Explorations in math and programming
David Lowry-Duda

The LMFDB will soon have a new section on modular curves. And as with modular forms, each curve will have a portrait or badge that gives a rough approximation to some of the characteristics of the curve.

I wrote a note on some of the technical observations and implementation details concerning these curves. This note can be found here. I've also added a link to it in the unpublished notes section of my research page.

Instead of going into details here, I'll refer to the details in the note. I'll give the core idea.

Each modular curve comes from a subgroup $H \subset \mathrm{GL}(2, \mathbb{Z}/N\mathbb{Z})$ for some $N$ called the level. To form a visualization, we compute cosets for $H \cap \mathrm{SL}(2, \mathbb{Z}/N\mathbb{Z})$ inside $\mathrm{SL}(2, \mathbb{Z}/N\mathbb{Z})$, lift these to nice elements in $\mathrm{SL}(2, \mathbb{Z})$, and then translate the standard fundamental domain of $\mathrm{SL}(2, \mathbb{Z}) \backslash \mathcal{H}$ by these cosets.

We show this on the Poincaré disk, to give a badge format similar to what we did for modular forms.

This is not a perfect representation, but it captures some of the character of the curve.

Here are a few of the images that we produce.

I had studied how to produce space efficient SVG files as well, though I did not go in this direction in the end. But I think these silhouettes are interesting, so I include them too.

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