Trigonal modular curves $X_0^*(N)$

Abstract

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.

On the cyclic torsion of elliptic curves over cubic number fields

“…The complete list of 2-gonal X 0 (N) was determined by Ogg [30], and that of 3-gonal ones by Hasegawa-Shimura [11]. The moduli interpretation of noncuspidal points of X 1 (N) are (E, ±P ), where E is an elliptic curve and P ∈ E is a point of order N. The moduli interpretation of noncuspidal points of X 0 (N) are (E, C), where E is an elliptic curve and C ⊂ E is a cyclic subgroup of order N. The map π : X 1 (N) −→ X 0 (N) send (E, ±P ) to (E, P ), where P is the cyclic subgroup generated by P .…”

“…And the actions of T 1 , · · · , T 6 on the Manin symbol (0, 1) in terms of the Manin basis are given in Table 2. (13,4), (13,5), (13,6), (13,7), (13,8), (13,9), (13,10), (13,11), (13, 12) 143 29…”

Points on Modular Curves Over Finite Fields

“…In this section, we consider a method to find the canonical embedding of X 0 (N ) which is described in [2,3]. The canonical embedding of X 0 (N ) is the embedding…”

“…We omit an explanation for the canonical curves whose defining ideals contain a cubic polynomial for which one can refer [2,3].…”

“…Ogg [4] determined all values of N for which X 0 (N ) is hyperelliptic, and Hasegawa and Shimura [2] determined all the trigonal curves X 0 (N ). A crucial instrument used in their proofs is #X 0 (N )(F p 2 ) which denote the number of points on the reduction of X 0 (N ) over the finite fields F p 2 where p is a prime with p N .…”