Let K be the fraction field of a strictly Henselian DVR of characteristic p ≥ 0 with algebraic closureK, and let α1, ..., α d ∈ P 1 K (K). In this paper, we give explicit generators and relations for the prime-to-pétale fundamental group of P 1 K {α1, ..., α d } that depend (solely) on their intersection behavior. This is done by a comparison theorem that relates this situation to a topological one. Namely, let a1, ..., a d be distinct power series in C[[x]] with the same intersection behavior as the αi's, converging on an open disk centered at 0, and choose a point z0 = 0 lying in this open disk. We compare the natural action of Gal(K/K) on the prime-to-pétale fundamental group of PK {α1, ..., α d } to the topological action of looping z0 around the origin on the fundamental group of P 1 C {a1(z0), ..., a d (z0)}. This latter action is, in turn, interpreted in terms of Dehn twists. A corollary of this result is that every prime-to-p G-Galois cover of P 1 K {α1, ..., α d } satisfies that its field of moduli (as a G-Galois cover) has degree over K dividing the exponent of G/Z(G).

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In this paper we consider the Suzuki curve y q + y = x q 0 (x q + x) over the field with q = 2 2m+1 elements. The automorphism group of this curve is known to be the Suzuki group Sz(q) with q 2 (q − 1)(q 2 + 1) elements. We construct AG codes over F q 4 from an Sz(q)-invariant divisor D, giving an explicit basis for the Riemann-Roch space L( D) for 0 < ≤ q 2 − 1. The full Suzuki group Sz(q) acts faithfully on each code. These families of codes have very good parameters and information rate close to 1. In addition, they are explicitly constructed. The dual codes of these families are of the same kind if 2g − 1 ≤ ≤ q 2 − 1.

Abstract. Let K be a field, let G be a finite group, and letX →Ȳ be a G-Galois branched cover of varieties over K sep . Given a mere cover model X → Y of this cover over K, in Part I of this paper I observe that there is a unique minimal field E over which X → Y becomes Galois, and I prove that E/K is Galois with group a subgroup of Aut(G). In Part II of this paper, by making the additional assumption that K is a field of definition (i.e., that there exists some Galois model over K), I am able to give an explicit description of the unique minimal field of Galois action for X → Y . Namely, if there exists a K-rational point of X above an unramified point P ∈ Y (K) then E is contained in the intersection of the specializations at P in the various different G-Galois models ofX →Ȳ over K. Using the same proof mechanism, I observe a reverse version of "The Twisting Lemma", which asserts that the behavior of the K-rational points on the various mere cover models over K, and the behavior of the specializations on the various G-Galois models over K, are all governed by a single equivalence relation (independent of the model) on the K-rational points of the base variety.