Tautological and non-tautological cycles on the moduli space of abelian varieties

The tautological Chow ring of the moduli space Ag of principally polarized abelian varieties of dimension g was defined and calculated by van der Geer in 1999. By studying the Torelli pullback of algebraic cycles classes from Ag to the moduli space Mgᶜᵗ of genus g of curves of compact type, we prove that the product class [A₁× A₅]∈ CH⁵(A₆) is non-tautological, the first construction of an interesting non-tautological algebraic class on the moduli spaces of abelian varieties. For our proof, we use the complete description of the the tautological ring R^*(M₆ᶜᵗ) in genus 6 conjectured by Pixton and recently proven by Canning-Larson-Schmitt. The tautological ring R^*(M₆ᶜᵗ) has a 1-dimensional Gorenstein kernel, which is geometrically explained by the Torelli pullback of [A₁× A₅]. More generally, the Torelli pullback of the difference between [A₁× Ag-1] and its tautological projection always lies in the Gorenstein kernel of R^*(Mgᶜᵗ). The product map A₁× Ag-1→ Ag is a Noether-Lefschetz locus with general Neron-Severi rank 2. A natural extension of van der Geer's tautological ring is obtained by including more general Noether-Lefschetz loci. Results and conjectures related to cycle classes of Noether-Lefschetz loci for all g are presented.