Geometry For Prime Addicts

Texed notes of my reading course last quarter advised by Yifeng Liu, with fellow participants Grisha Kondyrev and Jora Belousov. Our goal this reading course was understand some of Scholze’s recent work with perfectoid space techniques, in particular the proof of the monodromy weight conjecture.

Attached are only the notes I kept of my lectures, sometimes just outlines of what to talk about. I have had a few requests for these notes, and they are incomplete but I think they might still be helpful for others starting out in p-adic geometry.

During the reading course, I got the increasing feeling that we were just studying $G_Q$ as fast as our little legs could take us — our little legs being our knowledge of varieties (over various non-archimedian fields like $Q_p$ and $F_p((t))$. So, I finished with a talk on the Grothendieck-Teichmüller group — another approach to $G_Q$.

padicgeometry

The Height of a Formal Group Law in terms of the Symmetry of the Underlying CM Abelian Variety

This proof was made possible by a couple helpful and fabulous conversations with Yifeng Liu. All errors are mine and mine alone.

This is toward my understanding of the phrase “Why is height so important as an invariant? Because the height of a formal group law comes from the symmetry of the underlying variety.”

In short — high amount of symmetry in the underlying abelian variety implies a high height of its formal group law (the converse is NOT true, if this was true, Elkies’s supersingularity theorem would be false).

One method of getting lower dimensional formal group laws from abelian varieties of higher dimension is via using the theory of complex multiplication — splitting the abelian variety by splitting the prime (as I exposited in my paper here).

I show that for abelian varieties with CM, the height of the formal group law pieces are expressible as a formula in terms of the degree of some field extensions of $Q_p$ (one corresponding to each prime living over $p$) and the dimension of the rational endomorphism ring of the variety as a $Q$-vector space.

Notation:

  • $F$ is a formal group associated to a CM abelian variety $A$.
  • $\pi = \pi_A$, the geometric Frobenius of $A$
  • $L := Q(\pi)$ with $[L : \mathbb{Q}] = e$
  • $D := End^0(A)$ with $[D: L] = r^2$
  • $dim(A) = g = er/2$.
  • $L$ is the center of $D$
  • Assume that $\mathcal{O}_L \subset End(A)$ (which we may after replacing $A$ by an isogenous abelian variety).
  • Consider the set $\Sigma^{(p)}_L$ of discrete valuations of $L$ dividing the rational prime number $p$.


    Theorem:

  1. the decomposition $$D \otimes \mathbb{Q}_p = \prod_{w \in \Sigma^{(p)}_L} D_w$$ and $$\mathcal{O}_L = \prod \mathcal{O}_{L_w}$$
    gives a decomposition $$F = \prod_w F_w$$
  2. The height of $F_w$ equals $[L_w : \mathbb{Q}_p]\cdot$ r.

Continue reading The Height of a Formal Group Law in terms of the Symmetry of the Underlying CM Abelian Variety

Complex Analysis: Poles, Residues, and Child’s Drawings

Thanks to Laurens Gunnarsen for his superb pedagogy and for this amazing explanation on the incredible depth of connections springing from the Sperner lemma. All errors are mine not his. This started with a chain of events, sitting in on number theory seminars and encountering Abel’s differentials of the first and second kind, interest in the dessin, and led up to asking Laurens:

How do I understand poles and residues?

By understanding Riemann-Roch. But, first of all, you should know the zeroes and the poles of an analytic function, together with the residues of the function at each, pretty much tell you all you need to know about the function.

This is a little bit like saying that if you know the zeroes of a rational function — which is to say, the zeros of the polynomial that is its numerator — and the poles of a rational function — which is to say, the zeroes of the polynomial that is its denominator — then you basically know the rational function.  Continue reading Complex Analysis: Poles, Residues, and Child’s Drawings

Dirichlet Series and Homology Theories

Euler was a swashbuckler. He considered the following series, nevermind that it sums to infinity, let’s see what we can do with it!

I highly recommend that you follow along with a pen and paper to convince yourself that everything cancels as I say it will, if you’re a bit too lazy for that, or already familiar with the L-function, feel free to scroll ahead.

$x = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} +  … $

Let’s multiply by 1/2 and see what happens:

$\frac{1}{2}x = \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \frac{1}{10} +  …$

Then subtract this from the original for funsies.

$x -\frac{1}{2}x$

$=\frac{1}{1}$ + 1/2 $ + \frac{1}{3}$ + 1/4 $+ \frac{1}{5} +  …$ Continue reading Dirichlet Series and Homology Theories

p-adic numbers are Taylor expansions

Look at $P(x) = x-p$.  Fixing $p$ fixes the P(x).

$P(x)$ is a prime in the ring $\mathbb{C}[x]$.  We can build any polynomial out of these irreducible components.

Fix a prime, and expand your function:

$$f(x) = f(p) + f'(p)(x-p) + \frac{f”(p)}{2!}(x-p)^2 + …$$

If first few functions in the taylor series of two functions agree, than the functions are close, i.e., two functions are close if their difference is divisible by high powers of the monomial.

Let’s look at the integers, replacing primes of polynomials with primes of integers, fix a prime, and expand your function:

$$n = n(p) + n'(p)p + \frac{n”(p)}{2!}p^2 + …$$

Where $n(p), n'(p), \frac{n”(p)}{2!}, …$ are formal symbols, corresponding to the last coefficient, second to last coefficient, third to last, and so on and so forth.

Two integers are close if their expansions are divisible by high powers of p. The ring of all such integer expansions has only one maximal ideal, $p$.

This is how Hensel originally came upon the definition of the p-adics!