### number theory

The purpose of this post is to describe an application of the general intersection theory machinery (for curves on surfaces) developed in the previous posts: the Weil bound on points on a curve over a finite field.

1. Statement of the Weil bound

Let ${C}$ be a smooth, projective, geometrically irreducible curve over ${\mathbb{F}_q}$ of genus $g$. Then the Weil bound states that:

$\displaystyle |C(\mathbb{F}_q) - q - 1 | \leq 2 g \sqrt{q}.$

Weil’s proof of this bound is based on intersection theory on the surface ${C \times C}$. More precisely, let

$\displaystyle \overline{C} = C \times_{\mathbb{F}_q} \overline{\mathbb{F}_q},$

so that ${\overline{C}}$ is a smooth, connected, projective curve. It comes with a Frobenius map

$\displaystyle F: \overline{C} \rightarrow \overline{C}$

of ${\overline{\mathbb{F}_q}}$-varieties: in projective coordinates the Frobenius runs

$\displaystyle [x_0: \dots : x_n] \mapsto [x_0^q: \dots : x_n^q].$

In particular, the map has degree ${q}$. One has

$\displaystyle C( \mathbb{F}_q) = \mathrm{Fix}(F, \overline{C}(\overline{\mathbb{F}}_q))$

representing the ${\mathbb{F}_q}$-valued points of ${C}$ as the fixed points of the Frobenius (Galois) action on the ${\overline{\mathbb{F}_q}}$-valued points. So the strategy is to count fixed points, using intersection theory.

Using the (later) theory of ${l}$-adic cohomology, one represents the number of fixed points of the Frobenius as the Lefschetz number of ${F}$: the action of ${F}$ on ${H^0}$ and ${H^2}$ give the terms ${q+1}$. The fact that (remaining) action of ${F}$ on the ${2g}$-dimensional vector space ${H^1}$ can be bounded is one of the Weil conjectures, proved by Deligne for general varieties: here it states that ${F}$ has eigenvalues which are algebraic integers all of whose conjugates have absolute value ${\sqrt{q}}$. (more…)

This is the third in the series of posts intended to work through the proof of Lazard’s theorem, that the Lazard ring classifying the universal formal group law is actually a polynomial ring on a countable set of generators. In the first post, we reduced the result to an elementary but tricky “symmetric 2-cocycle lemma.” In the previous post, we proved most of the symmetric 2-cocycle lemma, except in characteristic zero. The case of characteristic zero is not harder than the cases we handled (it’s easier), but in this post we’ll complete the proof of that case by exhibiting a very direct construction of logarithms in characteristic zero. Next, I’ll describe an application in Lazard’s original paper, on “approximate” formal group laws.

After this, I’m going to try to move back to topology, and describe the proof of Quillen’s theorem on the formal group law of complex cobordism. The purely algebraic calculations of the past couple of posts will be necessary, though.

1. Formal group laws in characteristic zero

The last step missing in the proof of Lazard’s theorem was the claim that the map

$\displaystyle L \rightarrow \mathbb{Z}[b_1, b_2, \dots ]$

classifying the formal group law obtained from the additive one by “change-of-coordinates” by the exponential series ${\exp(x) = \sum b_i x^{i+1}}$ is an isomorphism mod torsion. In other words, we have an isomorphism

$\displaystyle L \otimes \mathbb{Q} \simeq \mathbb{Z}[b_1, b_2, \dots ] \otimes \mathbb{Q}.$

In fact, we didn’t really need this: we could have proved the homological 2-cocycle lemma in all cases, instead of just the finite field case, and it would have been easier. But I’d like to emphasize that the result is really something elementary here. In fact, what it is saying is that to give a formal group law over a ${\mathbb{Q}}$-algebra is equivalent to giving a choice of series ${\sum b_i x^{i+1}}$.

Definition 1 An exponential for a formal group law ${f(x,y) }$ is a power series ${\exp(x) = x + b_1 x^2 + \dots}$ such that

$\displaystyle f(x,y) = \exp( \exp^{-1}(x) + \exp^{-1}(y)).$

The inverse power series ${\exp^{-1}(x)}$ is called the logarithm.

That is, a logarithm is an isomorphism of ${f}$ with the additive formal group law.

So another way of phrasing this result is that:

Proposition 2 A formal group law over a ${\mathbb{Q}}$-algebra has a unique logarithm (i.e., is uniquely isomorphic to the additive one). (more…)

We are in the middle of proving an important result of Lazard:

Theorem 1 The Lazard ring ${L}$ over which the universal formal group law is defined is a polynomial ring in variables ${x_1, x_2, \dots, }$ of degree ${2i}$.

