Weil reciprocity for rigid analytic curves

We prove a version of Weil reciprocity for certain partially proper rigid analytic curves. This simultaneously generalizes Deligne's le symbole modere for compact complex-analytic curves with boundary and Garcia Lopez's reciprocity formula for rigid analytic curves that arise as smooth lifts of a projective curve over a finite field. We define the Robba ring of a general rigid analytic space; and when the construction is applied to partially proper curves, we obtain a generalization of the classical Robba ring of an open disk. For compactifiable curves, we show that there is a suitable lattice theory of Fréchet nuclear subspaces inside the Robba ring, and one can define a gerbe of determinant theories over it. In particular, any pair of functions inside the Robba ring that fix a lattice satisfy a Weil reciprocity formula. Finally, by studying the polar of the Robba ring, we show that the units in the Robba ring are Cartier self-dual in a suitable sense. Our result makes use of non-Archimedean functional analysis and categorical constructions in order to bypass a number of technical obstacles. In particular, our framework allows us to consider curves defined over non-spherically complete fields.Mathematic