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Lee, Jung-Jo
- Generalization of a Theorem of Hurwitz
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be the Eisenstein series of weight k attached to the full modular group. Let z be a CM point in the upper half-plane. Then there is a transcendental number Ωz such that
G2k(z)=Ω2kz. (an algebraic number).
Moreover, Ωz can be viewed as a fundamental period of a CM elliptic curve defined over the field of algebraic numbers. More generally, given any modular form f of weight k for the full modular group, and with algebraic Fourier coefficients, we prove that f(z)πk/Ωzk is algebraic for any CM point z lying in the upper half-plane. We also prove that for any automorphism σ of Gal (̅ℚ/ℚ), (f(z)πk/Ωzk)σ=fσ(z)πk/Ωzk.
Authors
Affiliations
1 Department of Mathematics, Kyungpook National University, Daegu-702-701, KR
2 Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario- K7L3N6, CA
3 Department of Mathematics, Yonsei University, 50 Yonsei-Ro, Seodaemun-Gu, Seoul-120-749, KR
1 Department of Mathematics, Kyungpook National University, Daegu-702-701, KR
2 Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario- K7L3N6, CA
3 Department of Mathematics, Yonsei University, 50 Yonsei-Ro, Seodaemun-Gu, Seoul-120-749, KR
Source
Journal of the Ramanujan Mathematical Society, Vol 31, No 3 (2016), Pagination: 215-226Abstract
This paper is an exposition of several classical results formulated and unified using more modern terminology. We generalize a classical theorem of Hurwitz and prove the following: let Gk (z)=∑1/(mz+n)kbe the Eisenstein series of weight k attached to the full modular group. Let z be a CM point in the upper half-plane. Then there is a transcendental number Ωz such that
G2k(z)=Ω2kz. (an algebraic number).
Moreover, Ωz can be viewed as a fundamental period of a CM elliptic curve defined over the field of algebraic numbers. More generally, given any modular form f of weight k for the full modular group, and with algebraic Fourier coefficients, we prove that f(z)πk/Ωzk is algebraic for any CM point z lying in the upper half-plane. We also prove that for any automorphism σ of Gal (̅ℚ/ℚ), (f(z)πk/Ωzk)σ=fσ(z)πk/Ωzk.
- Tame Ramification and Group Cohomology
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Authors
Affiliations
1 Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, IN
2 Department of Mathematics, Seoul National University, Shillim-dong, Gwanak-gu, Seoul 151-742, KP
1 Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, IN
2 Department of Mathematics, Seoul National University, Shillim-dong, Gwanak-gu, Seoul 151-742, KP
Source
Journal of the Ramanujan Mathematical Society, Vol 32, No 1 (2017), Pagination: 51–74Abstract
We give an intrinsic parametrisation of the set of tamely ramified extensions of a local field with finite residue field and bring to the fore the role played by group cohomology. We show that two natural definitions of the cohomology class of a tamely ramified finite galoisian extension coincide, and can be recovered from the parameter. We also give an elementary proof of Serre’s mass formula in the tame case and in the simplest wild case, and we classify tame galoisian extensions of degree the cube of a prime.- Ranks of Jacobians of Curves Related to Binary Forms
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Authors
Affiliations
1 Yonsei University, Shinchon-dong, Seodaemun-gu, Seoul 120-749, KR
1 Yonsei University, Shinchon-dong, Seodaemun-gu, Seoul 120-749, KR
Source
Journal of the Ramanujan Mathematical Society, Vol 27, No 2 (2012), Pagination: 119–126Abstract
Rubin and Silverberg reformulated the question of whether the ranks of the quadratic twists of an elliptic curve over Q are bounded into the question of whether a certain infinite series converges. Da¸browski and Je¸drzejak consider an analogue of this theorem for the family of Jacobian varieties of twisted Fermat curves xp + yp = m (p fixed odd prime), where m runs through pth power-free integers. In this paper, we consider the family of Jacobian varieties of curves f (x, y) = mzn, where f (x, y) ∈ Z[x, y] is an irreducible binary form of degree n ≥ 3 and m is an nth power-free integer.- Corrigendum to “Ranks of Elliptic Curves”
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Authors
Affiliations
1 Department of Mathematics, Queen’s University, Kingston, Ontario K7L 3N6, CA
1 Department of Mathematics, Queen’s University, Kingston, Ontario K7L 3N6, CA
Source
Journal of the Ramanujan Mathematical Society, Vol 21, No 4 (2006), Pagination: 297-298Abstract
Theorem 4.20 of [4] is incorrect as it stands. This was pointed out to me by several people. For example, the curve y2+y=x3+x2-142x+182 satisfies the condition in Theorem 4.20 of [4], but it has rank 5. We correct this here.- Ranks of Elliptic Curves
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Authors
Affiliations
1 Department of Mathematics, Queen’s University, Kingston, Ontario, K7L 3N6, CA
1 Department of Mathematics, Queen’s University, Kingston, Ontario, K7L 3N6, CA
Source
Journal of the Ramanujan Mathematical Society, Vol 17, No 4 (2002), Pagination: 267–285Abstract
Let E be an elliptic curve defined by a Weierstrass equation
y2 = x3 + Ax + B with A, B ∈ Q ------- (1) and
let E(D) be defined by
Dy2 = x3 + Ax + B with A, B ∈ Q