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Unformatted text preview: VARIETIES WITHOUT EXTRA AUTOMORPHISMS I: CURVES BJORN POONEN Abstract. For any field k and integer g 3, we exhibit a curve X over k of genus g such that X has no nontrivial 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 Aut X 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 Aut X = { 1 } , and a rigorous proof was provided by Baily [Ba]. The result was generalized to algebraically closed fields of arbitrary characteristic by Monsky [Mo]. The literature also contains some explicit constructions of curves with Aut X = { 1 } . Accola at the end of [Ac] observes that there exist triple branched covers X of P 1 C of genus g 5 with Aut X = { 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 Aut X = { 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 M g, 3 K 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 U g of M g, 3 K corresponding to the curves with trivial automorphism group. The result of Monsky above implies that U g meets every geometric fiber of M g, 3 K Spec Z . This, together with the LangWeil method, can be used to show that there exists N g > 0 such that for any field k with # k > N g (in particular, any infinite field), there exists a curve X of genus g over k with Aut X = { 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 Aut X = { 1 } . Remark. Our result gives an independent proof that U g meets every geometric fiber of M g, 3 K Spec Z . We cannot hope to prove Theorem 1 by writing down for each g a single equation with coefficients independent of p , because a curve over Q of positive genus must have bad re duction at some prime; this follows from the title result of [Fo]. Therefore subdivision into cases seems unavoidable. 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 DMS9801104, a Sloan Fellowship, and a Packard Fellowship....
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This note was uploaded on 10/26/2011 for the course MATH 8320 taught by Professor Clark during the Fall '10 term at University of Georgia Athens.
 Fall '10
 Clark
 Algebra

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