## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### Trigonal quotients of modular curves $X_{0}(N)$

#### Abstract

Let $W(N)$ be the group of Atkin-Lehner involutions on the modular curve $X_0(N)$. The purpose of this article is to give complementary result to [7, 8, 9]; namely, we determine trigonal curves of the form $X_0(N)/W'$, where $W'$ is a subgroup of $W(N)$ such that $2< |W'| < |W(N)|$.

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 82, Number 2 (2006), 15-17.

Dates
First available in Project Euclid: 2 March 2006

https://projecteuclid.org/euclid.pja/1141279058

Digital Object Identifier
doi:10.3792/pjaa.82.15

Mathematical Reviews number (MathSciNet)
MR2209765

Zentralblatt MATH identifier
1115.11031

#### Citation

Hasegawa, Yuji; Shimura, Mahoro. Trigonal quotients of modular curves $X_{0}(N)$. Proc. Japan Acad. Ser. A Math. Sci. 82 (2006), no. 2, 15--17. doi:10.3792/pjaa.82.15. https://projecteuclid.org/euclid.pja/1141279058

#### References

• A. O. L. Atkin and J. Lehner, Hecke operators on $\Gamma \sb{0}(m)$, Math. Ann. 185 (1970), 134–160.
• E. Arbarello, M. Cornalba, P. A. Griffith, and J. Harris, Geometry of algebraic curves. Vol. I, Springer, New York, 1985.
• M. Furumoto and Y. Hasegawa, Hyperelliptic quotients of modular curves $X_{0}(N)$, Tokyo J. Math. 22 (1999), no. 1, 105–125.
• R. Hartshorne, Algebraic geometry, Springer, New York, 1977.
• Y. Hasegawa, Hyperelliptic modular curves $X^{*}_{0}(N)$, Acta Arith. 81 (1997), no. 4, 369–385.
• Y. Hasegawa and K. Hashimoto, Hyperelliptic modular curves $X^{*}_{0}(N)$ with square-free levels, Acta Arith. 77 (1996), no. 2, 179–193.
• Y. Hasegawa and M. Shimura, Trigonal modular curves, Acta Arith. 88 (1999), no. 2, 129–140.
• Y. Hasegawa and M. Shimura, Trigonal modular curves $X^{+d}_{0}(N)$, Proc. Japan Acad. Ser. A Math. Sci. 75 (1999), no. 9, 172–175.
• Y. Hasegawa and M. Shimura, Trigonal modular curves $X^{\ast}_{0}(N)$, Proc. Japan Acad. Ser. A Math. Sci. 76 (2000), no. 6, 83–86.
• H. Hijikata, Explicit formula of the traces of Hecke operators for $\Gamma_{0}(N)$, J. Math. Soc. Japan 26 (1974), 56–82.
• M. Newman, Conjugacy, genus, and class numbers, Math. Ann. 196 (1972), 198–217.
• K. V. Nguyen and M.-H. Saito, D-gonality of modular curves and bounding torsions. (Preprint).
• A. P. Ogg, Hyperelliptic modular curves, Bull. Soc. Math. France 102 (1974), 449–462.
• M. Shimura, Defining equations of modular curves $X_{0}(N)$, Tokyo J. Math. 18 (1995), no. 2, 443–456.
• M. Yamauchi, On the traces of Hecke operators for a normalizer of $\Gamma_{0}(N)$, J. Math. Kyoto Univ. 13 (1973), 403–411.