From Notices of the AMS
The Quintic, the Icosahedron, and Elliptic Curves
by Bruce Bartlett
Communicated by William McCallum
There is a remarkable relationship between the roots of a quintic polynomial, the icosahedron, and elliptic curves. This discovery is principally due to Felix Klein (1878), but Klein's marvellous book [9] misses a trick or two, and doesn't tell the whole story. The purpose of this article is to present this relationship in a fresh, engaging, and concise way. We will see that there is a direct correspondence between:
- "Evenly ordered" roots $(x_1, \ldots, x_5)$ of a Brioschi quintic \begin{equation} X^5 + 10BX^3 + 45B^2X + B^2 = 0, \label{brioschi_intro} \end{equation}
- Points on the icosahedron, and
- Elliptic curves equipped with a primitive basis for their 5-torsion, up to isomorphism.
Moreover, this correspondence gives us a very efficient direct method to actually calculate the roots of a general quintic! For this, we'll need some tools both new and old, such as Cremona and Thongjunthug's complex arithmetic geometric mean [3], and the Rogers-Ramanujan continued fraction [5][12]. These tools are not found in Klein's book, as they had not been invented yet!
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