The fact that the Lazard ring is polynomial implies a number of results which are not a priori obvious: for instance, it shows that given a surjection of rings ${ A \twoheadrightarrow B}$, then any formal group law on ${B}$ can be lifted to one over ${A}$.

We began the proof of Lazard’s theorem last time: we produced a map

$\displaystyle L \rightarrow \mathbb{Z}[b_1, b_2, \dots ], \quad \deg b_i = 2i,$

classifying the formal group law obtained from the additive one ${x+y}$ by the “change of coordinates” ${ \exp(x) = \sum b_i x^{i+1}}$. We claimed that the map on indecomposables was injective, and that, in fact the image in the indecomposables of ${\mathbb{Z}[b_1, b_2, \dots ]}$ could be determined completely. I won’t get into the details of this (it was all in the previous post), because the purpose of this post is to prove a result to which we reduced last time.

Let ${A}$ be an abelian group. A symmetric 2-cocycle is a “polynomial” ${P(x,y) \in A[x, y] = A \otimes_{\mathbb{Z}} \mathbb{Z}[x, y]}$ with the properties:

$\displaystyle P(x, y) = P(y,x)$

and

$\displaystyle P(x, y+z) + P(y, z) = P(x,y) + P(x+y, z).$

These symmetric 2-cocycles come up when one tries to classify formal group laws over the ring ${\mathbb{Z} \oplus A}$, as we saw last time: in fact, we can think of them as “deformations” of the additive formal group law.

The main lemma which we stated last time was the following:

Theorem 2 (Symmetric 2-cocycle lemma) A homogeneous symmetric 2-cocycle of degree ${n}$ is a multiple of ${\frac{1}{d} ( ( x+y)^n - x^n - y^n )}$ where ${d =1}$ if ${n}$ is not a power of a prime, and ${d = p}$ if ${n = p^k}$.

For a direct combinatorial proof of this theorem, see Lurie’s notes. I want to describe a longer homological proof, which is apparently due to Mike Hopkins and which appears in the COCTALOS notes. The strategy is to interpret these symmetric 2-cocycles as actual cocycles in a cobar complex computing an ${\mathrm{Ext}}$ group. Then, the strategy is to compute this ${\mathrm{Ext}}$ group independently.

This argument is somewhat longer than the combinatorial one, but it has the benefit (for me) of engaging with some homological algebra (which I need to learn more about), as well as potentially generalizing in other directions.  (more…)

After describing the computation of ${\pi_* MU}$, I’d now like to handle the remaining half of the machinery that goes into Quillen’s theorem: the structure of the universal formal group law.

Let ${R}$ be a (commutative) ring. Recall that a formal group law (commutative and one-dimensional) is a power series ${f(x,y) \in R[[x,y]]}$ such that

1. ${f(x,y) = f(y,x)}$.
2. ${f(x, f(y,z)) = f(f(x,y), z)}$.
3. ${f(x,0) = f(0,x) = x}$.

It is automatic from this by a successive approximation argument that there exists an inverse power series ${i(x) \in R[[x]]}$ such that ${f(x, i(x)) = 0}$.

In particular, ${f}$ has the property that for any ${R}$-algebra ${S}$, the nilpotent elements of ${S}$ become an abelian group with addition given by ${f}$.

A key observation is that, given ${R}$, to specify a formal group law amounts to specifying a countable collection of elements ${c_{i,j}}$ to define the power series ${f(x,y) = \sum c_{i,j} x^i y^j}$. These ${c_{i,j}}$ are required to satisfy various polynomial constraints to ensure that the formal group identities hold. Consequently:

Theorem 1 There exists a universal ring ${L}$ together with a formal group law ${f_{univ}(x,y)}$ on ${L}$, such that any FGL ${f}$ on another ring ${R}$ determines a unique map ${L \rightarrow R}$ carrying ${f_{univ} \mapsto f}$. (more…)

Apologies for the lack of posts lately; it’s been a busy semester. This post is essentially my notes for a talk I gave in my analytic number theory class.

Our goal is to obtain bounds on the distribution of prime numbers, that is, on functions of the form ${\pi(x)}$. The closely related function

$\displaystyle \psi(x) = \sum_{n \leq x} \Lambda(n)$

turns out to be amenable to study by analytic means; here ${\Lambda(n)}$ is the von Mangolt function,

$\displaystyle \Lambda(n) = \begin{cases} \log p & \text{if } n = p^m, p \ \text{prime} \\ 0 & \text{otherwise} \end{cases}.$

Bounds on ${\psi(x)}$ will imply corresponding bounds on ${\pi(x)}$ by fairly straightforward arguments. For instance, the prime number theorem is equivalent to ${\psi(x) = x + o(x)}$.

