Abstracts

• Densities of 4-ranks of tame kernels, II, R. Osburn

  In this talk, we discuss recent results on computing the 4-rank of K_2(\mathcal{O}_F) for F a quadratic number field. These results lead to computations which might indicate densities in some generality.
  
  

• Densities of 4-ranks of tame kernels, I, R. Osburn

  The tame kernel of an algebraic number field with ring of integers O is the Milnor K-group K_2(O). The 4-rank of the finite abelian group K_2(O) for certain quadratic number fields was characterized by Conner and Hurrelbrink in terms of positive definite binary quadratic forms. Numerical calculations led to questions concerning possible density results of the 4-rank of tame kernels. The main results in this talk are affirmative answers to these questions.
  
  

• Stark-type conjectures "over Z", C. Popescu

  In the 1970s and 1980s Stark developed a remarkable conjecture aimed at interpreting the first non-vanishing derivative of an Artin $L$--function $L_{K/k, S}(s,\chi)$ at $s=0$ in terms of certain arithmetic properties of the Galois extension of global fields $K/k$. Work of Tate, Chinburg, and Stark himself has revealed far reaching applications of Stark's Conjecture to Hilbert's Twelfth Problem and the theory of Galois module structure of groups of units and ideal--class groups. In his search for new examples of Euler systems, Rubin has formulated in 1994 a strong version (``over {\bf Z}'', in Tate's terminology) of Stark's Conjecture for {\it abelian} $L$--functions of {\it arbitrary order of vanishing at $s=0$}. Our recent study of the functorial base--change behavior of Rubin's Conjecture led us to formulating a seemingly more natural Stark--type conjecture ``over {\bf Z}''. We will discuss and provide evidence for this new statement, as well as briefly describe the main goals of the conjectural program initiated by Stark.
  
  

• Kummer conjecture, quantum unique ergodicity and L-series, Y. Petridis

  Kummer conjectured an asymptotic formula for the first factor of the class number of a cyclotomic field. This has been investigated by Siegel, Ankeny and Chowla and Granville. On a different setting, the equidistribution of eigenfunctions of the Laplace operator on hyperbolic manifolds is a central question of Quantum Chaos. I will explain how the analytic theory of L-series is related to these problems and what answers it provides.
  
  

• Curves: can one find zeta without counting points?, H. J. Zhu

  Zeta functions of curves (over finite fields in my talk) contain important geometric and arithmetic data of the curves themselves and their Jacobians, so do their Newton polygons. The shape of the Newton polygon yields in a natural way a stratification on the moduli space of curves. Thanks to Weil, one can compute the zeta function of a curve of genus g by counting rational points on the curve over g many finite extensions. This straightforward algorithm becomes infeasible when one need to compute on some infinitely family of curves (for instance, a parametered family of curves). I will give an introductory survey of problems motivated by this area of research and describe what we can do by utilizing "higher Cartier operators" to approximate the zeta functions of some explicit families of curves (i.e., the general Artin-Schreier curves). This talk will be self-contained.
  
  

• Euler systems and circular distributions, Soogil Seo

  Euler Systems of Kolyvagin-Thaine have been applied to problems in algebraic number theory and Elliptic Curves by Rubin, Perrin-Riou, Kato..etc. We propose a problem of characterizing Euler Systems in the sense of Coleman. We will investigate these problems using Iwasawa theory,Cohomology of Number fields as well as the classical tools of algebraic number theory.
  
  

• The arithmetic of rigid Calabi-Yau threefolds, Helena A. Verrill

  I will start by describing some general motivation for the study of geometric Galois representations, and some of the well known results for elliptic curves, such as the modularity of elliptic curves. Then I will go on to talk about work on generalizations to higher dimensional varieties, and in particular to rigid Calabi-Yau threefolds, which may be considered to be a generalization of elliptic curves to a higher dimension.
  
  

• Elliptic curves and Galois representations, Susan Hammond Marshall

  We'll discuss elliptic curves (and their higher dimensional analogues, abelian varieties) as well as the Galois representations attached to them. We'll also explain how to use the theory of Galois representations to derive some formulae for an arithmetic invariant of an elliptic curve (or more generally, an abelian variety).
  
  

• Beilinson's conjectures and Fermat curves, Ramesh Sreekantan

  Over two hundred years ago, Euler and his contemporaries computed values of the function defined by the infinite series $$\zeta(s)=\sum_{n=1}^{\infty} \frac{1}{n^s}$$ for certain integer values of $s$. For example, he showed $\zeta(2)=\frac{\pi^2}{6}$. This function later came to be known as the {\em Riemann Zeta Function}. Dedekind generalized this function to number fields realizing the Riemann Zeta function as the `Zeta function of $\Q$'. Dirichlet proved a beautiful formula for the residue at $s=1$ of a meromorphic continuation of these `Dedekind Zeta functions' which explained why, for example, the residue of the Riemann Zeta function at $s=1$ is $1$. However, the values at other integer points seemed to be amusing curiosities.
  

  The notion of a Zeta function was later generalized to arbitrary algebraic varieties defined over number fields by Hasse and Weil. Beilinson, inspired by some work of Bloch, Borel and others, formulated some remarkable conjectures which explain the special values at integer points of these Zeta functions, and in particular, the special values of the Riemann Zeta function. However, they have been proved in very few cases.
  

  In this talk, I will describe one of these conjectures in some detail, illustrating it in some cases where it is known, and explain some of my work towards proving it in the case when the variety is a self-product of the Fermat curve, $${\mathbf F_N}: X^{N}+Y^{N}+Z^{N}=0$$.