On the number of Abelian subvarieties of reduced degree on an Abelian variety

Abstract

Due to Poincaré's Reducibility Theorem, we know that for an Abelian variety A there exists an isogeny decomposition of A into simple factors. However, this decomposition hides many interesting properties of the set of Abelian subvarieties of A. For example, it is not clear from the decomposition "how many" Abelian subvarieties A actually has. In this work, for a principally polarized Abelian variety (A;L) of dimension g in the moduli space Ag and some positive integers t and n, we de fine NA(t, n) := #f{S < ó = A : S is an abelian subvariety with dim S = n, X (L|S) < ó = t}, which counts the number of Abelian subvarieties of dimension n and reduce degree bounded by t. After some reductions of the problem of obtaining a bound for this number, we characterize the set of Abelian subvarieties of dimension n on Eg, for E an elliptic curve, in terms of the Grassmannian variety via the Stiefel variety. Finally, we use machinery from Diophantine Geometry to obtain an asymptotic estimation for NA(t, n) for any (A,L) Ag.