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OK everyone, buckle up. I have 12 minutes until I go to seminar, and am going to try to tell you in that time about my header photo. (1/n)
The first important thing to know about is elliptic curves. These are objects that mathematicians are interested in because they can carry three really important and different types of structure: (2/n)
(1) geometric structure: every elliptic curve can be written as the set of pairs of complex numbers (x,y) satisfying an equation of the form y^2 = x^3 + ax+b for some fixed complex numbers a and b. (3/n)
This means - among other things - that when you zoom in to an elliptic curve, it just looks like a little patch of the complex plane. (4/n)
(2) algebraic structure: there is a way to define the sum of two points on an elliptic curve which behaves just like you want addition to: there is a unique zero element, every point P has an inverse -P, and other nice features like P + Q = Q + P. (5/n)
(3) arithmetic structure: everything that I said is not just true for curves y^2 = x^3+ax+b with a,b complex numbers, but generally for any type of numbers a and b (appropriately modified in case (1)) - numbers in modular arithmetic, p-adic numbers, functions, and so on. (6/n)
So for example, if a and b are real numbers, then we can study the elliptic curve from the point of view of real numbers *and* the point of view of complex numbers. If integers, we can study the curve from the p-adic point of view and the complex. This is very powerful! (7/n)
Now, since we can add points on any elliptic curve (call it E), we can define a function from E to itself which takes a point P and maps it to P + P. This is just like the function f(x) = 2x in real numbers, except that our new notion of addition gives a new "times 2". (8/n)
If we repeatedly apply this function to a point, sometimes we'll get an infinite sequence of points, and sometimes we won't. Thinking of f(x) = 2x as a function of the real numbers, applying repeatedly to 1, we get
1 -> 2 -> 4 -> 8 -> 16 -> ...
an infinite sequence! (9/n)
On the other hand, applying repeatedly to 0 gives
0 -> 0 -> 0 -> 0 -> ...
a finite sequence. The points which give a finite sequence are called "torsion points". (10/n)
Understanding how these points - which are algebraically special - interact with the geometry and arithmetic of the curve gives insight into the structure of the curve itself.

So here's a question: (11/n)
if I take two different elliptic curves, say y^2 = x^3 + ax+b and y^2 = x^3 + cx + d, how many points (x,y) are torsion points for both curves? Well, this is actually easy - there's not many, because the curves share very few points at all, without the torsion restriction. (12/n)
But up to choice of sign, y is determined by x on an elliptic curve. So let's ask instead: how many values x have the property that x is the x-coordinate of a torsion point for both elliptic curves? (13/n)
THIS is what my header photo shows! It's taking two different elliptic curves, and putting blue dots at the x-coordinates of torsion points for either. Where two blue dots overlap, we have an x-value which is a torsion x-coordinate for both curves. (14/n)
In the paper just posted: arxiv.org/abs/1901.09945 my coauthors and I prove that there is an upper bound on how much overlap there can be, no matter which two elliptic curves you choose- you can't get an arbitrarily large amount of common torsion for two different curves. (15/15)
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