One of the reasons I’m writing the Dictionary of Mathematical Eponymy is to introduce myself to new ideas, and to mathematicians I didn’t know about. To things I wish I knew more about.
Elliptic curves are pretty high on that list.
What is the Lutz-Nagell theorem?
It’s sometimes - reasonably, since Trygve Nagell did discover it first - called the Nagell-Lutz theorem, but I reckon it’s less confusing this way round.
Here’s what it says:
- Let $a$, $b$ and $c$ be integers
- Let $E$ be an elliptic curve with equation $y^2 = x^3 + ax^2 + bx + c$
- The discriminant of $E$ is $D = -4a^3c + a^2b^2 + 18abc - 4b^3 - 27c^2$; suppose $D \ne 0$.
- Let $P = (X,Y)$ be a rational point of finite order on $E$
- $X$ and $Y$ are integers;
- Either $Y$ is a divisor of $D$, or $Y=0$ and the order of $P$ is 2.
A rational point is simply a point whose coordinates are rational; the idea of a point’s order is a bit more complicated.
Something you probably wouldn’t think to do with points, if you hadn’t come across elliptic curves before, is ‘add them together’. However, there’s an accepted way to add such points together: you draw a line through the two, and find where this line meets the curve again. Reflect this point in the $x$-axis, and the resulting point is the two initial points added together.
You can add a point to itself - getting $2P$ - by doing the same thing with the tangent to the curve at that point - find where it crosses the curve, reflect it in the axis, and that point is double the original point.
There’s one more exception: if you add two points with the same $x$-coordinate together, you get a vertical line. By convention, this intersects the curve again ‘at infinity’, and the point at infinity is considered to lie on the curve, and the line through a point and the point at infinity is simply the vertical line through the point.
The order of a point is $n$ such that $nP$ is the point at infinity. Since elliptic curves are smooth and symmetrical in the $x$-axis, if a point has a $Y$-coordinate of 0, the tangent to the curve there is vertical and $2P$ is the point at infinity - so $P$ is of order 2.
Why is it important?
The points on an elliptic curve form an abelian group, which makes them useful in several fields - most famously, Andrew Wiles’s breakthrough in proving Fermat’s Last Theorem was built on elliptic curves, and modern cryptography, rather than relying on factorising large numbers, relies on exchanging secrets using points on elliptic curves. (Given $P$ and $n$, it’s easy to find $nP$; given $nP$ and $n$, it’s very difficult to find $P$.)
The Lutz-Nagell theorem was an important step towards determining the structure of the so-called torsion groups of rational points on elliptic curves.
Who was Élisabeth Lutz?
Élisabeth Lutz was born in Thann, France in 1914 and studied under André Weil in Strasbourg in the 1930s. She earned her PhD in 1951 in Grenoble, where she later became a professor.
She died in 2008 in Grenoble.
* Edited 2019-12-06 to remove an unused variable. Thanks, @SciencePundit!
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