Arithmetic Algebraic Geometry pp 41-49 | Cite as

# The action of monodromy on torsion points of Jacobians

## Abstract

Let *C* _{ g } be the generic curve of genus *g* over an algebraically closed field **k** with field of definition *K* _{ g }. If \( {\overline K_g} \) is a separable closure of *K* _{ g } then we can consider the action of \( {\overline K_g}/{K_g} \)on the torsion of \( Pic({C_{{g,}}}_{{{{\overline K }_g}}}) \). One may ask for the image of this Galois group in the group of automorphisms of \( _{{tor}}Pic({C_{{g,}}}_{{{{\overline K }_g}}}) \). On the part which is of order prime to the characteristic the result is known, by transcendental methods, to be the full symplectic group. In this paper we will give an algebraic proof of this fact and also show that the action on the points of order a power of the characteristic *p* is the largest possible i.e. the full general linear group. Rather than dealing directly with the generic curve we will work with the corresponding moduli stacks. Using this reformulation Theorem 2.1 is the result giving the surjectivity for *p*-torsion points and Corollary 3.1 the result giving the surjectivity for all torsion points (this surjectivity is of course equivalent to the irreducibility statement of that theorem).

## Keywords

Elliptic Curf Generic Fibre Newton Polygon Picard Group Torsion Point## Preview

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