Equivariant Riemann-Roch theorems for curves over perfect fields

This thesis deals with the equivariant Riemann-Roch problem for curves over perfect fields, and with the related topic of geometric Galois module theory. We generalize Kock's work on the equivariant Riemann-Roch problem for curves over algebraically closed fields, proving a "weak" equivariant Riemann-Roch formula for arbitrarily ramified Galois covers of curves over perfect fields as well as a "strong" formula for weakly ramified covers. As an application of our results, we show that under certain conditions, the automorphism group of a geometric Goppa code acts faithfully on the code, meaning that the code has in some sense "maximal symmetry". In the last part of this thesis, we present an alternative proof for a result of Chinburg in geometric Galois module theory, describing the equivariant" Euler characteristic of the structure sheaf of a curve in terms of epsilon constants.