We prove equivariant Riemann-Roch formulae for algebraic curves over perfect fields. Our work generalizes similar results proved by the second author for curves over algebraically closed fields.
We use geometric and cohomological methods to show that given a degree bound for membership in ideals of a fixed degree type in the polynomial ring P = k[x 0 , . . . , x d ], one obtains a good generic degree bound for membership in the tight closure of an ideal of that degree type in any standard-graded k-algebra R of dimension d+1. This indicates that the tight closure of an ideal behaves more uniformly than the ideal itself. Moreover, if R is normal, one obtains a generic bound for membership in the Frobenius closure. If d ≤ 2, then the bound for ideal membership in P can be computed from the known cases of the Fröberg conjecture and yields explicit generic tight closure bounds.
Given a Galois cover of curves over F p , we relate the padic valuation of epsilon constants appearing in functional equations of Artin L-functions to an equivariant Euler characteristic. Our main theorem generalises a result of Chinburg from the tamely to the weakly ramified case. We furthermore apply Chinburg's result to obtain a 'weak' relation in the general case. In the Appendix, we study, in this arbitrarily wildly ramified case, the integrality of p-adic valuations of epsilon constants.
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