Construction techniques for Galois coverings of the affine line

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, pv) and the projective special linear groups PSL(2, pv) are realized for allp, and the Suzuki groups Sz(22v+1) are realized for p = 2. Jacobian varieties are used to realize certain extensions of realizable groups with abelian kernels