Ramification in division fields and sporadic points on modular curves

نویسندگان

چکیده

Consider an elliptic curve E over a number field K. Suppose that has supersingular reduction at some prime $$\mathfrak {p}$$ of K lying above the rational p. We completely classify valuations $$p^n$$ -torsion points by valuation coefficient $$p{\text {th}}$$ division polynomial. This classification corrects error in earlier work Lozano-Robledo. As application, we find minimum necessary ramification order for to have point exact . Using this bound show sporadic on modular $$X_1(p^n)$$ cannot correspond curves without canonical subgroup. generalize our methods $$X_1(N)$$ with N composite.

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ژورنال

عنوان ژورنال: Research in number theory

سال: 2023

ISSN: ['2363-9555', '2522-0160']

DOI: https://doi.org/10.1007/s40993-023-00424-2