arXiv subjects

Jigu Kim

Publications and source records attributed to Jigu Kim.

4 recordsLinked to original sources

arXiv2022

Congruences on the class numbers of $\mathbb{Q}(\sqrt{\pm 2p})$ for $p\equiv3$ $(\text{mod }4)$ a prime

For a prime $p\equiv 3$ $(\text{mod }4)$, let $h(-8p)$ and $h(8p)$ be the class numbers of $\mathbb{Q}(\sqrt{-2p})$ and $\mathbb{Q}(\sqrt{2p})$, respectively. Let $Ψ(ξ)$ be the Hirzebruch sum of a quadratic irrational $ξ$. We show that $h(-8p)\equiv h(8p)\Big(Ψ(2\sqrt{2p})/3-Ψ\big((1+\sqrt{2p})/2\big)/3\Big)$ $(\text{mod }16)$. Also, we show that $h(-8p)\equiv 2h(8p)Ψ(2\sqrt{2p})/3$ $(\text{mod }8)$ if $p\equiv 3$ $(\text{mod }8)$, and $h(-8p)\equiv \big(2h(8p)Ψ(2\sqrt{2p})/3\big)+4$ $(\text{mod }8)$ if $p\equiv 7$ $(\text{mod }8)$.

math.NT↗

arXiv2022

$p$-adic Properties for Taylor Coefficients of Half-integral Weight Modular Forms on $Γ_1(4)$

For a prime $p\equiv 3$ $(\text{mod }4)$ and $m\ge 2$, Romik raised a question about whether the Taylor coefficients around $\sqrt{-1}$ of the classical Jacobi theta function $θ_3$ eventually vanish modulo $p^m$. This question can be extended to a class of modular forms of half-integral weight on $Γ_1(4)$ and CM points; in this paper, we prove an affirmative answer to it for primes $p\ge5$. Our result is also a generalization of the results of Larson and Smith for modular forms of integral weight on $\mathrm{SL}_2(\mathbb{Z})$.

math.NT↗

arXiv2022

Congruences for odd class numbers of quadratic fields with odd discriminant

For any distinct two primes $p_1\equiv p_2\equiv 3$ $(\text{mod }4)$, let $h(-p_1)$, $h(-p_2)$ and $h(p_1p_2)$ be the class numbers of the quadratic fields $\mathbb{Q}(\sqrt{-p_1})$, $\mathbb{Q}(\sqrt{-p_2})$ and $\mathbb{Q}(\sqrt{p_1p_2})$, respectively. Let $ω_{p_1p_2}:=(1+\sqrt{p_1p_2})/2$ and let $Ψ(ω_{p_1p_2})$ be the Hirzebruch sum of $ω_{p_1p_2}$. We show that $h(-p_1)h(-p_2)\equiv h(p_1p_2)Ψ(ω_{p_1p_2})/n$ $(\text{mod }8)$, where $n=6$ (respectively, $n=2$) if $\min\{p_1,p_2\}>3$ (respectively, otherwise). We also consider the real quadratic order with conductor $2$ in $\mathbb{Q}(\sqrt{p_1p_2})$.

math.NT↗

arXiv2020

Distribution of root numbers of Hecke characters attached to some elliptic curves

In this paper, we show that an action on the set of elliptic curves with j= 1728 preserves a certain kind of symmetry on the local root number of Hecke characters attached to such elliptic curves. As a consequence, we give results on the distribution of the root numbers and their average of the aforementioned Hecke characters.

math.NT↗