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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

  1. X is not hyperelliptic;
  2. The monodromy map $G_K \rightarrow \mathbf{Sp}_{2g}(\mathbf{Z}_\ell)$ has open image;
  3. 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.

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  • $\begingroup$ Could you say more about the cycle class map you are using? Are you looking at the image of the algebraic cycle in Deligne cohomology, or something like that? $\endgroup$ Aug 21, 2014 at 13:13
  • $\begingroup$ I just mean etale cohomology -- I suppose this is "really" a motivic question but I don't think I need to invoke anything of that kind to get at what I'm interested in. $\endgroup$
    – JSE
    Aug 21, 2014 at 13:52
  • $\begingroup$ Just to clarify, when you say that the Ceresa cycle is homologically trivial, are you saying that the image of the cycle in etale cohomology of $J(X)\otimes_K \overline{K}$ is trivial? If so, are you asking about the cycle in etale cohomology of $J(X)$? $\endgroup$ Aug 21, 2014 at 17:43
  • $\begingroup$ I think it is unlikely that such a curve exists (assuming $K$ is finitely generated and $char(K) = 0$): it is generally believed that the Ceresa cycle is non-torsion modulo algebraic equivalence if $X$ is non-hyperelliptic; this would imply, assuming usual motivic conjectures, that the Ceresa class is also non-trivial $\endgroup$
    – naf
    Aug 22, 2014 at 6:32
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    $\begingroup$ @DanielLitt: I did see your paper on arXiv, the abstract looked very interesting but I have not yet read the paper. $\endgroup$
    – naf
    May 4, 2020 at 5:10

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