Algebraic-geometry/Number-Theory Seminar

September 16 (in Krg 413)
Cristian Popescu (University of California San Diego)
1-motives and special values of equivariant L-functions.
Abstract: We will discuss our recent proof (joint work with C. Greither) of a conjecture linking $\ell$--adic realizations of 1--motives and special values of equivariant $L$--functions in characteristic $p$, refining earlier results of Deligne and Tate. As a consequence, we will give proofs (in the characteristic $p$ setting) of various central classical conjectures on special values of $L$--functions, namely those due to Coates-Sinnott, Brumer-Stark, and Gross. Also, we will indicate how this theory can be extended to characteristic $0$.

September 23
Florin Ambro (Johns Hopkins University)
On toric singularities
Abstract: Singularities in birational geometry are expected to be separated into series according to a natural invariant, which is a rational number. In this talk I will discuss this invariant in the special case of toric singularities. It has an arithmetic flavour, being a local version of Minkowski's first minimum of a convex body. Using this idea, I will show that the above invariant is equivalent to the index of the toric singularity.

September 30
Nero Budur (U. of Notre Dame)
Singularities and Hodge filtrations
Abstract: This talk consists of two parts. In the first part we reviewvarious points of view on singularities: motivation, definitions, relations, and computability. In the second part we discuss recent work on the Hodge filtration for local systems. Applications include: computability of invariants of singularities in the hyperplane arrangement case, and polynomial periodicity of Hodge numbers for congruence covers.

November 4
Alan Huckleberry (Ruhr-University of Bochum)
K3-surfaces with special symmetry

November 11 (at 3PM, joint with the Complex-geometry seminar)
Robert Berman (Chalmers University)
TBA

February 3, 2009
J. Getz (Princeton U.)
Trace Formulae and Locally Symmetric Spaces

February 17, 2009
Zeev Rudnick (Tel Aviv University)
Statistics of the zeros of zeta functions in ensembles of hyperelliptic curves over a finite field

February 19, 2009
Ramin Takloo-Bighash (UIC)
Period Integrals and Special Values of L-functions

April 21, 2009 (at 3:30pm)
Donghoon Park (Brown U.)
1-motives with torsion and Cartier duality
Abstract: Deligne introduced the category of 1-motives and its realization functors. He showed Cartier duality for this category and also proved that a seminormal complex algebraic curve has a 1-motive over $\mathbb{C}$ whose realization is isomorphic to the first (singular, $l$-adic, and De Rham) cohomology group of this curve. For such a curve, Lichtenbaum defined three more 1-motives corresponding to its cohomology with compact support, homology, Borel-Moore homology. Ramachandran showed that cohomological and homological 1-motives are dual to each other. I will give the (additive) category of 1-motives with torsion and show Cartier duality for this category. Some people (Barbieri-Viale and Bertapelle) already considered such a question and defined an abelian category of 1-motives with torsion. I will also compare these two categories.