Suppose X/K is a curve over a field K, which we want to think of as non-algebraically closed, and let x be a point of X(K). The Ceresa cycle is defined as follows; you can embed X in Jac(X) by sending y to the divisor x-y. Of course you can also embed X in Jac(X) by sending y to y-x. This gives you two curves in the abelian variety Jac(X), whose difference is a homologically trivial cycle. And the cycle class map (combined with the polarization on Jac(X)) gives you a class, called the Ceresa class, in
$H^1(G_K, \wedge^3 H (-1) / H)$
where H is the first $\ell$-adic homology of the curve. (All this is taken from the introduction of Hain-Matsumoto's paper.) For fans of Shouwu Zhang and his collaborators, this ends up being more or less the same as the Gross-Schoen class whose arithmetic they study very extensively.
The Ceresa class is a really nice invariant of the curve, which, as Hain and Matsumoto explain, is intimately related with the Johnson filtration and the action of $G_K$ on the quotient of $\pi_1(X)$ by the third term of its lower central series.
So one thing we know (it's clear, in fact, from the Ceresa definition) is that the Ceresa class vanishes if X is a hyperelliptic curve. On the other hand, the Ceresa class is known to be nontrivial for the generic curve of genus g.
Question: Can you describe a curve X/K such that
- X is not hyperelliptic;
- The monodromy map $G_K \rightarrow \mathbf{Sp}_{2g}(\mathbf{Z}_\ell)$ has open image;
- The Ceresa class of X is trivial?
Or can we at least prove in some pure thought way that such an X exists? Or does it exist?
I'm more interested in the case where K is something like k(t) than the case where K is a number field, but I'd take either, to start with.