Modular curves with infinitely many quartic points
HTML articles powered by AMS MathViewer
- by WonTae Hwang and Daeyeol Jeon;
- Math. Comp. 93 (2024), 383-395
- DOI: https://doi.org/10.1090/mcom/3864
- Published electronically: August 18, 2023
- HTML | PDF | Request permission
Abstract:
In this work, we determine all modular curves $X_0(N)$ which admit infinitely many quartic points.References
- Dan Abramovich and Joe Harris, Abelian varieties and curves in $W_d(C)$, Compositio Math. 78 (1991), no. 2, 227–238. MR 1104789
- Francesc Bars, Bielliptic modular curves, J. Number Theory 76 (1999), no. 1, 154–165. MR 1688168, DOI 10.1006/jnth.1998.2343
- F. Bars, http://mat.uab.es/~francesc/publicac.html, 2023.
- Francesc Bars, Mohamed Kamel, and Andreas Schweizer, Bielliptic quotient modular curves of $X_0(N)$, Math. Comp. 92 (2023), no. 340, 895–929. MR 4524112, DOI 10.1090/mcom/3800
- Francesc Bars, Josep González, and Mohamed Kamel, Bielliptic quotient modular curves with $N$ square-free, J. Number Theory 216 (2020), 380–402. MR 4130087, DOI 10.1016/j.jnt.2020.03.010
- Matthew H. Baker, Enrique González-Jiménez, Josep González, and Bjorn Poonen, Finiteness results for modular curves of genus at least 2, Amer. J. Math. 127 (2005), no. 6, 1325–1387. MR 2183527, DOI 10.1353/ajm.2005.0037
- B. J. Birch and W. Kuyk, Modular functions of one variable. IV., Proceedings of the International Summer School on Modular Functions of One Variable and Arithmetical Applications, RUCA, University of Antwerp, Antwerp, July 17-August 3, 1972. Edited by B. J. Birch and W. Kuyk. Lecture Notes in Mathematics, Vol. 476, Springer-Verlag, Berlin-New York, (1975).
- Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, On the modularity of elliptic curves over $\mathbf Q$: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4, 843–939. MR 1839918, DOI 10.1090/S0894-0347-01-00370-8
- J. E. Cremona, Algorithms for modular elliptic curves, 2nd ed., Cambridge University Press, Cambridge, 1997. MR 1628193
- Olivier Debarre and Rachid Fahlaoui, Abelian varieties in $W^r_d(C)$ and points of bounded degree on algebraic curves, Compositio Math. 88 (1993), no. 3, 235–249. MR 1241949
- Fred Diamond and Jerry Shurman, A first course in modular forms, Graduate Texts in Mathematics, vol. 228, Springer-Verlag, New York, 2005. MR 2112196
- Gerd Faltings, Diophantine approximation on abelian varieties, Ann. of Math. (2) 133 (1991), no. 3, 549–576. MR 1109353, DOI 10.2307/2944319
- Gerd Faltings, The general case of S. Lang’s conjecture, Barsotti Symposium in Algebraic Geometry (Abano Terme, 1991) Perspect. Math., vol. 15, Academic Press, San Diego, CA, 1994, pp. 175–182. MR 1307396
- Charles R. Ferenbaugh, The genus-zero problem for $n|h$-type groups, Duke Math. J. 72 (1993), no. 1, 31–63. MR 1242878, DOI 10.1215/S0012-7094-93-07202-X
- Gerhard Frey, Curves with infinitely many points of fixed degree, Israel J. Math. 85 (1994), no. 1-3, 79–83. MR 1264340, DOI 10.1007/BF02758637
- Masahiro Furumoto and Yuji Hasegawa, Hyperelliptic quotients of modular curves $X_0(N)$, Tokyo J. Math. 22 (1999), no. 1, 105–125. MR 1692024, DOI 10.3836/tjm/1270041616
- Josep González and Joan-C. Lario, Rational and elliptic parametrizations of $\mathbf Q$-curves, J. Number Theory 72 (1998), no. 1, 13–31. MR 1643280, DOI 10.1006/jnth.1998.2259
- Yuji Hasegawa and Mahoro Shimura, Trigonal modular curves, Acta Arith. 88 (1999), no. 2, 129–140. MR 1700245, DOI 10.4064/aa-88-2-129-140
- Hiroaki Hijikata, Explicit formula of the traces of Hecke operators for $\Gamma _{0}(N)$, J. Math. Soc. Japan 26 (1974), 56–82. MR 337783, DOI 10.2969/jmsj/02610056
- Daeyeol Jeon, Modular curves with infinitely many cubic points, J. Number Theory 219 (2021), 344–355. MR 4177523, DOI 10.1016/j.jnt.2020.09.006
- Daeyeol Jeon, Bielliptic modular curves $X_0^+(N)$, J. Number Theory 185 (2018), 319–338. MR 3734352, DOI 10.1016/j.jnt.2017.09.006
- Daeyeol Jeon, Chang Heon Kim, and Euisung Park, On the torsion of elliptic curves over quartic number fields, J. London Math. Soc. (2) 74 (2006), no. 1, 1–12. MR 2254548, DOI 10.1112/S0024610706022940
- Daeyeol Jeon and Euisung Park, Tetragonal modular curves, Acta Arith. 120 (2005), no. 3, 307–312. MR 2188846, DOI 10.4064/aa120-3-6
- B. Kadets and I. Vogt, Subspace configurations and low degree points on curves, arXiv:2208.01067, 2022.
- F. Najman and P. Orlić, Gonality of the modular curve $X_0(N)$, to appear in Math. Comp., DOI 10.1090/mcom/3873. arXiv:2207.11650
- K. V. Nguyen and M.-H. Saito, $d$-gonality of modular curves and bounding torsions, arXiv:9603024, 1996.
- Andrew P. Ogg, Hyperelliptic modular curves, Bull. Soc. Math. France 102 (1974), 449–462. MR 364259
- Joseph H. Silverman, The arithmetic of elliptic curves, 2nd ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht, 2009. MR 2514094, DOI 10.1007/978-0-387-09494-6
- W. A. Stein, http://wstein.org/Tables/tables.html, 2023.
- Henning Stichtenoth, Algebraic function fields and codes, Universitext, Springer-Verlag, Berlin, 1993. MR 1251961
Bibliographic Information
- WonTae Hwang
- Affiliation: Department of Mathematics, Institute of Pure and Applied Mathematics, Jeonbuk National University, Baekje-daero, Deokjin-gu, Jeonju-si, Jeollabuk-do 54896, South Korea
- MR Author ID: 1335126
- Email: hwangwon@jbnu.ac.kr
- Daeyeol Jeon
- Affiliation: Department of Mathematics Education, Kongju National University, 56 Gongjudaehak-ro, Gongju-si, Chungcheongnam-do 314-701, South Korea
- MR Author ID: 658790
- Email: dyjeon@kongju.ac.kr
- Received by editor(s): April 19, 2022
- Received by editor(s) in revised form: February 23, 2023, and March 28, 2023
- Published electronically: August 18, 2023
- Additional Notes: The first author was supported by research funds for newly appointed professors of Jeonbuk National University in 2021. The second author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2019R1F1A1060149, 2022R1A2C1010487).
The second author is the corresponding author. - © Copyright 2023 American Mathematical Society
- Journal: Math. Comp. 93 (2024), 383-395
- MSC (2010): Primary 11G18, 11G30
- DOI: https://doi.org/10.1090/mcom/3864
- MathSciNet review: 4654626