Dongho Byeon

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t"> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding="application/x-tex">t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a square free integer. We shall show that there exist infinitely many positive fundamental discriminants <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D greater-th

Let D &gt;0 be the fundamental discriminant of a real quadratic field, and h ( D ) its class number. In this paper, by refining Ono's idea, we show that for any prime p &gt;3, [sharp ]{0&lt; D &lt; X | h ( D )[nequiv ]0(mod p)}&gt;&gt; p √( X )/log X .

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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s equals 1"> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">s=1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of Dedekind zeta functions attac

Let $g \geq 2$ and $n \geq 1$ be integers. In this paper, we shall show that there are infinitely many imaginary quadratic fields whose class number is divisible by $2g$ and whose discriminant has only two prime divisors. As a corollary, we shall show that there are infinitely many imaginary quadratic fields whose 2-class group is a cyclic group of order divisible by $2^{n}$.

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 [V1] and the first author [By] 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.