Tautological rings on Jacobian varieties of curves with automorphisms (Q1646623)

Let \(X\) be an abelian variety over \(\mathbb{C}\) of dimension \(g\). Let \(\text{A}^{\cdot}(X)\) denote the Chow ring with rational coefficients modulo algebraic equivalence. \textit{A. Beauville} showed in [Math. Ann. 273, 647--651 (1986; Zbl 0566.14003)] that there exists a bigradation on \(\text{A}(X)\) with \(\text{A}^p(X)=\bigoplus_{s=p-g}^p\text{A}^p(X)_{(s)}\). For a family of cycles \(\mathcal{J}\subset \text{A}(X)\), let \(\text{Taut}_X(\mathcal{J})\) denote the tautological ring generated by \(\mathcal{J}\), that is the smallest \(\mathbb{Q}\)-vector subspace of \(\text{A}(X)\) containing \(\mathcal{J}\) and closed under intersection and Pontryagin products, and operators \(k_*,k^*\) for all \(k\in\mathbb{Z}\). When \(X=J(C)\), the Jacobian of a smooth projective curve \(C\) of genus \(g\geq 1\), he proved that the \(\mathbb{Q}\)-algebra \(\text{Taut}_J(\{C\})\) is generated for the intersection product by the classes \(w^i\) of the subvarieties \(W_{g-i}\) parametrizing effective divisors on \(C\) of degree \(g-i\), and for the Pontryagin product by the Fourier transforms of the Newton polynomials \(N^i(w)\in\text{A}^i(J)_{(i-1)}\) in the \(w^i\), which are the components \(C_{(i)}\in \text{A}^{g-1}(J)_{(i)}\) appearing in Beauville's decomposition of \(C\in \text{A}^{g-1}(J)\). In the present paper the author considers a curve with a finite automorphism group \(G\), and investigates the structure of the tautological ring \(\text{Taut}_J(\{\pi_*C\in \text{A}(J);\pi\in\mathbb{Z}[G]\subset \text{End}(J)\})\). The main result shows that the ring is generated as \(\mathbb{Q}\)-subalgebra of \(\text{A}(J)\) (i) for the intersection product by all \(\pi^*N^i(w)\), (ii) for the Pontryagin product by all \(\pi_*C_{(i-1)}\) with \(\pi\in\mathbb{Z}[G]\) and \(i\in[1,g-1]\).