The Inverse Problem for P-adic Differential Equations
B. Heinrich Matzat
In the complete p-adic case the analogue of rational function fields are the fields of analytic elements K{t} over a complete p-adic field of constants K. Here we show that any connected linear group appears as differential Galois group over K{t}. Moreover, in case the residue field of K is algebraically closed, the inverse problem of differential Galois theory over K{t} can be solved completely (including non-connected groups). This result can be seen as a differential analogue of Harbater's solution of the finite inverse problem over p-adic function fields.