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Kim, Tae Kyung
기하학 수리물리 연구단
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A conjecture of Gross and Zagier: Case e (Q)tor ≃ Z/2Z ⊕ Z/2Z, Z/2Z ⊕z/4Z or Z/2Z ⊕ Z/6Z

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Title
A conjecture of Gross and Zagier: Case e (Q)tor ≃ Z/2Z ⊕ Z/2Z, Z/2Z ⊕z/4Z or Z/2Z ⊕ Z/6Z
Author(s)
Byeon D.; Kim, Taekyung; Yhee D.
Subject
Elliptic curves, ; Manin constant, ; Shafarevich-Tate group, ; Tamagawa number
Publication Date
2020-08
Journal
INTERNATIONAL JOURNAL OF NUMBER THEORY, v.16, no.7, pp.1567 - 1572
Publisher
WORLD SCIENTIFIC PUBL CO PTE LTD
Abstract
© 2020 World Scientific Publishing Company. 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)|1 2 is divisible by |E(Q)tor|. In this paper, we show that this conjecture is true if E (Q)tor ≃ Z/2Z ⊕ Z/2Z, Z/2Z ⊕Z/4Z or Z/2Z ⊕ Z/6Z
URI
https://pr.ibs.re.kr/handle/8788114/7718
DOI
10.1142/S1793042120500827
ISSN
1793-0421
Appears in Collections:
Center for Geometry and Physics(기하학 수리물리 연구단) > 1. Journal Papers (저널논문)
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