On nilpotent automorphism groups of function fields

Abstract

We study the automorphisms of a function field of genus g ≥ 2 over an algebraically closed field of characteristic p > 0. More precisely, we show that the order of a nilpotent subgroup G of its automorphism group is bounded by 16(g − 1) when G is not a p-group. We show that if |G| = 16(g − 1), then g − 1 is a power of 2. Furthermore, we provide an infinite family of function fields attaining the bound.

Plane curves with a large linear automorphism group in characteristic $p$

“…Case (iib) Assume that C has a point P in PG(2, q), and let s be a tangent to C at P . Arguing as before, if s is not a line of PG(2, q 2 ) then 1 3 q 2 (q − 1) 2 ≥ q 3 − q. Moreover, if s is a line of PG(2, q 2 ) \ PG(2, q) then P is a point of multiplicity at least q 2 − q.…”

“…For a point P ∈ PG(2, q), let s be a tangent to C at P . Arguing as before, if s is not a line of PG(2, q 2 ), then 1 3 q 2 (q − 1) 2 ≥ q 3 − q. Moreover, if s is a line of PG(2, q 2 ) \ PG(2, q) then deg(C) ≥ (q + 1)(q 2 − q) = q 3 − q by Bézout's theorem applied to (C, t).…”

“…For odd p, any Frobenius nonclassical curve is nonclassical. The converse does not hold in general, as C is Frobenius nonclassical if and only if (1) holds and there exists a homogeneous polynomial L(X, Y, Z) ∈ K[X, Y, Z] such that…”

“…Assume that C has a point Q in PG(2, q 2 ) \ PG(2, q), and let s be a tangent to C at Q. If s is not a line of PG(2, q 2 ) then there exist at least 1 3 q 2 (q − 1) 2 tangents to C at Q, and hence Q is a point with multiplicity at least 1 3 q 2 (q − 1) 2 > deg(C), a contradiction. If s is a line of PG(2, q 2 ) \ PG(2, q) then the previous argument shows that Q is at least a q 2 -fold point of C, whence deg(C) ≥ q 2 (q 2 − q) > q 3 − q follows by Bézout's theorem applied to (C, t), where Q ∈ t. It remains to consider the case where C has a unique tangent line at Q, namely the line t through Q.…”

“…To show (ii), it is useful to recall a couple of known facts on conics in PG(2, q), namely C 2 has exactly 1 2 q(q − 1) internal points, i.e. points not covered by tangents to C 2 at points in PG(2, q), and the same number of external lines, i.e.…”