On Weierstrass points of Riemann surfaces associated with $\Gamma(n,2n)$

Abstract

Introduction. Let f be a congruence subgroup of SL(2, R), and let R(F) be the Riemann surface associated with F, i.e., R(F) is the canonical compactification of F\H, where H is the upper half plane of C. If the genus of R(F) is not less than two, then we can ask the following problems: Problem 1. Is R(F) hyperelliptic ? Problem 2. Are the cusps of R(F) Weierstrass points? Historically, Problems 1 and 2 are completely solved for F = F(n) by H. Petersson [8] and by B. Schoeneberg [9] respectively. In the case of F0(n), partial solutions are given by J. Lehener and M. Newmann [6] and A.O.L. Atkin [2]. The purpose of this note Is to answer both problems in the case of F = F(n, 2n) (as for the definition of I\n, 2ri), see Definition 1). Our results are the following: 1. R(I\n, 2n)) is non-hyperelliptic for any positive integer ≪i^4 (see Theorem 4). 2. Every cusp of R(F(n, 2n)) is a Weierstrass point for any even integer n^4. But there is an example, where the opposite situation may occur if n is odd (see Theorem 6 and Remark 1). Notation. SL{2, R) (resp. SL(2, Z)) is the special linear group of degree two over the real number field R (resp. the rational integer ring Z), and PSL(2, R) is the projective special linear group of degree two over R. When J is a subgroup of SL(2, R), the image of J under the canonical homomorphism SL(2, R)―>PSL (2, R) is denoted by A. H* means the disjoint union of the upper half plane H, the rational numbers Q and {oo}.