View Full Document

The integral monodromy of hyperelliptic and trielliptic curves



View the full content.
View Full Document
View Full Document

20 views

Unformatted text preview:

The integral monodromy of hyperelliptic and trielliptic curves Jeffrey D Achter Rachel Pries Abstract We compute the Z and Z monodromy of every irreducible component of the moduli spaces of hyperelliptic and trielliptic curves In particular we provide a proof that the Z monodromy of the moduli space of hyperelliptic curves of genus g is the symplectic group Sp2g Z We prove that the Z monodromy of the moduli space of trielliptic curves with signature r s is the special unitary group SU r s Z Z 3 MSC 11G18 14D05 14H40 keywords monodromy hyperelliptic trigonal moduli Jacobian 1 Introduction If C S is a relative smooth proper curve of genus g 1 over an irreducible base then the torsion of the relative Jacobian of C encodes important information about the family Suppose is invertible on S and let s S be a geometric point The fundamental group 1 S s acts linearly on the fiber Pic0 C s Z 2g and one can consider the mod monodromy representation associated to C C S 1 S s Aut Pic0 C s GL2g Z Let M C S or simply M S be the image of this representation If a primitive th root of unity is defined globally on S then Pic0 C s is equipped with a skew symmetric form h i and M C S Sp Pic0 C s h i Sp2g Z If C S is a sufficiently general family of curves then M C S Sp2g Z 8 In this paper we compute M S when S is an irreducible component of the moduli space of hyperelliptic or trielliptic curves and C S is the tautological curve The first result implies that there is no restriction on the monodromy group in the hyperelliptic case other than that it preserve the symplectic pairing As a trielliptic curve is a Z 3 cover of a genus zero curve the Z 3 action constrains the monodromy group to lie in a unitary group associated to Z 3 The second result implies that this is the only additional restriction in the trielliptic case Theorem 3 4 Let be an odd prime and let k be an algebraically closed field in which 2 is invertible For g 1 M H g k Sp2g Z The second author was partially supported by



Access the best Study Guides, Lecture Notes and Practice Exams

Loading Unlocking...
Login

Join to view The integral monodromy of hyperelliptic and trielliptic curves and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view The integral monodromy of hyperelliptic and trielliptic curves and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?