Bounds on the number of automorphisms of curves over algebraically closed fields (Q1955732)

For any prime \(p\) and \(g \geq 2\), define \(\mu(g, p)\) to be the maximum possible number of automorphisms for a smooth projective curve of genus \(g\) over a field of characteristic \(p\). The classical Hurwitz bound shows that \(\mu(g, 0) \leq 84(g-1)\). In fact \[ 8(g+1) \leq \mu(g,0) \leq 84(g-1), \] and both inequalities are equalities for infinitely many \(g\). In characteristic \(p\), the behavior of \(\mu(g,p)\) is not as well understood. Oort asked if there exists a polynomial \(M_p(T) \in \mathbb{Q}[T]\) such that \(M_p(g)\) is a lower bound on \(\mu(g, p)\) which attains equality infinitely often. As a consequence of what has been written above, \(M_0(g) = 8(g+1)\). Oort asked further, is \(M_p(g) = 8(g+1)\)? In this note, it is shown that if \(g \geq 2\), then \(\mu(g, p) \geq 8(g+1)\) for almost all \(p\). If we restrict to those \(g\) with \(\mu(g, 0) = 8(g+1)\), then \(\mu(g, p) = 8(g+1)\) for almost all \(p\). Furthermore, there exist arbitrarily long increasing sequences of \(g_i \geq 2\) such that \(\mu(g_i, p) = \mu(g_i, 0) = 8(g+1)\) for almost al \(p\), and such that \(\mu(g_i, p) = \mu(g_i,0) = 84(g_i-1)\) for almost all \(p\). The proofs of these statements are fairly straightforward consequences of the fact that smooth curves in characteristic \(p\) with prime-to-\(p\) automorphism groups lift to characteristic \(0\) (along with the automorphism groups), that a curve in characteristic zero has good reduction at a set of valuations covering almost all primes as residue characteristics, and that automorphisms of curves of genus \(\geq 2\) with good reduction do not become trivial in the reduction.