Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

WEIERSTRASS POINTS ON HYPERELLIPTIC MODULAR CURVES

  • JEON, DAEYEOL (Department of Mathematics education Kongju National University)
  • Received : 2015.04.06
  • Published : 2015.10.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we find all Weierstrass points on the hyperelliptic modular curves $X_1(N)$.

Keywords

References

  1. A. O. L. Atkin, Weierstrass points at cusps ${\Gamma}_0(n)$, Ann of Math. (2) 85 (1967), no. 1, 42-45. https://doi.org/10.2307/1970524
  2. S. Choi, A Weierstrass point of ${\Gamma}_1(4p)$, J. Chungcheong Math. Soc. 21 (2008), no. 4, 467-470.
  3. F. Diamond and J. Shurman, A First Course in Modular Forms, Graduate Texts in Mathematics, 228. Springer-Verlag, New York, 2005. xvi+436 pp.
  4. N. Ishii and F. Momose, Hyperelliptic modular curves, Tsukuba J. Math. 15 (1991), no. 2, 413-423. https://doi.org/10.21099/tkbjm/1496161667
  5. D. Jeon, C. H. Kim, and A. Schweizer, Bielliptic intermediate modular curves, Preprint.
  6. K. Kilger, Weierstrass points on $X_0(pl)$ and arithmetic properties of Fourier coefficients of cusp forms, Ramanujan J. 17 (2008), no. 3, 321-330. https://doi.org/10.1007/s11139-007-9018-8
  7. W. Kohnen, Weierstrass points at cusps on special modular curves, Abh. Math. Sem. Univ. Hamburg 73 (2003), 241-251. https://doi.org/10.1007/BF02941280
  8. W. Kohnen, A short remark on Weierstrass points at infinity on $X_0(N)$, Monatsh. Math. 143 (2004), no. 2, 163-167. https://doi.org/10.1007/s00605-003-0054-1
  9. J. Lehner and M. Newman, Weierstrass points of ${\Gamma}_0(N)$, Ann of Math. (2) 79 (1964), 360-368. https://doi.org/10.2307/1970550
  10. J.-F. Mestre, Corps euclidiens, unites exceptionnelles et courbes elliptiques, J. Number Theory 13 (1981), no. 2, 123-137. https://doi.org/10.1016/0022-314X(81)90001-9
  11. A. P. Ogg, Rational points on certain elliptic modular curves, Analytic number theory(Pro. Sympos. Pure Math., Vol XXIV, St. Louis Univ., St. Louis, Mo., 1972), pp. 221-231. Amer. Math. Soc., Providence, R.I., 1973.
  12. A. P. Ogg, On the Weierstrass points of $X_0(N)$, Illinois J. Math. 22 (1978), no. 1, 31-35.
  13. K. Ono, The web of modularity: arithmetic of the coefficients of modular forms and q-series, CBMS Regional Conference Series in Mathematics 102, Amer. Math. Soc., Providence, RI. 2004.
  14. D. Rohrlich, Weierstrass points and modular forms, Illinois J. Math. 29 (1985), no. 1, 131-141.
  15. B. Schoneberg, Uber die Weierstrasspunkte in den Korpern der elliptischen Modulfunktionen, Abh. Math. Sem. Univ. Hamburg 17 (1951), 104-111. https://doi.org/10.1007/BF02950745