The function ${\psi(x)}$ is naturally connected to the ${\zeta}$-function in view of the formula

$\displaystyle - \frac{\zeta'(s)}{\zeta(s)} = \sum_{n=1}^\infty \Lambda(n) n^{-s}.$

In other words, ${- \frac{\zeta'}{\zeta}}$ is the Dirichlet series associated to the function ${\Lambda}$. Using the theory of Mellin inversion, we can recover partial sums ${\psi(x) = \sum_{n \leq x} \Lambda(x)}$ by integration of ${-\frac{\zeta'}{\zeta}}$ along a vertical line. That is, we have

$\displaystyle \psi(x) = \frac{1}{2\pi i} \int_{\sigma - i \infty}^{\sigma + i \infty} -\frac{\zeta'(s)}{\zeta(s)} \frac{x^s}{s} ds ,$

at least for ${\sigma > 1}$, in which case the integral converges. Under hypotheses on the poles of ${-\frac{\zeta'}{\zeta}}$ (equivalently, on the zeros of ${\zeta}$), we can shift the contour appropriately, and estimate the integral to derive the prime number theorem. (more…)

I’d like to explain what I think is a pretty piece of mathematics.

Deligne’s proof of the Weil conjectures, and his strengthenings of them, in his epic paper “La conjecture de Weil II” have had, needless to say, many applications—far more than the initial paper where he proved the last of the Weil conjectures. One of the applications is to bounding exponential sums. I’d like to begin sketching the idea today. I learned this from the article “Sommes trigonometriques” in SGA 4.5, which is very fun to read.

1. The trace formula

Let’s say you have some variety ${X_0}$ over a finite field ${\kappa}$, say separated. Suppose you have a ${l}$-adic sheaf ${\mathcal{F}_0}$ on ${X_0}$. I’ll denote by “dropping the zero” the base change to ${\overline{\kappa}}$, so ${\mathcal{F}}$ denotes the pull-back to ${X = X_0 \times_{\kappa} \overline{\kappa}}$.

For each point ${x \in X_0}$, we can take the “geometric stalk” ${\mathcal{F}_{\overline{x}}}$ (or ${\mathcal{F}_{0 \overline{x}}}$) which is given by finding a map ${\mathrm{Spec} \overline{\kappa} \rightarrow X_0}$ hitting ${x}$, and pulling ${\mathcal{F}_0}$ back to it. This “stalk” is automatically equivariant with respect to the Galois group ${\mathrm{Gal}(\overline{\kappa}/k(x))}$ for ${k(x)}$ the residue field, and as a result we can compute the trace of the geometric Frobenius ${F_x}$ of ${k(x) \hookrightarrow \overline{\kappa}}$—that’s the inverse of the usual Frobenius—and take its trace on ${\mathcal{F}_{\overline{x}}}$.

So the intuition here is that we’re taking local data of the sheaf ${\mathcal{F}_0}$: just the trace of its Frobenius at each point. One interesting interpretation of this procedure was discussed on this MO question: namely, the process is analogous to “integration over a contour.” Here the contour is ${\mathrm{Spec} k(x)}$, which has cohomological dimension one and is thus the étale version of a curve.

Now, the trace formula says that the local data of these traces all pieces into a simple piece of global data, which is the compactly supported cohomology. (more…)

The topic of this post is a curious functor, discovered by Deligne, on the category of sheaves over the affine line, which is a “sheafification” of the Fourier transform for functions.

Recall that the classical Fourier transform is an almost-involution of the Hilbert space ${L^2(\mathbb{R})}$. We shall now discuss the Fourier-Deligne transform, which is an almost-involution of the bounded derived category of ${l}$-adic sheaves, ${\mathbf{D}^b_c(\mathbb{A}^1_{\kappa}, \overline{\mathbb{Q}_l})}$. The Fourier transform is defined by multiplying a function with a character (which depends on a parameter) and integrating. Analogously, the Fourier-Deligne transform will twist an element of ${\mathbf{D}^b_c(\mathbb{A}^1_{\kappa}, \overline{\mathbb{Q}_l})}$ by a character depending on a parameter, and then take the cohomology.

More precisely, consider the following: let $G$ be a LCA group, $G^*$ its dual. We have a canonical character on $\phi: G \times G^* \to \mathbb{C}^*$ given by evaluation. To construct the Fourier transform $L^2(G) \to L^2(G^*)$, we start with a function $f: G \to \mathbb{C}$. We pull back to $G \times G^*$, multiply by the evaluation character $\phi$ defined above, and integrate along fibers to give a function on $G^*$.

