Journal of the Korean Mathematical Society (대한수학회지)

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

  • Dongho Byeon (Department of Mathematical Sciences Research Institute of Mathematics Seoul National University) ;
  • Taekyung Kim (Cryptolab) ;
  • Donggeon Yhee (Department of Mathematical Sciences Seoul National University)
  • Received : 2023.02.15
  • Accepted : 2023.07.11
  • Published : 2023.09.01
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Abstract

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.

Keywords

Acknowledgement

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (NRF-2023R1A2C1002612).

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