A conjecture of Gross and Zagier: Case E(ℚ)tor≅ℤ/3ℤ

2019 ◽  
Vol 15 (09) ◽  
pp. 1793-1800 ◽  
Author(s):  
Dongho Byeon ◽  
Taekyung Kim ◽  
Donggeon Yhee
Keyword(s):  
Elliptic Curve ◽  
Infinite Order ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Roots Of Unity ◽  
Prime Divisors ◽  
Heegner Point ◽  
Tate Group ◽  
Tamagawa Numbers

Let [Formula: see text] be an elliptic curve defined over [Formula: see text] of conductor [Formula: see text], [Formula: see text] the Manin constant of [Formula: see text], and [Formula: see text] the product of Tamagawa numbers of [Formula: see text] at prime divisors of [Formula: see text]. Let [Formula: see text] be an imaginary quadratic field where all prime divisors of [Formula: see text] split in [Formula: see text], [Formula: see text] the Heegner point in [Formula: see text], and [Formula: see text] the Shafarevich–Tate group of [Formula: see text] over [Formula: see text]. Let [Formula: see text] be the number of roots of unity contained in [Formula: see text]. Gross and Zagier conjectured that if [Formula: see text] has infinite order in [Formula: see text], then the integer [Formula: see text] is divisible by [Formula: see text]. In this paper, we show that this conjecture is true if [Formula: see text].

scholarly journals Ranks, 2-Selmer groups, and Tamagawa numbers of elliptic curves with ℤ∕2ℤ × ℤ∕8ℤ-torsion

2019 ◽  
Vol 2 (1) ◽  
pp. 173-189
Author(s):  
Stephanie Chan ◽  
Jeroen Hanselman ◽  
Wanlin Li
Keyword(s):  
Elliptic Curves ◽  
Selmer Groups ◽  
Tamagawa Numbers
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