Construction techniques for Galois coverings of the affine line

Abstract

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.