It is well known that the Quillen metric is defined on the determinant line bundle of a Riemann surface. I am wondering if there is any non-archimedean analog of this.
I remember this is one of the questions asked in Soule's book "Lectures on Arakelov Geometry" back in 1990s. Is there any progress on it since then? From non-archimedean point of view, a good homomorphism from line bundles over Spec(Z) should be "contracting" because we lose information when we push-forward. But this obviously has no counter part in archimedean places - a self-adjoint elliptic operator between sections of line bundles could have spectrum goes to $\infty$.
I was discussing this with one my colleague earlier this week, who suggest Quillen's metric is not "natural" from the point of view of one dimensional Arakelov theory (we need to introduce analytic torsion, which can be understand to be a formal way of computing the Jacobian of a transformation). I do not know if this is something trivial to the experts. A quick literature search did not show up anything. I am okay if this is done using more general framework like Berkovich spaces or adic spaces.
