Low Degree Places on the Modular Curve X1(N)

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arxiv 1202.4355 v5 pith:PVW4FT4H submitted 2012-02-20 math.NT

Mark van Hoeij This is my paper · ORCID

classification math.NT

keywords curvedefineddegreedegreesexamplesfieldsgivesgonality

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This short note gives questions and examples of points on X1(N) defined over number fields whose degrees are lower than the gonality.

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Rational isolated $j$-invariants from $X_1(\ell^n)$ and $X_0(\ell^n)$
math.NT 2025-06 accept novelty 7.0 of 10

For prime-power level, rational isolated j-invariants are exactly 15 values on X_1 (13 CM plus two non-CM) and 19 values on X_0 (13 CM plus six non-CM).
Prime order torsion on elliptic curves over number fields. Part I: Asymptotics
math.NT 2025-05 conditional novelty 7.0 of 10

Conditionally on sparse 'strange' newforms, the largest prime order of rational torsion on elliptic curves over degree-d fields is at most 3d+1 for large even d and o(d) for odd d.