Details
| Original language | English |
|---|---|
| Article number | 87 |
| Journal | Research in Number Theory |
| Volume | 8 |
| Issue number | 4 |
| Publication status | Published - 12 Oct 2022 |
Abstract
We complete the computation of all Q-rational points on all the 64 maximal Atkin-Lehner quotients X(N) ∗ such that the quotient is hyperelliptic. To achieve this, we use a combination of various methods, namely the classical Chabauty–Coleman, elliptic curve Chabauty, quadratic Chabauty, and the bielliptic quadratic Chabauty method (from a forthcoming preprint of the fourth-named author) combined with the Mordell-Weil sieve. Additionally, for square-free levels N, we classify all Q-rational points as cusps, CM points (including their CM field and j-invariants) and exceptional ones. We further indicate how to use this to compute the Q-rational points on all of their modular coverings.
ASJC Scopus subject areas
- Mathematics(all)
- Algebra and Number Theory
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In: Research in Number Theory, Vol. 8, No. 4, 87, 12.10.2022.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Rational points on hyperelliptic Atkin-Lehner quotients of modular curves and their coverings
AU - Adzaga, Nikola
AU - Chidambaram, Shiva
AU - Keller, Timo
AU - Padurariu, Oana
N1 - Funding Information: N. A. is supported by the Croatian Science Foundation under the project no. IP2018-01-1313. S. C. is supported by the Simons Foundation grant #550033. T. K. is supported by the Deutsche Forschungsgemeinschaft (DFG), Projektnummer STO 299/18-1, AOBJ: 667349 while working on this article. O. P. is supported by NSF Grant DMS-1945452 and Simons Foundation grant #550023.
PY - 2022/10/12
Y1 - 2022/10/12
N2 - We complete the computation of all Q-rational points on all the 64 maximal Atkin-Lehner quotients X(N) ∗ such that the quotient is hyperelliptic. To achieve this, we use a combination of various methods, namely the classical Chabauty–Coleman, elliptic curve Chabauty, quadratic Chabauty, and the bielliptic quadratic Chabauty method (from a forthcoming preprint of the fourth-named author) combined with the Mordell-Weil sieve. Additionally, for square-free levels N, we classify all Q-rational points as cusps, CM points (including their CM field and j-invariants) and exceptional ones. We further indicate how to use this to compute the Q-rational points on all of their modular coverings.
AB - We complete the computation of all Q-rational points on all the 64 maximal Atkin-Lehner quotients X(N) ∗ such that the quotient is hyperelliptic. To achieve this, we use a combination of various methods, namely the classical Chabauty–Coleman, elliptic curve Chabauty, quadratic Chabauty, and the bielliptic quadratic Chabauty method (from a forthcoming preprint of the fourth-named author) combined with the Mordell-Weil sieve. Additionally, for square-free levels N, we classify all Q-rational points as cusps, CM points (including their CM field and j-invariants) and exceptional ones. We further indicate how to use this to compute the Q-rational points on all of their modular coverings.
UR - http://www.scopus.com/inward/record.url?scp=85139757964&partnerID=8YFLogxK
U2 - 10.1007/s40993-022-00388-9
DO - 10.1007/s40993-022-00388-9
M3 - Article
VL - 8
JO - Research in Number Theory
JF - Research in Number Theory
IS - 4
M1 - 87
ER -