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.

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