Dongho Byeon

Indivisibility of Class Numbers and Iwasawa λ-Invariants of Real Quadratic Fields

Let D>0 be the fundamental discriminant of a real quadratic field, and h(D) its class number. In this paper, by refining Onos idea, we show that for any prime p>3, ♯{0>p√(X)/logX.

Rank-one quadratic twists of an infinite family of elliptic curves

Abstract A conjecture of Goldfeld implies that a positive proportion of quadratic twists of an elliptic curve E/ℚ has (analytic) rank 1. This assertion has been confirmed by Vatsal [Math. Ann. 311: 791–794, 1998] and the first author [Acta Arith. 114: 391–396, 2004] for only two elliptic curves. Here we confirm this assertion for infinitely many elliptic curves E/ℚ using the Heegner divisors, the 3-part of the class groups of quadratic fields, and a variant of the binary Goldbach problem for polynomials.

On the finiteness of certain Rabinowitsch polynomials II

Abstract Let m be a positive integer and fm(x) be a polynomial of the form fm(x)=x2+x−m. We call a polynomial fm(x) a Rabinowitsch polynomial if for t=[ m ] and consecutive integers x=x 0 , x 0 +1,…,x 0 +t−1, | f(x)| is either 1 or prime. In Byeon (J. Number Theory 94 (2002) 177), we showed that there are only finitely many Rabinowitsch polynomials fm(x) such that 1+4m is square free. In this note, we shall remove the condition that 1+4m is square free.

Class number 3 problem for the simplest cubic fields

We give some necessary conditions for class numbers of the simplest cubic fields to be 3 and using Lettl’s lower bounds of residues at s = 1 of Dedekind zeta functions attached to cyclic cubic fields [?], determine all the simplest cubic fields of class number 3.

A NOTE ON UNITS OF REAL QUADRATIC FIELDS

For a positive square-free integer , let and be positive integers such that is the fundamental unit of the real quadratic field , where if (mod 4) and otherwise For a given positive integer and a palindromic sequence of positive integers , , , we define the set := { > 0, }. We prove that for all square-free integer with one possible exception and apply it to Ankeny-Artin-Chowla conjecture and Mordell conjecture.

Existence of certain fundamental discriminants and class numbers of real quadratic fields

Let D be the fundamental discriminant of the quadratic field Q( √ D), h(D) its class number, and χD := ( D · ) the usual Kronecker character. Let p be prime, Zp the ring of p-adic integers, and λp(Q( √ D)) the Iwasawa λ-invariant of the cyclotomic Zp-extension of Q( √ D). Let Rp(D) denote the p-adic regulator of Q( √ D), and | · |p denote the usual multiplicative p-adic valuation normalized so that |p|p = 1 p . In [9], by applying Sturm’s theorem on the congruence of modular forms to Cohen’s half integral weight modular forms, Ono proved the following theorem.

On the cancellation problem of Zariski

Let K 1 and K 2 be extension fields over a field K with char K = p > 0. Assume L = K 1 ( x 1 ) = K 2 ( x 2 ) ⊃ K where x i is transcendental over K i , for i = 1, 2. In this paper we prove that if K 1 is a perfect field, then K 1 = K 2 .

Infinitely many elliptic curves of rank exactly two

In this paper, we construct an infinite family of elliptic curves whose rank is exactly two and the torsion subgroup is a cyclic group of order two or three, under the parity conjecture.

REAL QUADRATIC FUNCTION FIELDS OF MINIMAL TYPE

Abstract. In this paper, we will introduce the notion of the real qua-dratic function fields of minimal type, which is a function field analogue toKawamoto and Tomita’s notion of real quadratic fields of minimal type.As number field cases, we will show that there are exactly 6 real quadraticfunction fields of class number one that are not of minimal type. 1. IntroductionLet q be an odd prime and k = F q the finite field of order q. Let D ∈ k[x]be a monic square-free polynomial of even degree and K = k(x)(√D) the realquadratic function field over k. Let O K = k[x]+k[x]√D be the integral closureof k[x] in K, h(O K ) the ideal class number of O K , and ǫ K = t D + u D √D thefundamental unit of K.Let D ∈ k[x] be a monic square-free polynomial of even degree. Then√Dhas a continued fraction expansion√D = [a 0 ,a 1 ,a 2 ,...,a l−1 ,2a 0 ],where the sequence of non-constant polynomials a 1 ,...,a l−1 is palindromic,that is, a l−i = a i for 1 ≤ i ≤ l −1. Here l is the period of D.For a given positive integer l > 1 and a palindromic sequence of non-constant polynomials a