# Talk at the Festkolloquium to celebrate the Fields medal of Peter Scholze

## Slides of the talk

There is also a Video of the event

## TeX file

TeX file including image files: zip archive (this includes all the image files used for the slides, except for the logos of the University of Duisburg-Essen and the Essen Seminar for Algebraic Geometry and Arithmetic).

**Please note:** To compile the TeX file as is, you need `xelatex`

(propably `lualatex`

would also work, but I did not try), the `Arimo`

font and `gnuplot`

installed on your system. The command to compile the file then is

xelatex --shell-escape festkolloquium-scholze.tex

You can replace `xelatex`

by the common `pdflatex`

if you switch back to the standard font (comment out lines 52, 53, 56 in the TeX file).

The `--shell-escape`

parameter is required so that TeX can invoke gnuplot and thus produce the plots of graphs of polynomials on the slides (look at the `tikzpicture`

s in the TeX file).

## SageMath pictures

### Elliptic curve

Information about the elliptic curve $E:y^2 = x^3+x^2-x$ can be found in LMFDB

In SageMath, it is available and can be plotted as follows:

sage: E = EllipticCurve("20a2") sage: p = E.plot(xmin=-2, xmax=2, ymin=-3, ymax=3, thickness=4) sage: p.show(dpi=200)

### Modular form

The modular form corresponding to (the isogeny class of) $E$ is also in LMFDB

It can be pictured using the SageMath complex_plot method, as follows:

sage: f = Newforms(20,2,names="a")[0] sage: z = var('z') sage: g = 0 sage: for i, c in enumerate(f.coefficients(250)): ....: g += c * exp(2*pi*I*(i+1)*z) ....: sage: p1 = complex_plot(g, (-1.5, 1.5), (0, 1.2), plot_points=2000) sage: p1.show(dpi=600)

Depending on the desired accuracy of the picture, you may want to change some of the parameters. (For the above parameters, the computation taks a short while, but not very long, on my laptop.)

(In fact, there may be a more direct way to do this – please tell me if you know one …)

### Fundamental domain

There is also an easy way to plot the fundamental domain for $\Gamma_0(20)$ (the relevant congruence subgroup for the above $E$ and modular form). In the talk I did not use it, though.

sage: F = FareySymbol(Gamma0(20)).fundamental_domain(color_even='lightgray', ymax=1.5) sage: F.show()

## License

Feel free to distribute the above pdf as is, or to use what is described above or portions of the TeX file. The pictures of Urbano Monte’s map of the world are made available be the David Rumsey Map Collection under the CC BY-NC-SA license.

Most other pictures (Fields medal, …) were taken from Wikipedia and are in the public domain.

The newspaper snippets are screenshots.