Gaps at Weierstrass points for the modular group

Abstract

Let 5 be a compact Riemann surface of genus g^2, h: S-+S an automorphism of order N, and H the cyclic group of order N generated by h. One has a representation of H by letting it act on the g complex-dimensional space of abelian differentials of the first kind on 5 by h: g. In [3] the following was proved: (I) Suppose P = h(P) is a fixed point for h with gap sequence 7i> • • • > 7* and that h rotates at P by e, i.e., if z is a local parameter at P, z{P) = 0, then h(z) = ez+ • • • , e = 1. Then, with respect to a suitable basis for Ai, h is represented by the diagonal matrix (h) = d iag (e?i, €*», • • • , €**)• A corollary of this is (II) If P = h(P) is not a Weierstrass point then h has at most four fixed points. Thus if h has more than four fixed points all its fixed points are Weierstrass points. Let r be the inhomogeneous modular group, T(N) the principal congruence subgroup of level N>2, S(N) the compactified fundamental domain for T(N) which is a Riemann surface of genus g(N) = l+iV(iV--6)/24lLltf ( l~ l / £ 2 ) where the product is over primes dividing N. For details see Chapter 1 of [2]. T/T(N) is a group of automorphisms of S(N) whose fixed points are at three kinds of points. Firstly, parabolic points (cusps), equivalent under T/T(N) to 00 which is fixed under the cyclic group of order N generated by (the coset of) T: r—»r + l. Secondly, elliptic points of order 2, equivalent to i = V~-l which is fixed under the cyclic group of order 2 generated by S: r—> — 1/r. Thirdly, elliptic points of order 3, equivalent to p = e which is fixed under the cyclic group of order 3 gener1 Supported by NSF G-18929. The author wishes to acknowledge his gratitude to Professor H. E. Rauch for suggesting this area of research and helping to see it through and also to Professor D. J. Newman for several enlightening discussions.