Notes on Arithmetic Statistics

Interesting question in arithmetic statistics: for \(G \in {\mathsf{Grp}}\) finite, how many Galois extensions are there \(K/{\mathbb{Q}}\) with \(G = { \mathsf{Gal}} (K/{\mathbb{Q}})\) and \(\Delta \leq N\) (discriminant for some fixed \(N\)?

Example
 For \(G={\mathbb{Z}}/2\), it is \(O(N)\).
 For \(G={\mathbb{Z}}/3\), it is \(O(\sqrt N )\).

One can ask a similar question about \({ \operatorname{Cl}} (K)\) for \(G\in {\mathsf{Ab}}\), or replacing \({\mathbb{Q}}\) with Unsorted/function field \({\mathbb{F}}_q(t)\) for \(q=p^n\), and ask questions about frequency of primes ramified primes, split primes, or remaining inert primes.

Cool fact: there is an equivalence of categories between finitelygenerated extensions \(K/k\) with \(\operatorname{trdeg}(K/k) = 1\) and regular projective curves \(C_{/k}\).
 The (reverse) functor is the one sending a curve \(C\) to its function field \(k(C)\).

Hurwitz spaces come up here!

\({\varepsilon}\) is a \(q^i\) Weil number if \({\left\lvert { \iota({\varepsilon}) } \right\rvert} = q^{i/2}\) for any embedding \(\iota: { \mkern 1.5mu\overline{\mkern1.5mu \mathbb{Q} \mkern1.5mu}\mkern 1.5mu }\hookrightarrow{\mathbb{C}}\).
 Examples: eigenvalues of geometric Frobenius acting on \(H^i_c\).

As a general philosophy, one should expect that moduli space problems whose objects have nontrivial automorphisms are representable by Unsorted/stacks MOC, and those without nontrivial automorphisms are representable by scheme.

Weil Conjectures Talks
 Homological Stability Course Notes for Hurwitz spaces :
Old Notes: Erik Schreyer
Some old notes from March 10th, 2020
I talked to Erik Schreyer today about some of the research he did with his advisor Jason Cantarella, including his dissertation work (which he spoke about in the Geometry seminar last week) and a few other papers.
His dissertation work involved a cool way to represent arbitrary planar curves by piecewise circular arcs:
From what I understand, this involves fixing a curve (blue), choosing a collection of circles \(C_1, \cdots C_n\) (black) such that each \(C_i\) intersects \(C_{i+1}\) in at least one distinguished point \(p_i\) (pink). The curve traced out by following an arc on \(C_i\) and switching to circle \(C_{i+1}\) at \(p_i\) is intended to yield a good approximation to the original curve, with certain regularity conditions at the \(p_i\) (such as the first derivatives along both arcs agreeing at the point).
Erik’s work actually seems to go a bit farther – he has an algorithm (a curveclosing operator) that actually takes an open curve and produces a closed curve that is nearby in the \(C_1\) norm. He uses this to construct piecewise circular approximations that consist of circles of equal radii, along with some control over the \(C^1\) distance between the original curve and the approximation.
We also talked a bit about another problem Jason was working on, discussed in the following papers:

The symplectic geometry of closed equilateral random walks in 3space (Cantarella, Shonkwiler 2013)

A Fast Direct Sampling Algorithm for Equilateral Closed Polygons (Cantarella et al 2015)
Dirichlet’s Theorem
Dirichlet’s Theorem: An arithmetic progress with \((a, p) = 1\) contains infinitely many primes. As a corollary, one can always find a prime \(q\) that generates \({\mathbb{Z}}_p^{\times}\) for any prime \(p\).
A SES isomorphic to a direct sum that does not split
Not every sequence of the form \(0\to A \to A \oplus C \to C \to 0\) splits; take \begin{align*} 0 \to {\mathbb{Z}}\to {\mathbb{Z}}\oplus \bigoplus_{\mathbb{N}}{\mathbb{Z}}/(2) \to \bigoplus_{\mathbb{N}}{\mathbb{Z}}/(2) \to 0 \end{align*} where the first map is multiplication by 2, the second is the quotient map and a rightshift. This can’t split because \((1, 0, \cdots)\) has order 2 in the RHS but pulls back to \((1, 0) \oplus (2{\mathbb{Z}}\oplus 0)\) which has no element of order 2.
Cogroups
See cogroup.