Automorphism groups of Galois covers with prescribed p-groups

Determine, for a finite p-group G, the genera and p-ranks for which there exists a G-Galois cover X→P^1 defined over the prime field with full geometric automorphism group Aut(X)=G, including the case G=C_{p^n} of Artin–Schreier–Witt covers.

Background

The paper constructs Artin–Schreier curves whose full geometric automorphism group is the unavoidable deck group C_p, and completely determines the possible genus and p-rank pairs for this situation. The authors suggest extending this problem from cyclic groups of order p to arbitrary finite p-groups and requiring the full automorphism group to equal the prescribed group G.

For cyclic groups of higher p-power order, the relevant covers are Artin–Schreier–Witt covers. The paper identifies the absence of an analogue of the Valentini–Madan theorem for these covers as a principal obstruction, because the proof for C_p relies on uniqueness of the rational subfield of index p and descent of automorphisms to the base.

References

We end with two natural questions suggested by the method. Let $G$ be a finite $p$-group. For which genera and $p$-ranks does there exist a $G$-Galois cover $X\to$ defined over $$ with $Aut_{}(X)=G$? For $G=C_{pn}$ (Artin--Schreier--Witt covers) the descent step would require an analogue of the Valentini--Madan theorem.

— Artin-Schreier curves of prescribed p-rank without extra automorphisms  (2610.13279 - Tafazolian et al., 9 Oct 2026) in Section 6, final Question environment (Summary of the models and further remarks)