Dongho Byeon
Seoul National University · 数学
研究室紹介
Professor Dongho Byeon's research lab specializes in algebraic number theory, with a focus on class numbers, class groups, and fundamental units in real and imaginary quadratic fields. The lab investigates deep arithmetic properties of quadratic and cubic fields, including the distribution of class numbers modulo primes, the structure of 2-class groups, and the existence of fields with specific class group structures. Using tools from analytic number theory, L-functions, and modular forms, the lab also explores connections to elliptic curves, Heegner points, and Diophantine problems such as the Goldfeld conjecture and the binary Goldbach problem for polynomials.
Research Overview
Research Output Trend
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Selected Papers
15Let <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 >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 >3, [sharp ]{0< D < X | h ( D )[nequiv ]0(mod p)}>> 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
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.
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 $2^{n}$.