Questions tagged [formal-groups]
The formal-groups tag has no usage guidance.
9
votes
1answer
338 views
Nontrivial p-divisible groups over $\mathbb Z$ for general prime $p$
In Tate's famous paper about $p$-divisible groups, for a prime number $p$ he asks whether there exists a $p$-divisible group $G$ over $\mathbb Z$ such that $G$ is not a direct sum of $\mu_{p^\infty}$ ...
2
votes
0answers
75 views
Geometric intuition and computation for Cartier theory
I am learning Cartier theory of commutative formal groups by the book of Zink. It is a powerful tool but I don't understand it's motivation. The Cartier module of a formal group $G$ over a $\mathbb{Z}...
39
votes
3answers
1k views
Characterizing positivity of formal group laws
The formal group law associated with a generating function $f(x) = x + \sum_{n=2}^\infty a_n \frac{x^n}{n!}$ is $$f(f^{-1}(x) + f^{-1}(y)).$$ In my thesis, I found a large number of examples of ...
6
votes
0answers
88 views
Formal Group Laws in a lined topos
I am aware of the following: in the context of synthetic differential geometry (SDG) one obtains a Lie algebra by exponentiating a microlinear group by a standard infinitesimal object and taking the ...
8
votes
1answer
264 views
What is the essential image of $AbVar$ in $p-div$?
Given an abelian variety $A$ over a base scheme $\text{Spec } \mathcal{O}_{K_p}$, we define the functor $P$ as taking $A \mapsto \text{colim}_n A[p^n]$, its associated $p$-divisible group. What is the ...
8
votes
1answer
222 views
(Pre)orientation vs. formal completion
Let $\mathbb G$ be an abelian vatiety over an $\mathbb E_\infty$-ring $A$. That is to say, it consists of an abelian group object in the $\infty$-category of relative schemes $\mathbb G\to \...
4
votes
3answers
195 views
Morphisms of formal group laws $\,F_a \rightarrow F_m\,$ and $\,F_m\to F_m$
While studying cohomology theories on the stable homotopy setting, I have come up with the following basic question:
Consider the additive formal group law, $F_a$, and the multiplicative formal group ...
8
votes
1answer
446 views
deformation theory in positive characteristic
The idea "Formal deformation theory in characteristic zero is controlled by a differential graded Lie algebra (dgla)" goes back to Goldman-Millson, Deligne, Drinfeld among others; see Lurie's ICM talk....
3
votes
0answers
207 views
Rational cohomology of formal multiplicative group
Let $\hat{\mathbb G}$ be a formal group over a field $k$, and let $V$ be a finite dimensional algebraic representation of $\hat{\mathbb G}$ (meaning we have fixed a homomorphism of algebraic groups $\...
12
votes
0answers
342 views
Geometry underlying a comparison of Dieudonné theories
Maybe these hypotheses aren't necessary, but for me $\mathbb G$ will be a smooth formal group of dimension 1 and finite height over a perfect field $k$.
There are several presentations of the ...
10
votes
2answers
642 views
How to compute the formal group law of a Shimura variety (using its invariant differentials)?
I have a 3 dimensional abelian variety whose formal group law breaks into a formal summand where one of the pieces is one-dimensional.
I am desperately wondering how to compute the $p$-series of ...
4
votes
0answers
171 views
Definition of logarithm for universal vector extension
Let $S$ be a topological $\mathbb{Z}_p$-algebra and $R\to S$ a surjection (where $R$ has $p$ nilpotent) with topologically nilpotent kernel which has a PD structure.
We know that if $G/R$ is a $p$-...
2
votes
0answers
196 views
Reduction “modulo $p$” of $\mathfrak{p}$-torsion points of CM elliptic curves
Let $E/L$ be an elliptic curve defined over a number field $L$. Assume moreover that $E$ has complex multiplication by an imaginary quadratic field $K$. Let $\mathfrak{p}$ be a prime ideal of $K$. ...
2
votes
0answers
175 views
“Algebrazing” canonical subgroups of elliptic curves
I'm puzzled by a part of the construction of the canonical subgroup of a "not too supersingular" elliptic curve. In Katz's paper, one produces a subgroup of the formal group of the elliptic curve but ...
3
votes
1answer
125 views
Functional equations associated with addition theorems for elliptic functions
I'm trying to read the article "Functional equations associated with addition theorems for elliptic functions and two-valued algebraic groups" by Bukhshtaber,V. M. Russian Mathematical Surveys(1990),...
2
votes
0answers
157 views
Flat Quotients of Power Series Rings
I apologize if the question is too elementary. I did not get any response on math stackexchange.
I read the following statement in some algebraic topology notes and I want to know if it is true and, ...
3
votes
0answers
165 views
Dieudonne modules and Cartier-Dieudonne module of a formal group
As far as I understand, there are Dieudonne modules defined through the homomorphisms to Witt covector scheme and Cartier-Dieudonne modules defined by curves. Am I right that the latter sometimes (for ...
