For those attending the Haussdorf Summer School previously mentioned here, I followed up with the speakers to ask them a little about what background was optimal for getting the most out of their lectures. In particular, I asked them to send me a sentence along the following lines: it would be useful for participants in this course to know X and to have some familiarity with Y, but no knowledge of Z is assumed for various (set) values of X,Y, and Z. Here are the responses, which I hope will be useful for some of you. (Some light editing has taken place which may have introduced typos, as I’m pretty prone to those.)
- Arthur-Cesar le Bras and Gabriel Dospinescu on p-adic geometry.
It would be useful for participants in this course to know the basic formalism of adic spaces (most importantly the notion of continuous valuation, adic spectrum, rational domain and being aware of some of the perversities related to the structure presheaf not being a sheaf in general) and to have some familiarity with Fontaine rings (and what they are useful for; we recommend Berger’s paper An introduction to the theory of p-adic Galois representations, sections I-II for a brief overview) and p-divisible groups (the latter being most likely a prerequisite for other courses as well)
- George Boxer and Vincent Pilloni on Higher Hida theory.
A basic familiarity with modular forms, modular curves, Hecke operators, etc. together with basic familiarity with rigid analytic geometry would be enough to follow a significant part of the course.
- Patrick Allen and James Newton on Automorphy lifting.
It would be useful for participants in this course to know the basics of Galois cohomology and modular forms and to have some familiarity with automorphic representations and deformation theory of Galois representations (although we will give a quick summary of the deformation theory needed).
- Eva Viehmann and Cong Xue on Shtukas.
- Sophie Morel and Timo Richarz on Geometric Satake.
It would be useful to know about schemes, etale cohomology, the six functors, algebraic groups, root systems, highest weight theory, classical Satake isomorphism — for the last four topics, knowing the theory in the case of \(\mathrm{GL}_n\) or \(\mathrm{SL}_n\) is already pretty good. It would be helpful to have some familiarity with the Tannakian formalism, perverse sheaves, and G-bundles. No knowledge is assumed of loop groups or affine Grassmannians.
We will assume algebraic geometry and some familiarity with linear algebraic groups.