Research Proposal on Arithmetic Geometry

University Of Pennsylvania, Philadelphia PA

Investigators

Abstract

This project is in the field of arithmetic algebraic geometry and contains two parts: Neron models and the geometry of Shimura varieties. The focus of the first part is a numerical invariant, called the base change conductor, defined using the Neron models. This numerical invariant measures the difference between the Neron models of a semiabelian variety before and after making a finite base extension so that the semiabelian variety acquires semistable reduction. For an abelian variety over a number field, the base change conductor is equal to the decrease of Faltings height under stabilization. Recently E. de Shalit, J.-K. Yu and Chai proved that the base change conductor for a torus is equal to one half of the Artin conductor, using a congruence property for Neron models they discovered. This congruence property has been extended to abelian varieties by Chai. The explicit goals of the first project include: (a) Prove that the base change conductor for an abelian variety with potentially ordinary reduction is equal to the pairing between two central functions on the Galois group: the character of the Galois representation on the character group of a formal torus obtained from the Neron model, and a specific central function defined for every finite Galois extension. This specific central function has values in some cyclotomic extension of the field of p-adic numbers; it can be thought of as a "bisection of the Artin conductor" because the sum of this function with its complex conjugate is equal to the Artin character. (b) Prove an additivity property of the base change conductor. (c) Study the elementary divisors of the base change conductor. The second project is centered around the Hecke orbit problem and Oort's "foliation structure" for good reductions of Shimura variety. A notion of "Tate-linear" subvarieties of reduction of Shimura varieties with ordinary points will be investigated. (d) Verify in lower-rank cases the conjecture that every Tate-linear subvarieties is equal to the reduction of a Shimura subvariety. (e) Prove some cases of Oort's conjecture that the Zariski closure of a prime-to-p Hecke orbit is equal to the Zariski closure of a leaf in the foliation structure. This is a proposal in the area of mathematics known to as "Arithmetic Geometry." In this subject the problems and techniques of both Algebraic Geometry and Number Theory intermingle, to the benefit of both areas. Number theory is the oldest branch of mathematics. In recent years it has become an indispensable tools in areas such as communication systems, data transmission, and cryptology. A typical problem in Arithmetic Geometry concerns polynommial equations. For a system of polynomial equations, the degree of complexity of drops if one is allowed to use more general numbers, because it becomes easier to find solutions. The first part of this project studies a numerical invariant which measures how much the complexity drops. The second part of this project studies the of symmetries of a very special class of polynomial equations, called Shimura varieties, which are of central importance in Number Theory.