On some questions related to the Krichever correspondence

TL;DR

This paper explores new properties and explicit examples of the Krichever correspondence in one and two dimensions, introduces geometric analogs of Schur pairs, and investigates KP hierarchy analogs in two-dimensional local fields, deriving new PDEs.

Contribution

It provides explicit examples of the Krichever-Parshin map, introduces geometric analogs of Schur pairs, and develops new KP hierarchy analogs with associated PDEs in two-dimensional local fields.

Findings

Explicit examples of Krichever-Parshin map for plane curves

Introduction of geometrical Schur pair analogs in 2D local fields

Derivation of new PDEs for KP hierarchy analogs

Abstract

We investigate various new properties and examples of one-dimensional and two-dimensional Krichever correspondence developed by Parshin. In particular, we give explicit examples of the Krichever-Parshin map for various plane curves, we introduce analogs of the Schur pairs in a two-dimensional local field and show that they are oft geometrical. At the end we investigate analogs of the KP hierarchy for two-dimensional local skew-fields with arbitrary commutation law instead of the usual law of Weyl algebra. We derive for these hierarchies new partial differential equations, which coincide with the usual KP equation for certain values of parameters.