arXiv subjects

Dong Sung Yoon

Publications and source records attributed to Dong Sung Yoon.

22 records · Page 2Linked to original sources

arXiv2014

Construction of ray class fields by smaller generators and applications

We first generate ray class fields over imaginary quadratic fields in terms of Siegel-Ramachandra invariants, which would be an extension of Schertz's result. And, by making use of quotients of Siegel-Ramachandra invariants we also construct ray class invariants over imaginary quadratic fields whose minimal polynomials have relatively small coefficients, from which we are able to solve certain quadratic Diophantine equations.

math.NT↗

arXiv2011

Representations by $x_1^2+2x_2^2+x_3^2+x_4^2+x_1x_3+x_1x_4+x_2x_4$

Let $r_Q(n)$ be the representation number of a nonnegative integer $n$ by the quaternary quadratic form $Q=x_1^2+2x_2^2+x_3^2+x_4^2+x_1x_3+x_1x_4+x_2x_4$. We first prove the identity $r_Q(p^2n)=r_Q(p^2)r_Q(n)/r_Q(1)$ for any prime $p$ different from 13 and any positive integer $n$ prime to $p$, which was conjectured in [Eum et al, A modularity criterion for Klein forms, with an application to modular forms of level 13, J. Math. Anal. Appl. 375 (2011), 28--41]. And, we explicitly determine a concise formula for the number $r_Q(n^2)$ as well for any integer $n$.

math.NT↗

arXiv2010

Algebraic integers as special values of modular units

Let $φ(τ)=η((τ+1)/2)^2/\sqrt{2π}e^\frac{πi}{4}η(τ+1)$ where $η(τ)$ is the Dedekind eta-function. We show that if $τ_0$ is an imaginary quadratic number with $\mathrm{Im}(τ_0)>0$ and $m$ is an odd integer, then $\sqrt{m}φ(mτ_0)/φ(τ_0)$ is an algebraic integer dividing $\sqrt{m}$. This is a generalization of Theorem 4.4 given in [B. C. Berndt, H. H. Chan and L. C. Zhang, Ramanujan's remarkable product of theta-functions, Proc. Edinburgh Math. Soc. (2) 40 (1997), no. 3, 583-612]. On the other hand, let $K$ be an imaginary quadratic field and $θ_K$ be an element of $K$ with $\mathrm{Im}(θ_K)>0$ which generators the ring of integers of $K$ over $\mathbb{Z}$. We develop a sufficient condition of $m$ for $\sqrt{m}φ(mθ_K)/φ(θ_K)$ to become a unit.

math.NT↗

arXiv2010

Ray class fields generated by torsion points of certain elliptic curves

We first normalize the derivative Weierstrass $\wp'$-function appearing in Weierstrass equations which give rise to analytic parametrizations of elliptic curves by the Dedekind $η$-function. And, by making use of this normalization of $\wp'$ we associate certain elliptic curve to a given imaginary quadratic field $K$ and then generate an infinite family of ray class fields over $K$ by adjoining to $K$ torsion points of such elliptic curve. We further construct some ray class invariants of imaginary quadratic fields by utilizing singular values of the normalization of $\wp'$, as the $y$-coordinate in the Weierstrass equation of this elliptic curve, which would be a partial result for the Lang-Schertz conjecture of constructing ray class fields over $K$ by means of the Siegel-Ramachandra invariant.

math.NT↗