Journal of the Ramanujan Mathematical Society

Jung-Jo Lee ¹, M. Ram Murty ², Donghoon Park ³

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

Abstract

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 G_k (z)=∑1/(mz+n)^k
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 Ω_z such that
G_2k(z)=Ω^2k_z. (an algebraic number).
Moreover, Ω_z 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)π^k/Ω_z^k 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)π^k/Ω_z^k)^σ=f^σ(z)π^k/Ω_z^k.

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Chandan Singh Dalawat ¹, Jung-Jo Lee ²

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

Abstract

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.

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Jung-Jo Lee ¹

Affiliations
1 Yonsei University, Shinchon-dong, Seodaemun-gu, Seoul 120-749, KR

Abstract

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 x^p + y^p = 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.

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Jung-Jo Lee ¹

Affiliations
1 Department of Mathematics, Queen’s University, Kingston, Ontario K7L 3N6, CA

Abstract

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.

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Jung-Jo Lee ¹

Affiliations
1 Department of Mathematics, Queen’s University, Kingston, Ontario, K7L 3N6, CA

Abstract

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

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