Degrees of Hodge Loci

Abstract

We prove asymptotic estimates for the growth in the degree of the Hodge locus in terms of arithmetic properties of the integral vectors that define it. Our methods are general and apply to most variations of Hodge structures for which the Hodge locus is dense. As applications we give asymptotic formulas controlling the degrees of Noether-Lefschetz loci associated to smooth projective hypersurfaces in $\mathbb{P}^3$, and the degrees of subvarieties of the Torelli locus parameterizing Jacobians split up to isogeny.