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12 results on '"Yhee, Donggeon"'

1. On the Jacobian of hyperelliptic curves $y^2 = x^5 + m^2$

Author

Jeong, Keunyoung, Park, Junyeong, and Yhee, Donggeon

Subjects
Mathematics - Number Theory, 11G30, 11G10, 11F27
Abstract

In this paper, we study the algebraic rank and the analytic rank of the Jacobian of hyperelliptic curves $y^2 = x^5 + m^2$ for integers $m$. Namely, we first provide a condition on $m$ that gives a bound of the size of Selmer group and then we provide a condition on $m$ that makes $L$-functions non-vanishing. As a consequence, we construct a Jacobian that satisfies the rank part of the Birch--Swinnerton-Dyer conjecture., Comment: 17 pages

Published
2021

2. On a conjecture of Gross and Zagier

Author

Byeon, Dongho, Kim, Taekyung, and Yhee, Donggeon

Subjects
Mathematics - Number Theory, 11G05
Abstract

Gross and Zagier conjectured that if the analytic rank of a rational elliptic curve is 1, then the order of the rational torsion subgroup of the elliptic curve divides the product of Tamagawa number, Manin constant, and the square root of the order of Tate--Shafarevich group over an imaginary quadratic field. In this paper, we show that this conjecture is true., Comment: 22 pages; major revision: proved the whole conjecture without any other conjecture, deleted some routine computations (will be given upon request)

Published
2015

4. ON THE JACOBIAN OF A FAMILY OF HYPERELLIPTIC CURVES

Author

Jeong, Keunyoung, Park, Junyeong, and Yhee, Donggeon

Subjects
11G30, 11G10, 11F27
Abstract

In this paper, we study the algebraic rank and the analytic rank of the Jacobian of hyperelliptic curves y² = x⁵ + m² for integers m. Namely, we first provide a condition on m that gives a bound of the size of Selmer group and then we provide a condition on m that makes L-functions non-vanishing. As a consequence, we construct a Jacobian that satisfies the rank part of the Birch–Swinnerton-Dyer conjecture.

Published
2023

10. A conjecture of Gross and Zagier: Case E(ℚ)tor≅ℤ/2ℤ⊕ℤ/2ℤ, ℤ/2ℤ⊕ℤ/4ℤ or ℤ/2ℤ⊕ℤ/6ℤ.

Author

Byeon, Dongho, Kim, Taekyung, and Yhee, Donggeon

Subjects
DIVISIBILITY groups, PRIME numbers, QUADRATIC fields, DIVISOR theory, LOGICAL prediction, ELLIPTIC curves, INTEGERS
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 , P K the Heegner point in E (K) , and III (E / K) the Shafarevich–Tate group of E over K. Let 2 u K be the number of roots of unity contained in K. Gross and Zagier conjectured that if P K has infinite order in E (K) , then the integer c ⋅ m ⋅ u K ⋅ | 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 / 2 ℤ ⊕ ℤ / 2 ℤ , ℤ / 2 ℤ ⊕ ℤ / 4 ℤ or ℤ / 2 ℤ ⊕ ℤ / 6 ℤ. [ABSTRACT FROM AUTHOR]

Published
2020
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