For constructing un ramified coverings of the affine line in characteristicp, a general theorem about good reductions modulop of coverings of characteristic zero curves is proved. This is applied to modular curves to realize SL(2, ℤ/nℤ)/±1, with GCD(n, 6) = 1, as Galois groups of unramified coverings of the affine line in characteristicp, for p = 2 or 3. It is applied to the Klein curve to realize PSL(2, 7) for p = 2 or 3, and to the Macbeath curve to realize PSL(2, 8) for p = 3. By looking at curves with big automorphism groups, the projective special unitary groups PSU(3, p^v) and the projective special linear groups PSL(2, p^v) are realized for allp, and the Suzuki groups Sz(2^2v+1) are realized for p = 2. Jacobian varieties are used to realize certain extensions of realizable groups with abelian kernels.

The contact structure of two meromorphic curves gives a factorization of their jacobian.

A general technique is developed to enlarge the Galois group of an equation from a subgroup of a finite classical isometry group towards the corresponding similitude group.

We propound a descent principle by which previously constructed equations over GF(q_n)(X) may be deformed to have incarnations over GF(q)(X) without changing their Galois groups. Currently this is achieved by starting with a vectorial (= additive)q-polynomial ofq-degreem with Galois group GL(m, q) and then, under suitable conditions, enlarging its Galois group to GL(m, q_n) by forming its generalized iterate relative to an auxiliary irreducible polynomial of degreen. Elsewhere this was proved under certain conditions by using the classification of finite simple groups, and under some other conditions by using Kantor’s classification of linear groups containing a Singer cycle. Now under different conditions we prove it by using Cameron-Kantor’s classification of two-transitive linear groups.

We characterize the dicriticals of special pencils. We also initiate higher dimensional dicritical theory.

We study dicritical divisors and Rees valuations.