Varieties without Extra Automorphisms I: Curves

For any field k and integer g ≥ 3, we exhibit a curve X over k of genus g such that X has no non-trivial automorphisms over k. 1. Statement of the result Let k be a field, and let p be its characteristic, which may be zero. All our curves are smooth, projective, and geometrically integral over k. If X is a curve over k, let AutX denote the group of automorphisms of X over k. Hurwitz stated that for any g ≥ 3, there exists a curve of genus g over C such that AutX = {1}, and a rigorous proof was provided by Baily [Ba]. The result was generalized to algebraically closed fields of arbitrary characteristic by Monsky [Mo], and a simpler proof of this generalization was given later in [Popp]. The literature also contains some explicit constructions of curves with AutX = {1}. Accola at the end of [Ac] observes that there exist triple branched covers X of PC of genus g ≥ 5 with AutX = {1}. Mednyh [Me] constructs some other examples analytically, as quotients of the complex unit disk. Turbek [Tu] constructs explicit families of examples of X with AutX = {1}, over algebraically closed fields k of characteristic p 6= 2, and g = (m − 1)(n − 1)/2 for some integers m,n with (m,n) = 1, n > m + 1 > 3, and p not dividing (m−1)mn. He uses gap sequences at Weierstrass points to control automorphisms. Fix g ≥ 3, and let Mg,3K over Z denote the moduli space of curves equipped with a basis of the global sections of the third tensor power of the canonical bundle. Katz and Sarnak [KS, Lemma 10.6.13] show that there is a open subset Ug ofMg,3K corresponding to the curves with trivial automorphism group. The result proved by Monsky and Popp above implies that Ug meets every geometric fiber of Mg,3K → SpecZ. This, together with the Lang-Weil method, can be used to show that there exists Ng > 0 such that for any field k with #k > Ng (in particular, any infinite field), there exists a curve X of genus g over k with AutX = {1} [KS, Remark 10.6.24]. Our main result is that such curves exist even over small finite fields: Theorem 1. For any field k and integer g ≥ 3, there exists a curve X over k of genus g such that AutX = {1}. Remark . Our result gives an independent proof that Ug meets every geometric fiber of Mg,3K → SpecZ. Date: November 23, 1999. Most of this research was done while the author was at Princeton University supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship. The author is currently supported by NSF grant DMS-9801104, a Sloan Fellowship, and a Packard Fellowship. This article has been published in Math. Res. Letters 7 (2000), no. 1, 67–76. 1