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CURVES OF GIVEN p-RANK WITH TRIVIAL AUTOMORPHISM GROUP
 

Summary: CURVES OF GIVEN p-RANK WITH TRIVIAL
AUTOMORPHISM GROUP
JEFFREY D. ACHTER, DARREN GLASS, AND RACHEL PRIES
Abstract. Let k be an algebraically closed field of characteristic p > 0.
Suppose g 3 and 0 f g. We prove there is a smooth projective
k-curve of genus g and p-rank f with no non-trivial automorphisms.
In addition, we prove there is a smooth projective hyperelliptic k-curve
of genus g and p-rank f whose only non-trivial automorphism is the
hyperelliptic involution. The proof involves computations about the
dimension of the moduli space of (hyperelliptic) k-curves of genus g and
p-rank f with extra automorphisms.
1. Introduction
Let k be an algebraically closed field of characteristic p > 0. If g 3, there
exist a k-curve C of genus g with Aut(C) = {1} and a hyperelliptic k-curve
D of genus g with Aut(D) Z/2 (see, e.g., [16] and [8], respectively). In this
paper, we extend these results to curves with given genus and p-rank.
If C is a smooth projective k-curve of genus g with Jacobian Jac(C), the
p-rank of C is the integer fC such that the cardinality of Jac(C)[p](k) is pfC
.
It is known that 0 fC g. We prove the following:

  

Source: Achter, Jeff - Department of Mathematics, Colorado State University
Glass, Darren B. - Department of Mathematics, Gettysburg College
Pries, Rachel - Department of Mathematics, Colorado State University

 

Collections: Environmental Sciences and Ecology; Mathematics