Given a faithful action of a finite group G on an algebraic curve X of genus g_X > 1, we give explicit criteria for the induced action of G on the Riemann-Roch space H^0(X,O_X(D)) to be faithful, where D is a G-invariant divisor on X of degree at least 2g_X-2. This leads to a concise answer to the question when the action of G on the space H^0(X, \Omega_X^m) of global holomorphic polydifferentials of order m is faithful. If X is hyperelliptic, we furthermore provide an explicit basis of H^0(X, \Omega_X^m). Finally, we give applications in deformation theory and in coding theory and we discuss the analogous problem for the action of G on the first homology H_1(X, Z/mZ) if X is a Riemann surface.
- Pub Date:
- April 2014
- Mathematics - Algebraic Geometry;
- Mathematics - Number Theory;
- 14H30 (Primary) 30F30;
- 11R32 (Secondary)
- 27 pages, 1 figure, to appear in Canadian Journal of Mathematics