# 课程视频区

Contents

## Modular forms, Modularity Lifting theorem and Fermat last theorem

Notes on lecture by Wang https://yufanluo.com/wp-content/uploads/2021/01/Modularity-Lifting-by-Patrick-Allen.pdf

## Etale cohomology

Notes on Lecture I ：

1. 介绍了一下这门课和啥是etale cohomology
2. （Serre）不存在有理系数或者Q_p系数（p是base field的特征）的weil cohomology theory。这里必须强调一点，注意litt的claim中，那个schemes的base field是F_p的代数闭域！不是F_p！ 事实上，这个这个论证对于over包含F_｛p^2}的域k都是对的，特别地，对于F_p的代数闭域。但是，对于F_p，这个serre的论证是不对的！所以对于F_p，存在系数在Q_p的晶体上同调论（weil cohomology）与这里serre的论证不矛盾。
3. 介绍了weil conjecture，并且对1维，即曲线的情形，给出了有理函数部分的几何证明，函数方程部分作为exercise，RH部分没证明。关于这部分，i.e. 曲线的weil conj.的完整证明可参考 https://math.uchicago.edu/~may/VIGRE/VIGRE2007/REUPapers/FINALFULL/Raskin.pdf
4. 介绍了一般情形的weil conjecture的有理部分的证明，关键是发展一种系数在特征为0的域的weil cohomology给出zeta函数的上同调解释，其实就是Lefschetz trace formula for l adic cohomology。一旦你承认这样的cohomology的存在性，有理函数部分就是线性代数了。这部分可参考milne的 lectures on etale cohomology section 27。而对于dwork的原始证明，可参看这个https://arxiv.org/abs/2012.01809。多提一句，除了 l adic cohomology这个weil cohomology外，我们还有晶体上同调，也有Lefschetz trace formula ，所以晶体上同调也能证明这个有理函数部分。
5. 最后介绍了Serre’s Kähler analogue，将在下节课给出大概证明。Litt故意留很多gap，有很多exercise。

## $p$-adic geometry, perfectoid spaces,prismatic cohomology and geometrization of the local langlands program

A great preview of those lectures was given by Tao’s blog https://terrytao.wordpress.com/2019/03/19/prismatic-cohomology/. 一个很不错的笔记见 http://www.math.columbia.edu/~chaoli/doc/BhattEilenberg.html

This goal of the proposed summer school is twofold

1. Give an advanced introduction to Scholze’s theory.
2. To understand the relation between perfectoid spaces and some aspects of arithmetic of modular (or, more generally, automorphic) forms such as representations mod p, and lifting of modular forms, completed cohomology, local Langlands program and special values of L-functions.

https://www.bilibili.com/video/BV1tX4y1G7bM

## Higher category theory

This course will be an introduction to higher category theory focussing on the approach via $\infty$-categories.

## Rational points on varieties

This article surveys a few of the highlights in the arithmetic of curves: the proof of the Mordell Conjecture, and the more detailed theorythat has developed around the classes of curves most studied until now by number theorists: modular curves, Fermat curves, and elliptic curves. 讲义下载地址：http://www.claymath.org/library/proceedings/cmip08c.pdf

## An introduction to Langlands program

Berkeley. 课程主页见 http://wwwf.imperial.ac.uk/~buzzard/MSRI/ （底下有完整Latex笔记）

For notes, see: https://arxiv.org/pdf/2103.02329.pdf

## Condensed Mathematics

1. Condensed Seminar: Who cares about pyknosis https://www.msri.org/seminars/24809
2. Condensed Seminar: Pyknotic sets https://www.msri.org/seminars/24915#description
3. Condensed Seminar: Pyknotic abelian groups https://www.msri.org/seminars/24919
4. Condensed Seminar: Animating pyknotic sets (AKA pyknotic spaces) https://www.msri.org/seminars/24750
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