(Tokyo) 0. Introduction. For a positive integer N , let X 0 (N) (over C) be the modular curve corresponding to the modular group Γ 0 (N). It is known that there are only finitely many values of N for which X 0 (N) is sub-hyperelliptic, i.e., it admits a twofold covering onto the projective line P 1. These values are explicitly determined by Ogg [13]. A smooth projective curve C defined over an algebraically closed field k is called d-gonal if there exists a finite morphism C → P 1 over k of degree d. Thus, C is sub-hyperelliptic if and only if it is 2-gonal. Also, in the rest of the paper, we will use "trigonal", "tetragonal", "pentagonal" to mean "d-gonal" for d = 3, 4, 5, respectively. Recently, Nguyen and Saito [12] proved an analogue of the strong Uniform Boundedness Conjecture for elliptic curves defined over function fields of dimension one; if a base curve is d-gonal, they gave a bound of the orders of torsions of Mordell-Weil groups in term of d uniformly by connecting the problem with the problem of bounding the level N of d-gonal modular curves X 0 (N)/C. Therefore it is an interesting problem to give a sharp bound for the level N of d-gonal modular curves. In fact, there is a result of Zograf [17] which gives a linear bound of the level N of d-gonal modular curves X 0 (N)/C; see Theorem 4.3. (Nguyen and Saito [12] also gave a bound of N by a quartic polynomial in d by employing a purely algebraic method.) In this paper, we prove that X 0 (N) is trigonal if and only if it is of genus g ≤ 2 or is non-hyperelliptic of genus g = 3, 4 (Theorem 3.3; see also Remark 1.3). As a consequence, we have N ≤ 81 if X 0 (N) is trigonal. Also we will show that N ≤ 191 (resp. N ≤ 197) if d = 4 (resp. d = 5) (Proposition 4.4). These give a highly sharpened upper bound for 3 ≤ d ≤ 5 (cf. [17], [12]).
For a positive integer N , let X * 0 (N ) denote the quotient curve of X 0 (N ) by the Atkin-Lehmer involutions. In this paper, we determine the trigonality of X * 0 (N ) for all N . It turns out that there are seven values of N for which X * 0 (N ) is a non-trivial trigonal curve.
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