Time: 14h,  Tuesday, December 12, 2023

Venue: Room 301, Building A5, Institute of Mathematics

Abstract: Given a differential equation (DE) on a variety X, it is highly desired to compute or understand its Galois differential group. Over the complex field, this problem has been studied extensively by using complex analysis. We automatically face an algebraic problem when replacing the complex field with an arbitrary field k of characteristic 0. Thanks to Deligne who systematically treated the algebraic differential equations. This story went further under the work of Katz. Following Tannakian's (or Riemann-Hilbert correspondence) philosophy, starting with a differential equation, we want to find a representation of its Galois group (or a monodromy representation). To do so, we must define a fiber functor from the category of DEs to the category of vector spaces. The point is that: what if we don't have any point to take fiber? Here is an example.

Let k be a field of characteristic 0, and K = k((t)) the field of Laurent series. Let X be the affine line over k punctured at 0, and write D.E.(X/k) (and D.E.(K/k)) to be the category of differential equations on X/k (and K/k respectively). Then we have a natural inverse image functor from D.E.(X/k) to D.E.(K/k), and Katz showed that this is indeed an equivalence if we restrict the source to the full subcategory of "special" objects. As a corollary, we obtain a fiber functor from D.E.(K/k) to the category of k-vector spaces. That result is mainly based on the work of Turrittin-Levelt on the structure of an arbitrary object in D.E.(K/k).

In this talk, I will report our attempt to extend the above story to the case that the base field k is replaced by a complete discrete valuation ring. This is a work in progress with Prof. Phung Ho Hai.