Questions tagged [formal-groups]
The formal-groups tag has no usage guidance.
70
questions
6
votes
0answers
284 views
Coherent cohomology of the generic fiber of Lubin-Tate space vs. of Lubin-Tate space considered rationally?
I am trying to compare the coherent cohomology of the generic fiber of Lubin-Tate space to the coherent cohomology of Lubin-Tate space considered rationally, and I am going in circles! I would be very ...
1
vote
0answers
89 views
Cartier duality and Frobenius on Witt vector schemes
Suppose for simplicity we are working over $\mathbb{F}_p$. Cartier duality is an antiequivalence between formal groups and affine group schemes over $Spec(\mathbb{F}_p)$. Let $\mathbb{W}_p(-)$ denote ...
3
votes
1answer
312 views
Is the formal completion of an affine group necessarily a formal group?
Notation and Setting: let $\operatorname{Aff}$ denote the category of affine schemes whose objects are covariant representable functors $\operatorname{X}:\operatorname{Ring}\rightarrow\operatorname{...
0
votes
0answers
48 views
double-and-add algorithm with(z,w)-coordinates
At "The Arithmetic of Elliptic Curves" by Joseph H. Silverman
p.390 VI Example 6.7
given $E:y^2 = x^3+19x+112$ over ${F}_{127}$
points $P= (106,72)∈E$(${F}_{127}$), $Q= (12,121)∈E$(${F}_{127}$)
lifted ...
1
vote
0answers
113 views
Compatiblity of completion and fibre products. (Formal completion and formal groups)
Let $S$ be a scheme (not necessarily locally noetherian), $X$ a smooth separated group scheme over $S$, and $\hat{X}$ be the formal completion along with the identity section.
Then does the group ...
8
votes
1answer
261 views
Is there something “Koszul dual” to formal groups?
The Lie operad is Koszul dual to the commutative operad. In some sense, the data of a formal group is an "elaboration" of the data of a Lie algebra. Is there some corresponding "elaboration" of the ...
1
vote
0answers
118 views
Formal group as a limit of its finite subgroups
I'm reading Manin's article on formal groups and I have a problem with Lemma 1.1.
Consider $k$ a prefect ring of characteristic $p$ and $(A,m,k)$ a noetherian complete local ring of the same ...
7
votes
0answers
163 views
Meaning of Elliptic Irregular Primes
The Bernoulli numbers are defined by the equation
$$
\frac{t}{e^t-1}=\sum_k b_k \frac{t^k}{k!}.
$$
A prime number $p$ is irregular if it divides the numerator of one of the even Bernoulli numbers up ...
6
votes
0answers
106 views
Classification of one dimensional (non-commutative) formal group laws over $k[\epsilon]/(\epsilon^n)$
It's well-known that any one dimensional formal group law over a $\mathbb Q$-algebra or a reduced ring is commutative, but there are one dimensioal non-commutative formal group laws over rings like $k[...
5
votes
1answer
185 views
What are the modularity properties of Weierstrass sigma function?
I'm a little confused at the sigma orientation of tmf, see e.g. Witten genus and its references. The Weierstrass sigma function can be written as
$$\sigma_L(z)(q)=\frac{z}{\exp\left(\sum_{k\ge 2} G_k(...
2
votes
0answers
92 views
Exercise on formal group laws over an algebraically closed field
There is an exercise in Weinstein's notes on Lubin--Tate theory, namely show that there is a unique (up to isomorphism) one-dimensional formal group law of given finite height $h$ over an ...
2
votes
1answer
368 views
Formal group law and Koenigs function conjecture?
Let $f(x,y)$ be a symmetric real function and a formal group law
$$G(x + y) = f(G(x),G(y)). \tag{1}$$
Consider the equation
$$ h(2x) = f(h(x),h(x)) = A(h(x)). \tag{2}$$
This equation has many ...
