Trigonal modular curves

Abstract

(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]).

Torsion points on modular curves

“…The rest of the assertions are proved in Chapter 3 of [1]. We note that similar assertions have recently been proved by Hasegawa-Shimura (see [12], [13]) and by Nguyen-Saito (see [20]). …”

A Canonical Curve of Genus 17

“…In this paper we have chosen the presentation (1) as our definition of G. Using Gap one can check that the following gives us a representation of G as a permutation group P = (1, 13, 2, 11, 4, 5, 8) (3,10,6,14,7,9,12), (5) Q = (1, 7, 3, 4) (2,11,13,9,6,14,10,5),…”

“…form a basis for the ζ 2 -eigenspace of h 7B | π W 4 (I X 1 (2)) . 7 The generators of π W 10 (I X 1 (2))…”

“…Having the quadrics in the ideal available to do calculations this seemed to be a quick way to know the answer, however, this also turned out to be a very slow and long computation (at least with the algorithms that we used and I did not try to parallelize it, the tables in §10, §11 are meant to help speed up verification) and a simpler method would be desirable. Perhaps using a different interpretation of the curve one could apply tools similar to those used to determine the gonanality of modular curves (see for instance [7], [8] ). Nevertheless using the quadrics in the ideal to check the trigonality of modular curves is also among those tools (see §2 in [7] or §2.2 in [16]).…”

“…Perhaps using a different interpretation of the curve one could apply tools similar to those used to determine the gonanality of modular curves (see for instance [7], [8] ). Nevertheless using the quadrics in the ideal to check the trigonality of modular curves is also among those tools (see §2 in [7] or §2.2 in [16]).…”