Let \(X\) be a smooth, projective, geometrically integral curve over a field \(K\) of genus \(g \geq 3\), and \(x \in X(K)\) a rational point. The curve \(X\) is embedded in its Jacobian \(\mathrm{Jac}(X)\) via the Abel-Jacobi map \(P \mapsto[P -x]\). We denote by \( X^-\) the image of \(X\) under the negation map on the group \(\mathrm{Jac}(X)\). The Ceresa cycle is the homologically trivial algebraic cycle \(X - X^-\) in \(\mathrm{Jac}(X)\). It yields, via the \(\ell\)-adic cycle class map, a Galois cohomology class \[\mu(X,x) \in H^1(\mathrm{Gal}(\bar{K}/K),H_{\text{ét}}^{2g-3} (\mathrm{Jac}(X)\oplus\bar{K}, \mathbb{Z}_{\ell}(g - 1)))\] which only depends on the rational equivalence class of the Ceresa cycle. In [R. Hain and M. Matsumoto, J. Inst. Math. Jussieu 4, No. 3, 363–403 (2005; Zbl 1094.14013)], this class is reinterpreted in terms of the Galois action on the pro-\(\ell\) étale fundamental group of \(X\) and an analogous class \(\nu(X)\) is described which is base point-independent.
In this paper, two classes \(MD(X,x)\) and \(J(X)\) in Galois cohomology are defined called the modified diagonal and Johnson classes. Note that the Johnson class is basepoint-independent, and in case where \(X\) is smooth and projective, these classes are closely related to \(\mu(X,x)\) and \(\nu(X)\), respectively. This construction proceeds via abstract group theory, and in particular, it works for any pro-\(\ell\) group with torsion-free abelianization. Many of the fundamental properties of these classes are analysed, and their relationship to work of Hain and Matsumoto in the case where the curve is proper is discussed.
Let \(C\) be a curve of genus 7 over a field \(K\) of characteristic zero such that \(C_{\bar{K}}\) has automorphism group isomorphic to \(\mathrm{PSL}_2(8)\). Then, it is proved that the Johnson class of \(J(C)\) and hence the basepoint-independent Ceresa class \(\nu(X)\) are torsion. Furthermore, if \(\iota \in Aut(C)\) is any element of order 2, then the quotient \(C/\iota\) is non-hyperelliptic curve of genus 3 with \(J(C/\iota)\) and \(\nu(C/\iota)\) torsion. Note that such curves \(C\) exist [A. M. Macbeath, Proc. Lond. Math. Soc. (3) 15, 527–542 (1965; Zbl 0146.42705)].