Questions tagged [formal-groups]

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Is there anything special about the Honda formal group?

The "standard" Morava E-theory $E_n$ (at a prime $p$) is typically defined using the so-called "Honda formal group law", the unique FGL $\Gamma_n$ over $\mathbb{F}_{p^n}$ ...
Doron Grossman-Naples's user avatar
3 votes
0 answers
57 views

Conditions for a $p$-divisible group to be represented by a formal Lie group

Let $S$ be a scheme where $p$ is locally nilpotent and let $G$ be a $p$-divisible group over $S$. Is connectedness of $G$ equivalent to $G[p] := \ker(p : G \to G) \to S$ radicial (universally ...
kiwi's user avatar
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6 votes
1 answer
227 views

Generating the coordinate ring of the Lubin-Tate formal group

Let $K$ be a $p$-adic local field with uniformizer $\pi \in \mathcal{O}_{K}$ and residue field $k = \mathcal{O}_{K}/\pi$. Let $G$ be a Lubin-Tate formal $\mathcal{O}_{K}$-module and $G_{0}$ its ...
Piotr Pstrągowski's user avatar
2 votes
0 answers
67 views

Does there exists a "local slice" for an action $ \widehat{\mathbb{G}_{a}} $ on $ \operatorname{Spf}(\widehat{A}) $ (char zero)?

Every action $ \beta $ of $ \mathbb{G}_{a} $ on a variety $ \operatorname{Spec}(A) $ over a field of characteristic zero is obtained from a locally nilpotent derivation $ \delta $ via $ f(t_{0} \ast x)...
Schemer1's user avatar
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5 votes
0 answers
114 views

Faltings' Cartier duality for A-modules in terms of Hopf algebras

$\newcommand\dual{^{\text{dual}}}\newcommand\GrpSch{\mathrm{GrpSch}}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Vect{Vect}$If $G$ is a finite group scheme over a field $k$, we can define its ...
Homotopy theorist 's user avatar
0 votes
1 answer
129 views

Power series corresponding to $[a]\in \operatorname{End}(E)$ ($a \in R_K$) can be expressed as $[a](t)=at+\text{(term higher than degree $2$)}$?

Let $K$ be an imaginary quadratic field and $E/K$ be an elliptic curve which has complex multiplication on $K$. Let $R_K$ be ring of integers of $K$. Let $ \hat{E}$ be its formal group of $E$. Take $...
BrauerManinobstruction's user avatar
1 vote
0 answers
155 views

Homomorphism of formal group of elliptic curve corresponding to its endomorphism

Let $E$ be an elliptic curve and $ \hat{E}$ be its formal group. Rubin's lemma $3.7$ in 'Elliptic curves with complex multiplication' reads For arbitrary $φ∈End(E)$, there exists unique $φ(t)∈End( \...
BrauerManinobstruction's user avatar
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116 views

Proof of $[p](x)≡x^p\operatorname{mod}p \Bbb{Z}_p$ for formal group of elliptic curve

Let $E$ be an elliptic curve over $\Bbb{Q}_p$. Let $ \hat{E}$ be formal group of $E$. Let $[p](x)=x+_\hat{E}+・・・+_\hat{E}x$ (add by formal group law $p$ times). I want to know the proof of $[p](x)≡x^...
BrauerManinobstruction's user avatar
2 votes
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148 views

Relation between division point of elliptic curve and formal group of elliptic curve, $\Bbb{Q}_p(E[p])=\Bbb{Q}_p(\hat{E}[p])$

Let $E/ \Bbb{Q}_p$ be an elliptic curve over $ \Bbb{Q}_p$. $\hat{E}$ denote the corresponding formal group of $E$. I want to prove $\Bbb{Q}_p(E[p])=\Bbb{Q}_p(\hat{E}[p])$. $ \hat{E}[p]$ denotes $p$ ...
BrauerManinobstruction's user avatar
4 votes
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195 views

Is there any use for n-dimensional formal group laws in chromatic homotopy?

Chromatic homotopy tends to mainly focus on $1$-dimensional (commutative) FGLs. From a geometric perspective, this is because line bundles form a group and n-plane bundles don't, so the first Chern ...
Doron Grossman-Naples's user avatar
9 votes
2 answers
637 views

Can we use formal groups to recover Lie-theoretic representation theory in characteristic p?

