The arithmetic geometry of the Fields Medal 2018, perfectoid spaces (part 1/2, Affinoid Spaces)

Today, the fields medal was awarded. One of the recipients is the German director of the Max Planck institute Peter Scholze (he has only 30 years old!). He obtained his Ph.D. inventing and studying his notion of perfectoid spaces which is the topic that I will introduce, but first I will talk about affinoid spaces today.

I have been attending seminars with that is now ex-group of algebraic geometry at the university of Groningen where I made my Ph.D. during the past five months related to overconvergent sheaves and rigid geometry leaded by Jaap Top, Marius van der Put and Stephen Mueller. I learnt there about affinoid spaces and Frobenius structures. I will just define and give some motivation of their invention here and then I will try to give a very brief and informal description of a perfectoid space tomorrow in a continuation of this post.


Why?

One of the main purposes of all these mathematics is to study points on algebraic varieties. For example, is well known by Gerd Faltings  (see here) that the number of solutions of an algebraic curve defined by $latex C(x,y)=0$ of genus $latex g \geq 2$ has a finite number of solutions over any finite extension of $latex \mathbb{Q}$, which we denote by $latex k$. In other words, there is just a finite number of $latex (\tfrac{a}{b},\tfrac{c}{d})$ where $latex a,b,c,d\in O_k$ and $latex c,d\neq 0$ and $latex C(\tfrac{a}{b},\tfrac{c}{d})=0$.

There are bounds for the number of points of $latex C/k$, in fact, if we fix $latex k=\mathbb{Q}$, $latex O_k=\mathbb{Z}$ and $latex C/\mathbb{Q}$ is hyperelliptic, one has that #$latex C(k)\leq 8rg+33(g-1)+1$ where $latex r=\text{rank}(J)$ and $latex J$ is the algebraic group variety associated to the curve $latex C$ which must have rank at most $latex g-3$.  This bound was presented to me in Groningen by professor Michael Stoll from the University of Bayreuth (see here). He uses Chabauty-Coleman integration to obtain this.

The problem is that this bound is very restricted, does not work for varieties of arbitrary dimension, and only works for a very well behaved family of curves (hyperelliptic). For the restriction of $latex r\leq g-3$ there is current hot develpments by Bas Edixhoven, Rene Schoof and others in Leiden using Poincare Torsors on the Neron Severi group of $latex J$.


p-adic numbers (recall)

Consider the non-archimidean field $latex \mathbb{Q}_p$  (here you can construct them using the Cauchy completion of $latex \mathbb{Q}$ under the weird absolute value $latex |x|_p=|p^n\tfrac{a}{b} |_p=p^{-n}$ with $latex p$ prime and $latex a,b\in\mathbb{Z}$ (just as you construct $latex \mathbb{R}$ from the convergent sequences over $latex \mathbb{Q}$. It is handy to represent the elements of $latex \mathbb{Q}_p$ as the "Laurent series" of the form $latex \sum_{i=-m}^\infty a_ip^i\in\mathbb{Q}_p$ where $latex a_i\in\{0,...,p-1\}$ and $latex m\in\mathbb{Z}^+$ (see Theorem 3.4 here).

There is very nice theory that we wont expose here, such that taking the curve $latex C/\mathbb{Q}_p$ can tell us information of $latex C/\mathbb{Q}$. You can imagine this field $latex \mathbb{Q}_p$ as being something that approximates a solution point of $latex C$ over $latex \mathbb{Q}$, since its elements are power series in p, and we will use this "imagination" which is very informal, to indeed, pursue this objective. For the algebrist it is easier to define these numbers. We start with the inverse limit of the rings $latex \mathbb{Z}/p^n\mathbb{Z}$ which we denote by $latex \mathbb{Z}_p$. A p-adic integer $latex m$ is then a sequence $latex (a_n)_{n\geq 1}$ such that $latex a_n$ is in $latex \mathbb{Z}/p^n\mathbb{Z}$. If $latex n \leq m$, then $latex a_n \equiv a_m (\bmod p^n)$. It is easy to check that the ring of fractions of $latex \mathbb{Z}_p$ is in fact what we described before, the field $latex \mathbb{Q}_p$ which has characteristic 0.

Affinoid algebras and spaces.

Consider the field $latex \mathbb{Q}_p$ for some prime $latex p$. Consider the ring of formal power series in the variables $latex x_1,...,x_n$ and the subring $latex \tau_n\subset \mathbb{Q}_p[[x_1,...,x_n]]$ of all the strictly convergent series, that is, if we denote the power series in multi index notation as $latex \sigma:=\sum_{I} c_I x^I$,  then $latex \sigma\in\tau_n$ if and only if $latex |c_I|_p=0$ as $latex I\to \infty$. This give us a feature of $latex \tau_n$, first,... it is a ring and an algebra (prove it). Moreover, every element  $latex \sigma\in\tau_n$ has a norm, $latex ||\sigma||=\sup\{c_I\}_{I}$, making $latex \tau_n$ a Banach $latex \mathbb{Q}_p$-algebra of countable type. So here, we reduced the horribly big ring of formal power series to something which cohomologically will be more "manageable", in fact $latex \tau_n$ is a unique factorization domain with Krull dimension $latex n$.

An affinoid algebra is any $latex A:=\tau_n/(J)$ where $latex J$ is a closed ideal of $latex A$.  An affinoid space $latex X$ is the maximal spectrum of this ring, that is $latex X=\text{Specm}(A)$ (set of maximal ideals of $latex A$, which can be regarded as points).

Imagine that to study $latex C/\mathbb{Q}$ you will consider the space of points $latex X_C:=\text{Specm}\;\tau_2/(C(x,y))$, which can be given many topologies, like the Zariski (very coarse unfortunately), Grothendieck or étale Topology and others.


Tomorrow (I hope) I will continue with perfectoid spaces and I will try to develop an example of how to work with these spaces which allow us to work with "mixed characteristic", namely, characteristic 0 and p.


The geometry of curves over $latex K:= \mathbb{Q}_p$ is very interesting, here, as a matter of morbosity, I leave you of the Berkovich projective line using these ideas for $latex \mathbb{P}^1(K)$.

Eduardo Ruiz Duarte
Twitter: @torandom
PGP: FEE7 F2A0