This research project on the moduli of abelian varieties contains three components: the Hecke orbit problem in a positive characteristic p, CM lifting of abelian varieties over finite fields and Hecke symmetry in the p-adic topology. Here p denotes a prime number. The PI's previous sponsored research has established the Hecke orbit conjecture for the Siegel modular varieties. Several methods developed for the Hecke orbit problem have proved useful for other question. Research on the Hecke symmetry, both in characteristic p and in mixed characteristics (0,p), will likely lead to new tools and perhaps new insight on the geometry of modular varieties. The effect of Hecke symmetry in the p-adic topology has not been systematically studied before. Progress on the expected non-density statement for the union of all Hecke translates of a lower-dimensional subvariety will provide an affirmative answer to a question posed by N. Katz on the existence of abelian varieties of dimension 3 or higher over the field of all algebraic numbers which are not isogenous to any Jacobian variety. A necessary and sufficient condition for the existence of a CM lifting up to isogeny over a normal domain of characteristic 0 is known from prior sponsored research. The focus in this part of the project is the existence of a CM lifting up to isogeny over a not-necessarily normal local domain of characteristics (0,p).
A projective algebraic variety is basically a subset of the projective space defined by a system of homogeneous polynomial equations, with the surrounding projective space stripped off and only the abstract mathematical structure left. An abelian variety is a very special kind of projective algebraic variety such that one can perform addition and substraction on it and the standard rules of arithmetic are satisfied. A moduli space considered above is an algebraic variety whose points parametrizes abelian varieties with a fixed dimension and shape of symmetry. Such a moduli space does not have many symmetries of the usual kind, those which to every point of the moduli space associate another point of the moduli space. Instead there are many symmetries of another sort on these moduli spaces, known as Hecke correspondences, which to every point of the moduli space associates several points of the moduli space. A basic question about Hecke symmetry is the following. Start with either one point or more generally a subvariety of a moduli space, then spread it around using all Hecke correspondences. Does the resulting subset almost fill the whole moduli space, so that every point of the moduli space has many points of the spread-out subset nearby and as close to it as one wants? This question is of interest mainly in one of two situations: the polynomial equations are considered over a ring such that either a prime number p times 1 is equal to 0, or over a ring which contains the standard integers but elements divisible by a high power of p is close to 0. The CM lifting question involves both kinds of rings above. Abelian varieties are extensively used in modern number theory, and properties of the moduli spaces over either kind of rings above are often of key importance for application in number theory. The PI's prior sponsored research has made Hecke symmetry a useful tool for number theory and algebraic geometry, producing short new proofs of known theorems as well as results inaccessible by previous methods. This project, when carried out, is likely to increase our knowledge about geometry of the moduli spaces of abelian varieties and may generate additional methods applicable to other areas of mathematics.