7
votes
2answers
463 views
$p$-torsion of an abelian variety of $p$-rank $0$
Let $k$ be an algebraically closed field of characteristic $p > 0$ and let $A$ be an abelian variety over $k$ such that $A[p](k) = 0$, i.e., such that $A$ has $p$-rank $0$. If I am not mistaken, ...
13
votes
2answers
477 views
Infinitesimal deformations of the formal group of $\mathbb{G}_m$
For a commutative ring $R$, consider the formal group $\widehat{\mathbb{G}}_m$ over $R$ that is the completion of $\mathbb{G}_{m, R}$ along its identity section (naively, $\widehat{\mathbb{G}}_m$ is ...
3
votes
0answers
155 views
Efficiently computing (plethysm-like?)substitutions of symmetric functions
This is a rather technical question, it arose in connection of some calculations that I need to have better grasp of the question Formal group law over $\mathbb{F}_p$ and my own older one What is ...
9
votes
0answers
228 views
Unicity of Johnson-Wilson Theories
Let $E$ be a ring spectrum with a $p$-typical complex orientation. Then we call $E$ a form of $BP\langle n\rangle$ or a generalized $BP\langle n\rangle$ if the induced map
$$\mathbb{Z}_{(p)}[v_1,\...
5
votes
0answers
249 views
Legendre polynomials and formal groups
Let $P_n(x)$ be Legendre polynomials:
$$\frac{1}{\sqrt{1-2tx+t^2}}=\sum\limits_{n=0}^{\infty}P_n(x)t^n.$$
Usual arguments from the theory of formal groups allow to
prove that for any $n$
$$P_n(x)=Q_n(...
13
votes
2answers
1k views
Formal group law over $\mathbb{F}_p$
Let $p$ be a prime. For each $n > 0$ there is a unique 1-dimensional commutative formal group law $F$ over $\mathbf{Z}$, $F(X, Y) = X + Y + \dots \in \mathbf{Z}[[X, Y]]$, whose logarithm function ...
1
vote
0answers
130 views
Skew symmetry for the Hilbert symbol
Let $K$ be a local field containing the group $\mu_n$ of $n$th roots of 1 and the $\theta_K:K^*\to G_K^{ab}$ be the reciprocity map. The we know that the Hilbert symbol $$K^*\times K^*\to \mu_n$$ $$(a,...
2
votes
0answers
64 views
Continuity of the solutions of an isogeny in a formal group
Notation for the problem:
Let $E/\mathbb{Q}_P$ be a local field, and $\mu_E$ its maximal ideal. Let $K=E\{\{T\}\}$ be the standard 2-dimensional local field equipped with the Parshin topology and let ...
12
votes
2answers
882 views
Formal group law is a group object in …?
A formal group law over a commutative ring $R$, (by nLab) is a sequence of power srires
$$
f_1,...,f_n\in R[[x_1,...,x_n,y_1,...,y_n]]
$$
such that, using the notation
$$
x=(x_1,...,x_n),y=(y_1,.....
22
votes
3answers
2k views
Is there a better proof of this fact in number theory/formal group theory?
Let $\Phi_n$ be the $n$'th cyclotomic polynomial, and put
\begin{align*}
a_n &= \Phi_n(1) \\
b_n &= \gcd\left(\left(\begin{array}{c} n \\ 1\end{array}\right),\dotsc,\left(\begin{array}{c} n ...
4
votes
0answers
165 views
Continuity of the Hilbert pairing
I would like to know if the Kummer pairing (or the analogue of the Hilbert Symbol) for a one dimensional group defined over the ring of integers of a higher-dimensional local field is continuous (with ...
4
votes
0answers
514 views
formal group laws of Abelian varieties in positive characteristic
Let $G$ be an algebraic group defined over an (algebraically closed) field $k$. Then one can obtain a formal group law by completing the multiplication map $m: G \times G \to G$ at the unit of $G$.
...
10
votes
0answers
314 views
Are automorphisms of abelian varieties detected by the formal group?
Let $A$ be an abelian variety of dimension $g$ over an algebraically closed field $k$.
Assume $k$ has characteristic $p$ and denote by $A(p)$ the $p$-divisible group of dimension $g$ associated with ...
2
votes
0answers
174 views
Formal n-buds from BU(n) rather than SU(n)
It's known, from Ravenel's green book, as well as other sources, that we build formal group laws over a ring from n-buds, where an n-bud is essentially a truncated formal group law (sometimes called a ...
39
votes
2answers
3k views
What is known about the sum x^{n^2}/n?
It follows from a general theorem of Honda that the formal group with the logarithm
$$
x+x^{2^s}/2+x^{3^s}/3+x^{4^s}/4+\cdots
$$
has integer coefficients. I became interested in it because its $p$-...
8
votes
2answers
709 views
Is it possible to construct a formal group law from a Lie group without choosing coordinates?
There is a three-way correspondence between:
Real (connected and simply connected) Lie groups of dimension $n$;
$\mathbb R$-Lie algebras of dimension $n$;
Formal group laws in $n$ variables over the ...
