Semi-direct Galois covers of the affine line [unknown]

Linda Gruendken, Laura Hall-Seelig, Bo-Hae Im, Ekin Ozman, Rachel Pries, Katherine Stevenson
2011 WIN—Women in Numbers   unpublished
Let k be an algebraically closed field of characteristic p > 0. Let G be a semi-direct product of the form (Z/ Z) b Z/pZ where b is a positive integer and is a prime distinct from p. In this paper, we study Galois covers ψ : Z → P 1 k ramified only over ∞ with Galois group G. We find the minimal genus of a curve Z which admits a covering map of this form and we give an explicit formula for this genus in terms of and p. The minimal genus occurs when b equals the order a of modulo b and we also
more » ... ulo b and we also prove that the number of curves Z of this minimal genus which admit such a covering map is at most
doi:10.1090/fic/060/10 fatcat:ejpsjerpnraktpbrsgemykd5d4