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 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
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).
--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
tikzpictures in the TeX file).
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)
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") 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 …)
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()
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.
LMFDB – The L-functions and Modular Forms Database