The Fellowship of the Ring seminar is held on **Friday**
from **10:30 - 12:00 am (NOTE NEW TIME!!) **
at Brandeis University, room Goldsmith 226. For directions to the Brandeis
Mathematics Department, click
here.

- January 27, 10:30 am

Paul Monsky, Brandeis University

"Hilbert-Kunz functions for plane curves "

Abstract: Let k be an algebraically closed field of characteristic p, and f in k[x,y,z] be an irreducible form of degree d. If q=p^n, e_n denotes the k-dimension of k[x,y,z]/(x^q,y^q,z^q,f). When the plane curve defined by f is non-singular, there are very precise results about the e_n, proved by Brenner and by Trivedi, and completely explicit results in various special cases, due to Teixeira and me. When the curve is singular, much less is known. I'll describe what I believe the general result to be in this situation, and present partial results and computer generated evidence for my beliefs. - February 3, 10:30 am

No meeting

- February 10, 10:30 am

Kevin Buzzard, Imperial College, Harvard

"Everything I know about the 2-adic eigencurve"

Abstract: In papers with Calegari and Kilford (both of which just recently appeared in Compositio) and a preprint with Calegari, I try to get a real "handle" on the 2-adic level 1 eigencurve, a geometric object parametrising certain modular forms. I'll talk about the results of these papers, try to draw some pictures, and give some hints as to what goes into the proofs. - February 17, 10:30 am

Dan Abramovich, Brown

"Reductive group-schemes, tame stacks, and stable maps"

Abstract: Joint work with M. Olsson and A. Vistoli. We describe finite linearly reductive group schemes in any characteristics, and use them to define tame Artin stacks, arguably an appropriate replacement in characteristic p for Deligne--Mumford stacks in characteristic 0. In particular there is a good and useful theory of stable maps into tame Artin stacks. - February 24, 10:30 am

No meeting

- March 3, 10:30 am

Keith Conrad, Univ. of Connecticut

"Root numbers and ranks in positive characteristic"

Abstract: We will discuss the construction of a family of elliptic curves in characteristic p where the ranks are systematically higher than expected based on root number calculations (though not contradicting any well-known conjectures). An analogue of this particular phenomenon in characteristic 0 is only expected to occur for isotrivial families. The construction in characteristic p is not isotrivial and exploits a surprising distinction between the Mobius function in characteristic 0 and characteristic p. This is joint work with B. Conrad and H. Helfgott. - March 10, 10:30 am

No meeting

- March 17, 10:30 am

Gaetan Chenevier, Univ.~Paris 13, Harvard

"On number fields with given ramification"

Abstract: Let p be a prime number, K a maximal algebraic extension of Q unramified outside p. I will talk about the injectivity of the natural maps Gal(Qp^bar/Qp) --> Gal(K/Q). - March 24, 10:30 am

Anish Ghosh, Brandeis

'Diophantine approximation on manifolds : The dynamical technique"

Abstract: We will present a survey of probabilistic results in Diophantine approximation. A dynamical technique due to Kleinbock and Margulis will then be introduced and will be used to obtain some new results on Diophantine properties of affine subspaces and their submanifolds. - March 31, 10:30 am,
**ROOM 317, THESIS DEFENSE**

Aftab Pande, Brandeis

"Local constancy of dimensions of Hecke eigenspaces of automorphic forms" - April 7, 10:30 am

Mark Kisin, Chicago, Harvard

"Galois representations and Artin motives"

Abstract: The unramified Fontaine-Mazur conjecture predicts that a global p-adic Galois representation which is unramified at p and unramified outside finitely many primes has finite image. I will explain a proof of this conjecture for representations which come from geometry. The proof, which is not difficult, uses some of the deepest things we know about motives, such as compatible systems, the Weil conjectures, and the p-adic comparison isomorphism. This is joint work with Sigrid Wortmann. - April 11, 10:30 am
(Note this is a
**TUESDAY**for which Brandeis is on a Friday schedule)

Samit Dasgupta, Harvard

"Shintani zeta-functions and Gross-Stark units for totally real fields"

Abstract: Let F be a totally real number field and let p be a finite prime of F, such that p splits completely in the finite abelian extension H of F. Stark has proposed a conjecture stating the existence of a p-unit in F with absolute values at the places above p specified in terms of the values at zero of the partial zeta functions associated to H/F. Gross proposed a refinement of Stark's conjecture which gives a conjectural formula for the image of Stark's unit in F_p*/E, where F_p denotes the completion of F at p and E denotes the topological closure of the group of totally positive units of F. We propose a further refinement of Gross' conjecture by proposing a conjectural formula for the exact value of Stark's unit in F_p*. Our formula may be viewed as an explicit class field theory for F. - April 21, 10:30 am

No meeting

- April 28, 10:30 am

Matthew Emerton, Northwestern, Harvard

"Elliptic curves of odd modular degree"

Abstract: A celebrated theorem of Breuil, Conrad, Diamond, and Taylor (building on the ground-breaking ideas of Wiles) shows that any elliptic curve E over Q is modular; that is, for some integer N there is a surjective map X_0(N) --> E (where X_0(N) denotes the modular curve of level N). In fact there are many such maps, but there is an essentially unique minimal one: here minimal means that both N and the degree m of the map are taken to be as small as possible. Thus to any elliptic curve E over Q, we have two integer invariants attached, namely N and m. The first of these is well understood -- it is the so called conductor of E. The second of these, known as the modular degree of E, is less well understood. In this talk, I will describe joint work with Frank Calegari, which gives some information on the parity of m. - May 5, 10:30 am

Ben Howard, Boston College

"Anticyclotomic Iwasawa theory of CM elliptic curves"

Abstract: Iwasawa theory in the anticyclotomic Z_p-extension of a quadratic imaginary field can be more exotic than the corresponding cyclotomic theory, as functional equations force L-functions to vanish frequently. I will discuss the Iwasawa theory of a CM elliptic curve over Q as one ascends the anticyclotomic Z_p extension of the CM field. The behavior of the p-Selmer group depends on the sign in the functional equation over Q and whether p is a prime of ordinary or supersingular reduction; the rank of the Selmer group grows in different ways in each case. This is joint work with A. Agboola. - May 10, 11:00 am (
**WEDNESDAY**Note exceptional date/time.)

Matthew Szczesny, Boston University

"The chiral de Rham complex, elliptic genera, and supersymmetry"

Abstract: I will discuss the chiral de Rham complex (introduced by Malikov, Schechtman, and Vaintrob in '98) and its relation to the physical sigma-model. I will also discuss how various geometric predictions made by physics (the appearance of the elliptic genus, mirror symmetry, and enhanced supersymmetry on manifolds with special holonomy) appears in the picture.

Schedule of Talks, 2001-2005.