12
votes
1answer
555 views
Cohomology of Formal Groups
Lubin and Tate, in discussing moduli of 1-dimensional formal groups construct a cohomology theory of formal groups, at least in degrees 0,1 and 2. Does their result about deformations actually follow ...
1
vote
0answers
125 views
Isogenies in multidimensional formal groups
Let $K/\mathbb{Q}_p$ be a local field, $A$ the ring of integers of K, $\pi$ a uniformizer element for $A$, $F$ an n-dimensional formal group with coefficients in $A$ and $f$ an endomorphism of $F$. ...
9
votes
1answer
497 views
Generalizing detropicalization
Given an identity in max,plus arithmetic, are there ways to turn it into an ordinary algebraic identity it other than by replacing addition by multiplication and replacing max by series-plus or by ...
2
votes
0answers
108 views
Lazard's $\Gamma_n(f)$ as cocycle
In Michel Lazard's "Commutative Formal Groups" Springer Lecture Notes, he defines an operator on a polynomial 3-cochain $f$ denoted $\Gamma_n(f)$, which defines as the $n^{th}$ homogeneous piece of ...
7
votes
0answers
163 views
Schwede's DB spectra and MU
In Stefan Schewede's paper Formal groups and stable homotopy of commutative rings, he introduces $\Gamma$-rings (ring spectra) $DB$ for any commutative ring $B$ such that the the set of 1-dimensional ...
1
vote
1answer
224 views
Is every (one dimensional) n-bud of total degree n also a formal group law?
This is essentially a request for counterexamples, since I know so few $n$-buds (or as some might say, formal group law $n$-chunks). One notices that the only $1$-bud of maximal degree 1 is the ...
8
votes
0answers
321 views
Non-commutative Formal Group Laws
Does anyone know of a good, complete reference for non-commutative formal group laws (i.e. construction of a "Lazard ring," discussion of non-commutative formal groups, perhaps some discussion of ...
15
votes
1answer
1k views
What do formal group laws of height $\geq 3$ look like?
By the classification of formal groups in characteristic $p$, we know that isomorphism classes of connected smooth $1$-dimensional formal groups, equivalently group scheme structures on $\operatorname{...
0
votes
0answers
243 views
The formal Group of the dual Abelian Variety
For an abelian variety $A$ with formal group $F$, how is the formal group $F^\ast$ of the dual abelian variety $A^{\vee}$ related to $F$? In general, for a formal group $F$, is there a concept of dual ...
14
votes
1answer
670 views
Is there an algebro-geometric description of $\nu$?
Motivation: According to the "chromatic" picture of stable homotopy, we should think of the moduli stack $M_{FG}$ of formal groups as a "good approximation" to the stable homotopy category (more ...
4
votes
0answers
285 views
Formal non-CM in local fields
An elliptic curve $E$ with complex multiplication by an imaginary quadratic field $F$ has $\ell$-adic Galois representations that essentially encode the class field theory of $F$ - in other words, the ...
16
votes
0answers
640 views
Lubin-Tate vs cohomological approach to local CFT
Local class field theory ("local CFT") can be developed in various ways, among them is a cohomological approach and an explicit approach due to Lubin and Tate (both can be found in Milne's CFT notes ...
11
votes
0answers
332 views
Galois invariants in a ring of fractional power series over a finite field
Let $\mathbf{F}_q$ be a finite field, and let $A=\mathbf{F}_q [[ x^{1/q^\infty} ]]$ be the completion of $\mathbf{F}_q[x^{1/q^\infty}]$ with respect to the $x$-adic topology. Then the $q$th power ...
2
votes
2answers
276 views
Different Lie group structures on a vector space with the same Lie algebra structure
This is an eccentric question: recall that a smooth Lie group structure on $\mathbb R^n$ is uniquely identified by a triple $(\mu,\iota,e)$ where $\mu:\mathbb R^n\times\mathbb R^n\to\mathbb R^n$ is ...
5
votes
2answers
495 views
Reference request: Spec A_* is the automorphism group of the additive formal group law
Dear all,
I'm seeking a reference for a claim made in lecture 8 of Jacob Lurie's chromatic homotopy theory notes (http://www.math.harvard.edu/~lurie/252xnotes/Lecture8.pdf). More particularly, ...
5
votes
0answers
557 views
Formal groups in the supersingular reduction case
Let $E/\mathbb{Q}$ be an elliptic curve with potential good supersingular reduction at $p$. Thus, there is a finite extension $K/\mathbb{Q}_p$ such that $E/K$ has good supersingular reduction. Let us ...
13
votes
1answer
603 views
Obstructions to formally integrating vector fields in characteristic p?
Let $M$ be a smooth scheme over some field $k$ of characteristic $p$, and $\vec X$ a vector field on it. Equivalently, $\vec X$ gives a map $Spec\ k[\epsilon]/\langle \epsilon^2 \rangle \times M \to M$...
