#### Document Type

Article - Open Access

#### Editor

Alina-Carmen Cojocaru, et al.

#### Publication Title

WIN- Women in Numbers: Research Directions in Number Theory

#### Publisher

American Mathematical Society

#### Publication Date

2011

#### Abstract/ Summary

Let k be an algebraically closed field of characteristic p > 0. Let G be a semi-direct product of the form (Z/`Z) b o 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 prove that the number of curves Z of this minimal genus which admit such a covering map is at most (p − 1)/a when p is odd.

#### Repository Citation

Gruendken, L.,
Hall-Seelig, L. L.,
Im, B.,
Ozman, E.,
Pries, R.,
&
Stevenson, K.
(2011). Semi-Direct Galois Covers of the Affine Line. *WIN- Women in Numbers: Research Directions in Number Theory*, 201-211.

Available at: https://scholarworks.merrimack.edu/mth_facpub/2