A question on primes

I recently encountered a result which seems to be analogous to the following result of Dirichlet, which I wrote in a few common forms to be more suggestive.

Theorem (Dirichlet on Arithmetic Progressions). Let a​ and b​ be relatively prime positive integers, then there are infinitely many primes p $\equiv$ a mod b​ (i.e., the progression $a + b \mathbb{N}$​ contains infinitely many primes).

Corollary 1. There are infinitely many primes such that a​ is not a square and quadratic residue (mod p) and infinitely many primes such that a​ is a quadratic non-residue (mod p). In nicer terms, given n, there are infinitely many primes such that the Legendre symbol $(\frac{-n}{p})$ takes on each value for infinitely many p (for p ≠ 2​).

Corollary 2. Something like: Given a prime $p$​, there’s a fifty-fifty chance that the image of the prime splits (through an order two extension) [remains inert/ramify when passed to the extension]. This is saying half are quadratic non-residues and half are quadratic residues, respectively, that is, half are squares mod p, and the other half aren’t. For example, $2 \mod 5$ is not a square, but $1 \mod 5$  and $3 \mod 5$ are squares.

A result that tastes like Corollary 2 is the existence of infinitely many super singular primes for every elliptic curve over $\mathbb{Q}$: Continue reading A question on primes

Notes on One-Parameter Deformations of Cohomology Theories

Thanks to Dr. Lubin for graciously helping me derive and understand one-parameter families of formal group laws, thanks to Eric Peterson for introducing me to Morava’s Forms of K-theory, thanks to Agnes Beaudry for pointing out that there was a neater way to check for Landweber-exactness.

Today, I want to discuss the opposite procedures of deformations and contractions of complex-orientable cohomology theories.

Really, all I want is to illustrate the fact that, with the appropriate combinations of both procedures, we obtain new cohomology theories. So, we’re going to  examine the construction of one-parameter families of cohomology theories via one-parameter deformations of formal group laws — of particular interest is the case where a continuous deformation causes an increase of chromatic height.

Motivation and Story Leading To This Construction

I was playing with the concept of group contraction, and thinking about the construction of elliptic cohomology theories.

I accidentally constructed a model of Morava E-theory of height 2 at the prime 3. (We didn’t realize that it was $E_{(2)}(3)$ at first, we just thought it was some weird cohomology theory.)

I found this kind of enlightening and so I want to show you how I came across it.

Morava constructed a family of elliptic cohomology theories by deforming K-theory (well, by deforming the multiplicative group associated to K-theory). His construction can be viewed as a recipe:

e.g., $\mathbb{C}^\times$
e.g., $y^3 = x^3 – x$ over $\mathbb{F}_3$
2. deform that point (create a family of algebraic groups indexed by one parameter)
e.g., $\mathbb{C}^\times/q^{\mathbb{Z}}$ where $q:= e^{2\pi i}$, we vary the norm of $0 \leq |q| < 1$.
e.g., $y^3 = x^3 + tx^2 – x$ over $\mathbb{F}_3[[t]]$
3. look at the formal group laws associated to your family, this is still indexed by one parameter (in fact, they can be viewed as ONE formal group law, if you keep the parameter formal)
4. either apply the Landweber exact functor theorem to the whole family stalkwise (specializing the parameter), or apply the Landweber exact functor theorem to the ONE formal group law (keeping the parameter formal).

Let me say this again, because when I explain this to people they like me to say it twice.

Morava’s deformation method is a recipe which consists of 4 steps:

1. construct a continuous family of smooth algebraic groups indexed by q
2. construct a continuous family of formal group laws indexed by q
3. construct a family indexed (indexed by q) of contra-functors from Top \to AbGrp (“potential” cohomology theories — we don’t know if they are exact yet)
4. prove that each member of this family of contravariant functors is exact (or treat the family as one functor, keeping the variable formal, and prove that this functor is exact)

Spf $E^*[[x]]$: Your walk through a flower garden

Inspired by the extraordinary expository style of Dr. Kazuya Kato, I’ve started reading parts of a (translated) Japanese children’s book when I’m stuck on a tough paper or concept — revisiting the concept with such a dreamlike world in mind usually unfolds an illustrative perspective. A misty world which begs to be put into firm ground via prolonged formal and concrete afterthought.

He embraces that teaching can be poetic and tantalizing, providing not a definition but a deep and creative hint that causes an exploratory shift in perspective, allowing you to walk down the path to the conclusion yourself. I wanted to try to exposit with this philosophy: confusion is expected and encouraged as impetus for reaching understanding. With that in mind, step into your flower garden.

Planted in a line of earth ($\text{Spec }R$)
there are flowers, $C$, whose heads are smooth projective genus 1 curves
with stems that can retract into the ground,
s.t. the flower meets the earth at one point (a marked point).

Bordism with singularities construction of elliptic homology

This post assumes familiarity with the Landweber exact functor theorem, elliptic genera, and bordism theories.

An ongoing desire of mine is to geometrically approach elliptic spectra.

Note that I’m not talking about geometric cocycles for tmf. I’m talking about the vague goal of understanding elliptic spectra in their own right using “geometric” techniques (properties of the object that are invariant under a chosen collection of transformations of that object).

There is a presentation of an elliptic homology theory as a bordism theory with singularities outlined by Landweber in Elliptic Cohomology and Modular Forms (based on this paper which presents $H_*(-;\mathbb{Z})$ as a bordism theory with singularities).

This seemed like the beginning of the answer to my desire, but I now think that this construction is basically a less intuitive version of $MSO_*(-) \otimes_{MSO_*} R$, thinking about “tensoring out” classes in $MSO$ as coning them off. I’ll explain what I mean by “coning them off” in a bit, first let me outline the construction:

Outline:

1. Start with an elliptic genus $\pi_*(MSO) \xrightarrow{\phi} R$
2. Mod out the ring spectrum $MSO$ by $ker(\phi)$ to get a spectrum “$MSO/ker(\phi)$” whose homotopy groups are $R$
i.e., construct $F: \pi_*(MSO/ker(\phi)) \to R$ s.t. $ker(F) = 0$
3. Check that the spectrum $MSO/ker(\phi)$ is a ring spectrum

Let’s go through it! Continue reading Bordism with singularities construction of elliptic homology