9
votes
1answer
501 views
Nontrivial p-divisible groups over $\mathbb Z$ for general prime $p$
In Tate's famous paper about $p$-divisible groups, for a prime number $p$ he asks whether there exists a $p$-divisible group $G$ over $\mathbb Z$ such that $G$ is not a direct sum of $\mu_{p^\infty}$ ...
1
vote
0answers
108 views
Geometric intuition and computation for Cartier theory
I am learning Cartier theory of commutative formal groups by the book of Zink. It is a powerful tool but I don't understand it's motivation. The Cartier module of a formal group $G$ over a $\mathbb{Z}...
40
votes
3answers
1k views
Characterizing positivity of formal group laws
The formal group law associated with a generating function $f(x) = x + \sum_{n=2}^\infty a_n \frac{x^n}{n!}$ is $$f(f^{-1}(x) + f^{-1}(y)).$$ In my thesis, I found a large number of examples of ...
6
votes
0answers
107 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 ...
9
votes
1answer
314 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 ...
9
votes
1answer
268 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 \...
5
votes
3answers
225 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 ...
7
votes
1answer
604 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
236 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 $\...
13
votes
0answers
381 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
686 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
219 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
249 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
217 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
134 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
181 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
195 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
561 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
517 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 ...
4
votes
0answers
177 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
239 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,\...
6
votes
0answers
263 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 $n$
$$P_n(x)=Q_n(...
14
votes
2answers
1k views
Formal group law over $\mathbb{F}_p$
Let $p$ be a prime. For each $n > 0$ there is a unique 1-dimensional commutative formal group law $F$ over $\mathbf{Z}$, $F(X, Y) = X + Y + \dots \in \mathbf{Z}[[X, Y]]$, whose logarithm function ...
1
vote
0answers
138 views
Skew symmetry for the Hilbert symbol
Let $K$ be a local field containing the group $\mu_n$ of $n$th roots of 1 and the $\theta_K:K^*\to G_K^{ab}$ be the reciprocity map. The we know that the Hilbert symbol $$K^*\times K^*\to \mu_n$$ $$(a,...
2
votes
0answers
67 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 ...
15
votes
2answers
1k 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
$$
x=(x_1,...,x_n),y=(y_1,.....
22
votes
3answers
2k views
Is there a better proof of this fact in number theory/formal group theory?
Let $\Phi_n$ be the $n$'th cyclotomic polynomial, and put
\begin{align*}
a_n &= \Phi_n(1) \\
b_n &= \gcd\left(\left(\begin{array}{c} n \\ 1\end{array}\right),\dotsc,\left(\begin{array}{c} n ...
4
votes
0answers
188 views
Continuity of the Hilbert pairing
I would like to know if the Kummer pairing (or the analogue of the Hilbert Symbol) for a one dimensional group defined over the ring of integers of a higher-dimensional local field is continuous (with ...
5
votes
0answers
615 views
formal group laws of Abelian varieties in positive characteristic
Let $G$ be an algebraic group defined over an (algebraically closed) field $k$. Then one can obtain a formal group law by completing the multiplication map $m: G \times G \to G$ at the unit of $G$.
...
11
votes
0answers
385 views
Are automorphisms of abelian varieties detected by the formal group?
Let $A$ be an abelian variety of dimension $g$ over an algebraically closed field $k$.
Assume $k$ has characteristic $p$ and denote by $A(p)$ the $p$-divisible group of dimension $g$ associated with ...
3
votes
0answers
201 views
Formal n-buds from BU(n) rather than SU(n)
It's known, from Ravenel's green book, as well as other sources, that we build formal group laws over a ring from n-buds, where an n-bud is essentially a truncated formal group law (sometimes called a ...
40
votes
2answers
4k views
What is known about the sum x^{n^2}/n?
It follows from a general theorem of Honda that the formal group with the logarithm
$$
x+x^{2^s}/2+x^{3^s}/3+x^{4^s}/4+\cdots
$$
has integer coefficients. I became interested in it because its $p$-...
10
votes
2answers
845 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 ...
13
votes
1answer
617 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
528 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
113 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
168 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 ...
