# Arithmetic and Geometry Seminar, Fall 2001

Friday afternoons in South Hall, Room 4607B

Organisers: A. Agboola, J. McKernan

October 5, 2001, 2pm-4pm  October 12, 2001, 2pm-4pm September 28, 2001, 2pm Organisational meeting Abstract: None Luis Finotti, UCSB Degrees of the Elliptic Teichm\"uller Abstract: An ordinary elliptic curve over a perfect field of characteristic $p>0$ has a canonical lift'' to an elliptic curve over the ring of Witt vectors of that field. The map that lifts points, $$\tau(x_0,y_0)=((x_0,x_1, \dots),(y_0,y_1, \dots)),$$ called the elliptic Teichm\"uller, can be used in the construction of error-correcting codes, and the degrees (or, equivalently, the order at infinity) of the coordinate functions of this map is important to decide how good the codes obtained are. The following bounds will be proved: $$\ord_0 (x_n) \geq -(n+2) p^n+n p^{n-1}, \quad \ord_0 (y_n) \geq -(n+3) p^n+np^{n-1}.$$ It will also be proved that the bound for $x_n$ is not the exact order if, and only if, $p$ divides $(n+1)$, and the bound for $y_n$ is not the exact order if, and only if, $p$ divides $(n+1)(n+2)/2$. Finally, I will give an algorithm to compute the reduction modulo $p^3$ of the canonical lift for $p \not=2,3$. A preprint of a paper with the results above can be found at: http://www.math.ucsb.edu/~finotti/can_lifts.html Xi Chen, UCSB Families of rationally connected varieties Abstract:We will go over the paper Families of rationally connected varieties'' by T. Graber, J. Harris and J. Starr. A projective variety $X$ is called rationally connected if any two points of $X$ can be joined by a chain of rational curves. For example, the vanishing locus of a homogeous polynomial in $n$ variables is rationally connected if it has degree less than $n$. The main result of this paper says that a rationally connected variety over a function field of transcendence degree one has a rational point. This is an amazing result since the same is certainly not true over number fields. It is easy to find a quadratic polynomial with rational coefficients that does not have any rational zeroes at all. Jiayuan Lin, UCSB Families of rationally connected varieties II Abstract: We will continue to go over the paper Families of rationally connected varieties', by Tom Graber, Joe harris, and Jason Starr (available on the Los Alamos preprint server at math.AG/0109220. See also the paper Every rationally connected variety over the function field of a curve has a rational point', by Johan de Jong and Jason Starr, available at http://www-math.mit.edu/~dejong/, for a proof that also works in characteristic p.). Ozlem Imamoglu, UCSB Kronecker limit formulas Abstract: In this talk we will start with a review of classical Kronecker limit formulas and show how they give rise to functions which are defined as infinite products, namely the Dedekind eta function and Siegel functions. Next we will look at the elliptic dilogarithm function and reprove a classical result of Bloch along the lines of a Kronecker limit formula. David Pollack, Caltech A Generalization of Serre's Conjecture Abstract: Serre's Conjecture predicts the existence of a modular form mod p associated to each odd continous, irreducible two-dimensional mod p representation of the absolute Galois group of Q. Moreover, Serre gives explicit predictions for the weight,level and nebentype of the associated modular form. I will discuss a generalization of this conjecture to suitable n-dimensional representations of arbitrary niveau, and present some evidence supporting it. This is joint work with Avner Ash and Darrin Doud. No Seminar This Week TBA Abstract: Darren Long, UCSB Pseudomodular Surfaces Abstract: We discuss some connexions between hyperbolic geometry and number theory. Thanksgiving Holiday Boas Erez, University of Utah Invariants of bilinear forms over schemes: example of a computation. Abstract: The talk will be in two parts. In the first part we review some basic facts about bilinear forms over rings, e.g. the theory of the Witt ring and cohomological invariants for forms over fields. We shall emphasize the notion of metabolic form which will play an important role in the second part. In the second part we present joint work with Ph. Cassou-Nogu\`es and M.J. Taylor which builds upon formulae of Serre and Esnault-Kahn-Viehweg, and show how to give an expression for the Hasse-Witt invariants of a form obtained from the trace form attached to a tame covering of schemes with odd ramification. After recalling the necessary background, we explain how to use recent work on duality in triangulated categories, by Balmer, Saito and Walter, to reduce one of the main steps in the proof to local computations. TBA TBA Abstract: