Torsion points on 𝑋₀(𝑁)

Torsion points on J 0(N) and Galois representations

“…The main theorem of this article complements the results of Coleman-Kaskel-Ribet [2] on the "cuspidal torsion packet" of X 0 (N ). Recall that X 0 (N ) has two cusps, customarily denoted 0 and ∞.…”

“…As we point out in [2], the results of [10] imply that the intersection of X 0 (N ) and C (computed in J 0 (N )) consists of the two cusps 0 and ∞. In words, to prove that a torsion point P of X 0 (N ) is a cusp is to prove that it lies in the group C. For this, it is useful to decompose P into its primary parts: If P is a torsion point P of J 0 (N ) and is a prime number, we let P be the -primary part of P .…”

“…The article [2] introduces a strategy for identifying Ω precisely. Clearly, Ω contains the two cusps 0 and ∞ of X 0 (N ), whose images under ι have order n := num N −1 12 and 1, respectively [9, p. 98].…”

“…Clearly, Ω contains the two cusps 0 and ∞ of X 0 (N ), whose images under ι have order n := num N −1 12 and 1, respectively [9, p. 98]. Further, in the special case when X 0 (N ) is hyperelliptic, we note in [2] that the hyperelliptic branch points of X 0 (N ) belong to Ω if and only if N is different from 37. (Results of Ogg [11,12] show that X 0 (N ) is hyperelliptic if and only if N lies in the set { 23, 29, 31, 37, 41, 47, 59, 71 }.)…”

“…In [2], we advance the idea that Ω might contain only of the points we have just catalogued: Guess 1.1. Suppose that P is a point on X 0 (N ) whose image in J 0 (N ) has finite order.…”