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arXiv · 1408.0995

A Genus Two Curve Related to the Class Number One Problem

Viet K. Nguyen
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Abstract

We give another solution to the class number one problem by showing that imaginary quadratic fields $\Q(\sqrt{-d})$ with class number $h(-d)=1$ correspond to integral points on a genus two curve $\mscrK_3$. In fact one can find all rational points on $\mscrK_3$. The curve $\mscrK_3$ arises naturally via certain coverings of curves:\ $\mscrK_3\rg\mscrK_6$,\ $\mscrK_1\rg\mscrK_2$\ with $\mscrK_2\colon y^2=2x(x^3-1)$ denoting the Heegner curve, also in connection with the so-called Heegner-Stark covering $\mscrK_1\rg\mscrK_s$.

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Viet K. Nguyen. 2014-08-05. A Genus Two Curve Related to the Class Number One Problem. https://arxiv.org/abs/1408.0995

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