TY - JOUR
T1 - Computing Quadratic Points on Modular Curves X0(N)
AU - Adžaga, Nikola
AU - Keller, Timo
AU - Michaud-Jacobs, Philippe
AU - Najman, Filip
AU - Ozman, Ekin
AU - Vukorepa, Borna
N1 - Funding Information:
The second author was supported by the Deutsche Forschungsgemeinschaft (DFG), Projektnummer STO 299/18-1, AOBJ: 667349 while working on this article. The third author was supported by an EPSRC studentship EP/R513374/1 and had previously used the surname Michaud- Rodgers. The fourth and sixth authors were supported by QuantiXLie Centre of Excellence, a project co-financed by the Croatian Government and European Union through the European Regional Development Fund - the Competitiveness and Cohesion Operational Programme (Grant KK.01.1.1.01.0004). The fifth author was partially supported by TUBITAK Project No 122F413.
PY - 2024/5
Y1 - 2024/5
N2 - In this paper we improve on existing methods to compute quadratic points on modular curves and apply them to successfully find all the quadratic points on all modular curves X0(N) of genus up to 8, and genus up to 10 with N prime, for which they were previously unknown. The values of N we consider are contained in the set L = {58, 68, 74, 76, 80, 85, 97, 98, 100, 103, 107, 109, 113, 121, 127}. We obtain that all the non-cuspidal quadratic points on X0(N) for N ∈ L are complex multiplication (CM) points, except for one pair of Galois conjugate points on X0(103) defined over Q(√2885). We also compute the j-invariants of the elliptic curves parametrised by these points, and for the CM points determine their geometric endomorphism rings.
AB - In this paper we improve on existing methods to compute quadratic points on modular curves and apply them to successfully find all the quadratic points on all modular curves X0(N) of genus up to 8, and genus up to 10 with N prime, for which they were previously unknown. The values of N we consider are contained in the set L = {58, 68, 74, 76, 80, 85, 97, 98, 100, 103, 107, 109, 113, 121, 127}. We obtain that all the non-cuspidal quadratic points on X0(N) for N ∈ L are complex multiplication (CM) points, except for one pair of Galois conjugate points on X0(103) defined over Q(√2885). We also compute the j-invariants of the elliptic curves parametrised by these points, and for the CM points determine their geometric endomorphism rings.
KW - elliptic curves
KW - Jacobians
KW - Modular curves
KW - Mordell–Weil sieve
KW - quadratic points
KW - symmetric Chabauty
UR - http://www.scopus.com/inward/record.url?scp=85187143570&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2303.12566
DO - 10.48550/arXiv.2303.12566
M3 - Article
VL - 93
SP - 1371
EP - 1397
JO - Mathematics of Computation
JF - Mathematics of Computation
SN - 0025-5718
IS - 347
ER -