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arXiv · 1008.0473

Algebraic integers as special values of modular units

Ja Kyung Koo·Dong Hwa Shin·Dong Sung Yoon
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Abstract

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.

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Ja Kyung Koo, Dong Hwa Shin, Dong Sung Yoon. 2010-08-08. Algebraic integers as special values of modular units. https://arxiv.org/abs/1008.0473

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