• Title/Summary/Keyword: Manin constant

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A CONJECTURE OF GROSS AND ZAGIER: CASE E(ℚ)tor ≅ ℤ/2ℤ OR ℤ/4ℤ

  • Dongho Byeon;Taekyung Kim;Donggeon Yhee
    • Journal of the Korean Mathematical Society
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    • v.60 no.5
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    • pp.1087-1107
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    • 2023
  • Let E be an elliptic curve defined over ℚ of conductor N, c the Manin constant of E, and m the product of Tamagawa numbers of E at prime divisors of N. Let K be an imaginary quadratic field where all prime divisors of N split in K, PK the Heegner point in E(K), and III(E/K) the Shafarevich-Tate group of E over K. Let 2uK be the number of roots of unity contained in K. Gross and Zagier conjectured that if PK has infinite order in E(K), then the integer c · m · uK · |III(E/K)| $\frac{1}{2}$ is divisible by |E(ℚ)tor|. In this paper, we prove that this conjecture is true if E(ℚ)tor ≅ ℤ/2ℤ or ℤ/4ℤ except for two explicit families of curves. Further, we show these exceptions can be removed under Stein-Watkins conjecture.