TY - JOUR
T1 - On the level of modular curves that give rise to isolated j-invariants
AU - Bourdon, Abbey
AU - Ejder, Özlem
AU - Liu, Yuan
AU - Odumodu, Frances
AU - Viray, Bianca
N1 - Funding Information:
This project was started at the Women in Numbers 4 conference, which was held at the Banff International Research Station. We thank BIRS for the excellent working conditions and the organizers, Jennifer Balakrishnan, Chantal David, Michelle Manes, and the last author, for their support. We also thank the other funders of the conference: the Association for Women in Mathematics (NSF ADVANCE grant HRD-1500481), the Clay Mathematics Institute, Microsoft Research, the National Science Foundation (NSF grant DMS-1712938), the Number Theory Foundation, and the Pacific Institute for Mathematics Sciences.We thank Jeremy Rouse, Drew Sutherland, and David Zureick-Brown for helpful conversations. We also thank Nigel Boston, Pete L. Clark, Loïc Merel, Filip Najman, Paul Pollack, and the anonymous referees for comments on an earlier draft. The third author was partially supported by NSF grants DMS-1652116 and DMS-1301690 and the last author was partially supported by NSF CAREER grant DMS-1553459.
Publisher Copyright:
© 2019 Elsevier Inc.
PY - 2019/12/1
Y1 - 2019/12/1
N2 - We say a closed point x on a curve C is sporadic if C has only finitely many closed points of degree at most deg(x) and that x is isolated if it is not in a family of effective degree d divisors parametrized by P1 or a positive rank abelian variety (see Section 4 for more precise definitions and a proof that sporadic points are isolated). Motivated by well-known classification problems concerning rational torsion of elliptic curves, we study sporadic and isolated points on the modular curves X1(N). In particular, we show that any non-cuspidal non-CM sporadic, respectively isolated, point x∈X1(N) maps down to a sporadic, respectively isolated, point on a modular curve X1(d), where d is bounded by a constant depending only on j(x). Conditionally, we show that d is bounded by a constant depending only on the degree of Q(j(x)), so in particular there are only finitely many j-invariants of bounded degree that give rise to sporadic or isolated points.
AB - We say a closed point x on a curve C is sporadic if C has only finitely many closed points of degree at most deg(x) and that x is isolated if it is not in a family of effective degree d divisors parametrized by P1 or a positive rank abelian variety (see Section 4 for more precise definitions and a proof that sporadic points are isolated). Motivated by well-known classification problems concerning rational torsion of elliptic curves, we study sporadic and isolated points on the modular curves X1(N). In particular, we show that any non-cuspidal non-CM sporadic, respectively isolated, point x∈X1(N) maps down to a sporadic, respectively isolated, point on a modular curve X1(d), where d is bounded by a constant depending only on j(x). Conditionally, we show that d is bounded by a constant depending only on the degree of Q(j(x)), so in particular there are only finitely many j-invariants of bounded degree that give rise to sporadic or isolated points.
KW - Faltings's theorem
KW - Isolated points
KW - Merel's uniform boundedness theorem
KW - Modular curves
KW - Serre's uniformity conjecture
KW - Sporadic points
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U2 - 10.1016/j.aim.2019.106824
DO - 10.1016/j.aim.2019.106824
M3 - Article
AN - SCOPUS:85072973496
SN - 0001-8708
VL - 357
JO - Advances in Mathematics
JF - Advances in Mathematics
M1 - 106824
ER -