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Orders of the torsion of points of curves of genus 1

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Let K be an algebraic number field of degree n; let be the number of divisor classes of the field K; y: v2=u4+au2+B is the Jacobian curve over\(K; B\left( {a^2 - 48} \right) = c^2 \prod\limits_{i - 1}^N {q_i } \) where C is an integral divisor, q1, ..., qN are distinct prime divisors. One proves that there exists an effectively computable constant c=c(n, h(K), N), such that the order m of the torsion of any primitive K-point on

is bounded by it: m⩽C.

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Literature cited

  1. 1.

    E. Galois, Works [Russian translation], Moscow-Leningrad (1936).

  2. 2.

    V. A. Dem'yanenko, “The torsion of analytic curves,” Izv. Akad. Nauk SSSR, Ser. Mat.,35, 280–307 (1971).

  3. 3.

    B. Mazur, “Rational points on modular curves,” in: Lect. Notes Math.,601, 107–148. Springer-Verlag, Berlin (1977).

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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, AN SSSR, Vol. 82, pp. 5–28, 1979.

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Dem'yanenko, V.A. Orders of the torsion of points of curves of genus 1. J Math Sci 18, 843–861 (1982). https://doi.org/10.1007/BF01763959

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