The tautological Chow ring of the moduli space $\mathcal{A}_{g}$ 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 $\mathcal{A}_{g}$ to the moduli space $\mathcal{M}_{g}^{\operatorname{ct}}$ of genus $g$ of curves of compact type, we prove that the product class $[\mathcal{A}_{1}\times \mathcal{A}_{5}]\in \mathsf{CH}^{5}( \mathcal{A}_{6})$ 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 tautological ring $\mathsf{R}^{*}(\mathcal{M}_{6}^{\operatorname{ct}})$ in genus 6 conjectured by Pixton and recently proven by Canning-Larson-Schmitt. The tautological ring $\mathsf{R}^{*}(\mathcal{M}_{6}^{\operatorname{ct}})$ has a 1-dimensional Gorenstein kernel, which is geometrically explained by the Torelli pullback of $[\mathcal{A}_{1}\times \mathcal{A}_{5}]$ . More generally, the Torelli pullback of the difference between $[\mathcal{A}_{1}\times \mathcal{A}_{g-1}]$ and its tautological projection always lies in the Gorenstein kernel of $\mathsf{R}^{*}(\mathcal{M}_{g}^{\operatorname{ct}})$ . The product map $\mathcal{A}_{1}\times \mathcal{A}_{g-1}\rightarrow \mathcal{A}_{g}$ 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.
