over Q,                       and where obstacles lie ahead even for GL_2 over number fields.                    In particular, I'll discuss work of Emerton making the local-global                    compatility statement precise for GL_2 over Q, and work of                    Breuil and Paskunas illustrating the difficulty of constructing                    a mod p local Langlands correspondence for GL_2 of a p-adic                     field.     Speaker: Bas Edixhoven     Title: Fast computation of Ramanujan's tau-function     Abstract:  Joint work with J-M. Couveignes, R. de Jong, F. Merkl and J. Bosman.                      See http://arxiv.org/abs/math.NT/0605244 for details. If time                      permits, an application to theta series of lattices will be given.       Speaker: Jordan Ellenberg (unable to attend)     Title: Problems of arithmetic distribution and stable cohomology of moduli spaces     Abstract: Number theorists have long been interested in questions                     about distribution of arithmetic objects: what is the frequency that                     the class group of a random quadratic imaginary field contains                     (Z/3Z)^r? How many extensions K/Q with Galois group S_5 have                     discriminant less than N? How many elliptic curves over Q are there                     with conductor at most N? In general, little is known about such                     questions. It turns out that the analysis of such questions over                     function fields over finite fields reveals surprising connections                     with problems of current interest in topology and algebraic geometry.                     In particular, we explain how a purely topological theorem about                     stabilization of cohomology for families of congruence subgroups of                     mapping class groups (or, in the algebraic geometer's notation,                     families of Hurwitz spaces over C) would imply function-field                     versions of several distributional conjectures in number theory (and                     several other statements which have not previously been conjectured),                     and we describe some partial progress towards the desired topological                     results. Joint work with Akshay Venkatesh and Craig Westerland.     Speaker: Eknath Ghate     Title: Nearly ordinary deformations and the splitting question     Abstract: The local representation attached to a p-ordinary form of weight at least two                      is known to be reducible. We describe a criterion for this representation                      to be split in terms of whether the underlying form has CM. This is joint                      work with Hida, and builds on earlier work with Vatsal.                          Speaker: Minhyong Kim     Title: Fundamental groups, principal bundles, and rational points     Abstract: In the early 20th century, Weil gave an algebraic construction of                      the Jacobian as a tool for studying Diophantine problems on algebraic                      curves. After unsuccessful attempts at proving the Mordell conjecture,                      he contemplated the importance of constructing non-abelian analogues                      of the Jacobian, and developed the theory of vector bundles on                      algebraic curves to this end.                      We will discuss this history a little bit, and describe how some progress                      along these lines can be made through the use of non-abelian                      fundamental groups and associated principal bundles.       Speaker: Shin-ichi Kobayashi     Title: p-adic L-functions at supersingular primes for CM elliptic curves     Abstract: We discuss about an integral structure on p-adic Fourier theory                     by Schneider-Teitelbaum and we generalize Amice's result on                     locally analytic functions on Z_p  to that on the integer ring                     of a general local field K.                     Then, as an application, we construct certain p-adic distributions                     on O_K which recover cyclotomic p-adic L-functions at supersingular                     primes for elliptic curves with CM by O_K.     Speaker: Tadashi Ochiai     Title: On p-adic L-functions for Hida families of Hilbert modular forms     Abstract: To generalize Iwasawa theory it is important to generalize                      constructions of p-adic L-functions attached to various p-adic                      families of Galois representations. After giving a brief introduction                      to an ongoing project to generalize Iwasawa theory, I explain the                      constuction of a several variable p-adic L-function associated to                      a Hida family of Hilbert modular forms. This is a joint work with                      Mladen Dimitrov.     Speaker: C. S. Rajan     Title: Weil etale topology and Lichtenbaum conjectures on zeta values         Abstract:     Speaker: V. Suresh     Speaker: B. Ramakrishnan     Title: Twisted Averages of L-functions     Abstract: In this talk we discuss some results on twisted averages of                     L-functions of modular forms of integral and half-integral weight     Speaker: V. Suresh     Title: Galois cohomology of function fields of p-adic curves     Abstract: In this talk we describe the structure of the second and third                      Galois cohomology groups of function fields of p-adic curves.                       We also study the structure of division algebras and give an                      application to quadratic forms  over such fields.     Speaker: Jacques Tilouine     Title: Companion forms for GSp(4) and the BGG complex     Abstract: We have formulated with F. Herzig a Serre conjecture for GSp(4). Among                      the (generic) 20 potential Serre weights, we can explain conjecturally 4                      through the coherent dual BGG complex theory and 4 others through                     Urban's analytic dual BGG complex theory. This explanation provides a                      hierarchy of complexity among these potential companion forms. Using                      Faltings' mod p comparison theorem, we can prove the existence of the                      first non trivial of the 4 companion forms in the coherent case.