# Arithmetic and Geometry Seminar, Winter 2002

Friday afternoons in South Hall, Room 4607B

Organisers: A. Agboola, J. McKernan

January 18, 2002  January 25, 2002 January 11, 2002 Organisational Meeting TBA Abstract: Bisi Agboola, UCSB Line bundles, rational points, and ideal classes Abstract: To what extent are line bundles on arithmetic varieties characterised by their restriction to integral points? I shall discuss an Arakelov variant' of the above question. Luis Finotti, UCSB Minimal Degree Liftings of Hyperelliptic Curves Abstract: The main goals of the talk will be to analyse the properties of lifts of hyperelliptic curves over perfect fields of characteristic $p>2$ (to hyperelliptic curves over the ring of Witt vectors) that have the coordinate functions of the lift of points having minimal degrees. I will try to go over the following topics (depending on time): upper and lower bounds for the minimal degrees, a necessary condition to achieve the lower bounds, lifting of the Frobenius, computational aspects, an explicit formula for derivatives of coordinate functions of the elliptic Teichm\"uller lift, analyse the case of ordinary elliptic curves in more detail and finally, give a possible example for Mochizuki's theory of `canonical liftings'' in the case of genus 2 curves. A preprint of a paper containing these topics can be found at: http://www.math.ucsb.edu/~finotti/ (A reviewed version should be there by Wednesday morning.) James McKernan, UCSB 3-fold and 4-fold Log Flips after Shokurov I Abstract: These talks will give an introduction to Shokurov's recent preprint where he gives an inductive method to construct log flips. In particular Shokurov's preprint has a relatively simple proof of the existence of 3-fold flips. This is especially exciting, as the only other proof of 3-fold flips, combines two monumental papers; Mori's original proof of 3-fold flips (for which he earnt the field's medal) and Shokurov's proof of existence of 3-fold log flips. Moreover both of these papers relied heavily on classification results, which can only feasibly be carried out in very low dimensions. Shokurov's proof does not rely on an explicit classification and provides the first strong evidence for the existence of flips in higher dimensions. Roughly speaking, Shokurov's proof proceeds in two steps. It is well known that the existence of the flip is equivalent to finite generation of an associated algebra (known as a divisorial algebra). Moreover, using Shokurov's original paper, it is known how to reduce to the case of log flips with non-empty reduced boundary. It is then straightforward to prove that the original algebra is finitely generated iff the restricted algebra is finitely generated. The induction would be complete, except for the unfortunate fact that the restricted algebra need not be a divisorial algebra. However Shokurov has isolated two key properties of the restricted algebra: (1) the restricted algebra is a subalgebra of a divisorial algebra (2) the restricted algebra is "asymptotically log canonically saturated" The first condition is self-explanatory. The second condition is part of the many new ideas that appear in Shokurov's preprint and will be explained in the lectures. One of the major contributions of Shokurov's preprint is a conjecture to the effect that algebras satisfying (1) and (2) are finitely generated and the proof that this conjecture holds in dimension two and that a restricted version holds in dimension three to prove the existence of log flips in dimension four. In the two lectures I hope to give an introduction to this exciting new direction of research. I also hope to indicate some of the ideas and definitions that appear in the proof that will almost certainly have a major influence on the future direction of research in higher dimensional geometry. Contents: Lecture I: History of flips, survey of known results, outline of Shokurov's recent proof of existence of 3-fold and 4-fold log flips, introduction to bi-divisors and their associated linear systems and algebras, some elementary reduction steps. Lecture II: Log canonical saturation, existence of a limit, proof in the one dimensional case via approximation, descent and the obstruction bi-divisor, predictions, the CBS conjecture, proof of the CBS conjecture in dimension two, outline of the four dimensional case. Relevant web links: http://www.maths.warwick.ac.uk/~miles/3folds/ has links to parts of Shokurov's recent preprint. http://can.dpmms.cam.ac.uk/~corti/shok.ps.gz gives a nice introduction to Shokurov's preprint, containing a reasonably short, essentially self-contained, proof of three fold log flips. James McKernan, UCSB 3-fold and 4-fold Log Flips after Shokurov II Abstract: See abstract above. Lynne Walling, University of Colorado Some applications of Hecke operators on Siegel modular forms Abstract: No seminar this week Abstract: Andrew Kresch, University of Pennsylvania Arithmetic standard conjectures on Grassmannians Abstract: We describe joint work with Harry Tamvakis in which we prove, for the Grassmannian varieties G(2,N), the arithmetic standard conjectures (Hard Lefschetz and Hodge Index) proposed by Henri Gillet and Christophe Soule'. The work shows how the primitive cohomology of the varieties in some sense controls the complexity of the conjectures. The conjectures are translated by means of arithmetic Schubert calculus into a system of inequalities. These then lead to a bound (conjectural, but proved in a suitably averaged sense) on values of certain orthogonal polynomials called Racah polynomials. Winfried Kohnen, University of Heidelberg Lifting modular forms of half-integral weight to Siegel modular forms Abstract: In 1999, T. Ikeda proved a conjecture due to Duke-Imamoglu about the existence of a lifting from elliptic Hecke eigenforms to Siegel Hecke eigenforms. The purpose of this talk is to give an explicit linear version of this lifting map. No pre-knowledge on Siegel modular forms will be supposed. No seminar this week Abstract: Sinai Robins, Temple University (visiting UC San Diego) A new class of zeta functions attached to cones and polytopes Abstract: There is a fascinating connection between polyhedra and zeta functions defined over them. These new zeta functions are extensions of the Riemann zeta function, to several variables. The functions we study are similar to Shintani's zeta functions but in fact possess functional equations between the (simplicial) cone they are defined over, and its dual cone. From the perspective of combinatorial geometry, these functions help enumerate weighted lattice points in objects, and from the perspective of number theory these functions are meromorphic extensions of the Riemann zeta function. Edray Goins, Caltech Icosahedral Q-Curve Extensions Abstract: Given a complex 2-dimensional Galois representation over $\mathbb Q$ with projective image being the simple group of order 60 (that is, a so-called icosahedral Galois representation), one may use Felix Klein's work to realize the representation as one coming from the 5-torsion of an elliptic curve defined over a quadratic extension of $\mathbb Q(\sqrt{5})$. This yields an unsatisfactory theory since one would like to consider abelian varieties over $\mathbb Q$ instead. I'll explain how this will indeed happen for a special class of icosahedral Galois representations, where I'll present the especially nice properties of the associated elliptic curves. Arpad Toth, Fordham University Holomorphic automorphisms of affine homogeneous spaces Abstract: The complex analytic geometry of bounded symmetric domains is well understood, due to the facts that 1. these domains are homogeneous spaces under some Lie group G, that acts through holomorphic automorphism of the domain, and 2. the group G contains all bi-holomorphic automorphisms of the domain. For a general homogeneous space, the second property no longer holds, (e.g. C^n), but one can still approximate any biholomorphic self-map by simple ones that arise from the group action. I will briefly review this theorem, and explain its connection to the complex analytic Abhyankar-Moh problem.