Abstracts of Talks
Local Structure of the Singularities of the Theta Divisor
Samuel Grushevsky
We study the infinitesimal analog of the Andreotti-Mayer loci - instead of looking at the dimension of the singular set of the theta divisor of an abelian variety, we look at the jet of the theta divisor at a singular point (i.e. at the derivatives of the theta function). In particular we prove a conjecture of Hershel Farkas that 4-dimensional Jacobians with a vanishing theta-null are characterized among all 4-dimensional abelian varieties by having a singular point of order two on the theta divisor that is not an ordinary double point (i.e. has a degenerate tangent cone). We also discuss the higher-dimensional analog of this condition and its relation to the intersections of the irreducible components of the Andreotti-Mayer loci. This is joint work with Riccardo Salvati Manni.
Theta functions, Arakelov invariants, and hyperbolic heat kernels
Jay Jorgenson
The analytic aspect of Arakelov theory of algebraic curves involve a number of invariants including: Green's functions, the canonical metric, the Arakelov metric, and Faltings's delta function. In the early development of the theory, these quantities were expressed in terms of the Riemann theta function. In recent work with Jurg Kramer, we have developed a connection with hyperbolic geometry through the hyperbolic heat kernel. As a result, an number of questions have been addressed, including the behavior of these invariants for certain families of algebraic curves, such as modular curves. As time allows, we will discuss a connection with certain trace formula from analytic number theory, namely the Petersson and Kutznetzov formula.
Theta Constants Identities for Jacobians of Cyclic 3-Sheeted Covers of the Sphere and Representations of the Symmetric Group
Yaacov Kopeliovich
We find identities between cubic powers of theta constants with rational characteristics evaluated at the period matrix τR, for R a cyclic 3 sheeted cover of the Sphere with 3k branch points λ1 … λ3k. These identities follow from Thomae's formula. This formula expresses sixth powers of theta constants as polynomilas in λ1 … λk3. We apply the representation of the symmetric group to find relations between the polynomials and hence a relation between cubic powers of the associated theta constants.
Some Problems in Abelian Functions
Alan Mayer
Discussion of issues that distinguish Jacobians among principally polarized abelian varieties. The discussion will propose untraditional or conjectural properties related to: the theta divisor, the Riemann-Roch theorem, and modular correspondences.
Discussion: Higher Weierstrass Points on the Klein Curve; Gauss' AgM; Prym Varieties as Spectral Manifolds; the Higher-Rank Heat Equation
Eleanor Farrington (Ph.D. student, Boston University) will review modular properties of the Weierstrass points of the Klein curve, the genus-3 curve with 168 automorphisms, and explore the extent to which the fixed points of these automorphisms are higher-order Weierstrass points. Emma Previato will report on the status of the higher-genus AgM as it relates to the Klein curve; then, review the genus-2 constructions of spectral Pryms and the heat equation for rank-2 bundles posing questions in higher genus.
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