The a-number and the Ekedahl-Oort types of Jacobians of curves

In this thesis we deal with the a-numbers of curves in characteristic p and with the Ekedahl-Oort strata on the moduli spaces of curves, mostly in low genus or for small characteristic.
One of our results is an improvement of a result of R. Re. We give an upper bound on the genus of a curve for which the Cartier operator has rank 1. Re dealt with the case of rank 0. We do this by studying linear systems on the curve and the related action of the Cartier operator.
Other results deal with the moduli of hyperelliptic curves of genus 4 and characteristic 3. To study the induced stratification we give an explicit description of the de Rham cohomology and how Verschie-bung acts on it. Using cyclic covers of the projective line, in genus 4 we show that certain strata inside the p-rank 0 locus only do occur if certain congruence conditions on the characteristics are fulfilled. We study the dimensions of all strata for the moduli of genus 4 in characteristic 3. We also look at curves of various special types, like Artin-Schreier curves, hyperelliptic curves and trigonal curves. By studying the action of the Cartier operator on differential forms explicitly we are able to construct Artin-Schreier curves with large a-numbers. For hyperelliptic curves we use the Hasse-Witt matrix to give bounds on the a-numbers for the case where the genus is larger than the characteristic. Similarly for trigonal curves in small characteristics, we bound their a-numbers.