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I'm currently teaching a short summer course on cryptography to high school students. Today, I taught them about elliptic curves. After spending some time playing around with their graphs over $\mathbb{R}$, we used Sage to graph elliptic curves over various prime finite fields. This was a ton of fun, and the students asked a lot of great questions. One of them asked how many loops such an elliptic curve would have if you just connected the dots, and how that would change as you increase the prime $p$. That inspired me to ask the following:

Take an elliptic curve $E$ over a prime finite field $\mathbb{F}_p$. Graph the set $E(\mathbb{F}_p)$ on the integer lattice $\lbrace 0,\ldots,p-1\rbrace^2$. Is anything known about the persistent homology of such sets of points? For example, what happens if you fix $E$ and vary $p$? What if you fix $p$ and compute the barcode diagrams for all elliptic curves over $\mathbb{F}_p$?

I have no reason to expect anything particularly interesting beyond the curiosity of my students, but that's enough to make me hope for a cool fact that I can tell them.

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  • $\begingroup$ I fear this has little to do with elliptic curves. (Stritckly speaking you consider the affine solutions to a fixed Weierstrass equation, you would get different pictures for isomorphic curves.) So this is an affine cubic with only one point at infinity. But the same question can be ask for any affine curve and any ordering of the points of $\mathbb{F}_p$. Or is there some reason you believe the particular shape of the group law has some influence on the question? $\endgroup$
    – Chris Wuthrich
    Commented Jul 6, 2023 at 8:40
  • $\begingroup$ I would also be surprised if the group law had anything to do with this, or if the points of an affine cubic were anything less than random (subject to symmetry and the degree bound coming from Bézout's theorem). But random algebraic geometry is pretty far from my work, so I have no reason to trust my intuition. $\endgroup$
    – Stephen McKean
    Commented Aug 1, 2023 at 13:25

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I think that going all the way to barcodes and persistent homology is a big leap, and probably not one where there will be something interesting. But maybe there are interesting things if you just go a baby step in that direction.

You're looking at $E({\mathbb F}_p)$ as a subset of ${\mathbb R}^2$. But I'd say it is better to look on the torus ${\mathbb R}^2 / (p {\mathbb Z})^2$ to start, so that the distance from $0$ to $1$ equals the distance from $p-1$ to $0$, wrapping around.

Then you can start "thickening" your dots on the torus, and see what happens. The very first interesting things (colliding dots) will happen when you have two points $(x,y)$ and $(x',y')$ with $x'-x = \pm 1$ and $y' = y$, or with $x'=x$ and $y'-y = \pm 1$. (Modulo $p$ that is.)

When do these happen? One can look at the variety $X \subset E \times E$, whose points are pairs of points $((x,y),(x',y')) \in E \times E$ satisfying such a pair of conditions like $x' - x = 1$ and $y' = y$. The number of such points is bounded by 6 if my scribbles are correct, and easy enough to relate to the number of solutions to a cubic equation over ${\mathbb F}_p$ anyways.

But I think this is not particularly interesting in its current form, because one could look at any such pair of conditions like $x' - x = u$ and $y' - y = v$. The number of solutions shouldn't depend on the archimedean size of $u$ and $v$, as far as I can tell. So if you find a point $(x,y)$ in $E({\mathbb F}_p)$, I think that other points on $E({\mathbb F}_p)$ will not strongly prefer to be close (on the torus or in ${\mathbb R}^2$ to $(x,y)$ or far from $(x,y)$ -- the only big influence would be the $y \rightarrow -y$ symmetry.

I don't really know though. I might start by asking the question: to what extent does $E({\mathbb F}_p)$ behave like a random symmetric (with respect to $y \rightarrow -y$) subset of the torus ${\mathbb R}^2 / (p {\mathbb Z})^2$. Perhaps someone has looked at this before, varying $p$ and/or varying the curve $E$. If it is indeed "random", then geometric invariants won't see more than the background distribution. (I don't know the expectations for persistent homology of a random finite subset of the torus of a given cardinality either!)

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  • $\begingroup$ Thanks for your thoughts! I suppose that E(F_p) won't behave totally like a random symmetric subset of the torus, because Bézout's theorem says that any line hits at most 3 elements of E(F_p). But maybe E(F_p) behaves like a random symmetric subset of the torus that also satisfies this "bounded sampling by lines" behavior. $\endgroup$
    – Stephen McKean
    Commented Aug 1, 2023 at 13:18
  • $\begingroup$ @StephenMcKean Oh right -- this is getting kind of neat! Let me know if you end up looking at subsets of the torus computationally. It could be an interesting student project too. Feel free to drop me an email if you compute things or if you want me to ask some students. $\endgroup$
    – Marty
    Commented Aug 2, 2023 at 22:37

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