Everything we’ve done here has a sheaf-theoretic analog, however: pulling back a function corresponds to the functorial pull-back of sheaves, multiplication by a character corresponds to tensoring with a suitable line bundle, and integration along fibers corresponds to the lower shriek push-forward. Much of the classical formalism goes over to the sheaf-theoretic case. One can prove an “inversion formula” analogous to the Fourier inversion formula (with a Tate twist).

Why should we care? Well, Laumon interpreted the Fourier transform as a suitable “deformation” of the cohomology of a suitable sheaf on the affine line, and used it to give a simplified proof of the main results of Weil II, without using scary things like vanishing cycles and Picard-Lefschetz theory. The Fourier transform also behaves very well with respect to perverse sheaves: it is an auto-equivalence of the category of perverse sheaves, because of the careful way in which it is calibrated. Its careful use can be used to simplify some of the arguments in BBD that also rely on other scary things.

I’d like to take a quick (one-post) break from simplicial methods. This summer, I will be studying étale cohomology and the proofs of the Weil conjectures through the HCRP program. I have currently been going through the basic computations in étale cohomology, and, to help myself understand one point better, would like to mention a very pretty and elementary argument I recently learned from Johan de Jong’s course notes on the subject (which are a chapter in the stacks project).

1. Motivation via étale cohomology

When doing the basic computations of the étale cohomology of curves, one of the important steps is the computation of the sheaf ${\mathcal{O}_X^*}$ (that is, the multiplicative group of units), and in doing this one needs to know the cohomology of the generic point. That is, one needs to compute

$\displaystyle H^*(X_{et}, \mathcal{O}_X^*)$

where ${X = \mathrm{Spec} K}$ for ${K}$ a field of transcendence degree one over the algebraically closed ground field, and the “et” subscript means étale cohomology. Now, ultimately, whenever you have the étale cohomology of a field, it turns out to be the same as Galois cohomology. In other words, if ${X = \mathrm{Spec} K}$, then the small étale site of ${X}$ is equivalent to the site of continuous ${G = \mathrm{Gal}(K^{sep}/K)}$-sets, and consequently the category of abelian sheaves on this site turns out to be equivalent to the category of continuous ${G}$-modules. Taking the étale cohomology of this sheaf then turns out to be the same as taking the group cohomology of the associated ${G}$-module. So, if you’re interested in étale cohomology, then you’re interested in Galois cohomology. In particular, you are interested in things like group cohomologies of the form

$\displaystyle H^2(\mathrm{Gal}(K^{sep}/K), (K^{sep})^*).$ (more…)

I asked a question on what Tate’s thesis was really about on math.SE, and Matthew Emerton has posted a very thoughtful and detailed answer. You should go read it. Actually, I recommend looking at all his answers on the website, which are easily some of the best answers there.

(Apparently I am not the first person to notice this.)

Last time, we showed that prime numbers admit succinct certificates. Given a number containing ${k}$ bits, one could produce a “proof” that the number is prime in polynomial in ${k}$ space. In other words, the language ${PRIMES}$ containing all primes (represented in the binary expansion, say) lies in the complexity class ${\mathbf{NP}}$.

In practice, though, being in ${\mathbf{NP}}$ does not give a good way of solving the problem by itself; for that, we want the problem to be in ${\mathbf{P}}$. Perhaps, though, that is too strict. One relaxation is to consider classes of problems that can be solved by randomized algorithms that run in polynomial time—randomized meaning that the right output comes out with high probability, that is. This is the class ${\mathbf{BPP}}$: by definition, something belongs to the class ${\mathbf{BPP}}$ if there is a polynomial-time nondeterministic Turing machine such that on any computation, at least ${2/3}$ of the computation tree leads to the right answer. So the idea of ${\mathbf{BPP}}$ is that, to decide whether a string ${x}$ belongs to our language ${\mathcal{L}}$, we generate a bunch of random bits and then run a deterministic algorithm on the string ${x}$ together with the random bits. Most likely, if the random bits were not too bad, we will get the right answer.

In practice, people tend to believe that ${\mathbf{BPP} = \mathbf{P}}$ while ${\mathbf{P} \neq \mathbf{NP}}$. But, in any case, probabilistic algorithms may be more efficient than deterministic ones, so even if the former equality holds they are interesting.

Today, I want to describe a probabilistic algorithm for testing whether a number ${N}$ is prime. As before, the algorithm will rely on a little elementary number theory. (more…)

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