In differential geometry, Lie's theorems allow us to integrate any Lie algebra representation to a Lie group representation. The algebraic version of this is more complicated (and I'm not terribly ...
Doron Grossman-Naples's user avatar
1 vote
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64 views

Definition of formal group functors

In the book by Demazure "Lectures on $p$-Divisible Groups" a formal group functor over a field $k$ is defined in II.4 as a functor $\operatorname{Mf}_k \to \operatorname{Grp}$ where $\...
kiwi's user avatar
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5 votes
1 answer
354 views

Question about log and exp of a formal group law

Let $K$ be a finite extension of $\mathbb{Q}_p$. Let $F$ be a Lubin–Tate formal group law defined over $K$ with endomorphism $f(T)$ corresponding to $\pi$ (a uniformizer of $K$). Then one can define ...
just someone local's user avatar
12 votes
1 answer
250 views

Is every complex oriented ring spectrum with additive FGL an Eilenberg-Maclane spectrum?

Suppose $E$ is a complex-oriented ring spectrum whose formal group law is isomorphic to the additive one. As the title suggests, we might as well change the complex orientation so that the formal ...
kiran's user avatar
  • 1,883
10 votes
0 answers
278 views

Classification of derived formal group laws

Denote by $SCR$ the $\infty$-category of "simplicial commutative rings" (i.e. the nonabelian derived category of the category of finitely generated polynomial rings). Given $R \in SCR$, one ...
Tom Bachmann's user avatar
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4 votes
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Ramification behavior of field given by adjoining $p$-torsion point on formal group of abelian variety

Setup. Let $p > 2$ be a prime, let $K$ be the completion of the maximal unramified extension of $\mathbb{Q}_p$, and fix an algebraic closure $\overline{K}$ of $K$. Let $A/K$ be an abelian variety ...
Jackson Morrow's user avatar
4 votes
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The unit root subspace of a genus-2 odd degree hyperelliptic curve of semistable reduction

Let $K$ be a finite extension of $\mathbb{Q}_p$. If $A_K$ is a semistable Abelian variety over $K$, then we have a Frobenius endomorphism on $H_{dR}^1(A_K)$, whose definition depends on a choice of a ...
E. Kaya's user avatar
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6 votes
0 answers
272 views

Group-like elements of universal enveloping algebra

Suppose $\mathfrak{g}$ is a finite-dimensional Lie algebra over $\mathbb C$. Take $A=U(\mathfrak g[[t]])$, a universal enveloping algebra of $\mathfrak g[[t]]$ over $\mathbb C[[t]]$. Then we may ...
Troshkin Michael's user avatar
4 votes
0 answers
144 views

Preorientation of additive formal group

In "A Survey of Elliptic Cohomology", Section 3.2, Lurie asserts that the preorientations of the additive formal group $\widehat{\mathbf G}_a$ over $\mathbf Z$ are classified by the $\mathbb ...
A Rock and a Hard Place's user avatar
4 votes
1 answer
1k views

Completed tensor product and power series rings

I want to know if the notion of completed tensor product in Stacks Project Tag 0AMU is the one that yields $$k[[x]] \widehat{\otimes} k[[y]]≅k[[x,y]].$$ Here I should be considering the inverse limit ...
Minkowski's user avatar
  • 499
3 votes
2 answers
322 views

Some special subgroups of formal groups

Let $G$ be a 1-dimensional, commutative formal group over a ring $R$. Give $G$ a coordinate $x$ and let $A\subset R$ be the subring generated by the coefficients of the corresponding formal group law $...
kiran's user avatar
  • 1,883
2 votes
0 answers
125 views

Reference for the $3$-series of an elliptic formal group law

The $3$-series of the formal group law of the Weierstrass curve $y^2 = x^3 + a_2 x^2 + a_4 x$ begins $$ [3](z) = 3 z - 8 a_2 z^3 + (24 a_2^2 - 96 a_4) z^5 - (72 a_2^3 - 288 a_2 a_4) z^7 + (216 a_2^4 - ...
John Rognes's user avatar
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4 votes
1 answer
387 views

When is a formal group smooth?

This is a question that I suspect is simply a matter of technical issues written down or clarified somewhere in the literature, but which I can't find. Suppose we're working over an arbitrary base ...
xir's user avatar
  • 1,712
6 votes
0 answers
347 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 ...
Catherine Ray's user avatar
2 votes
0 answers
229 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 ...
user237334's user avatar
3 votes
1 answer
427 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{...
sagirot's user avatar
  • 435
1 vote
0 answers
275 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 ...
k.j.'s user avatar
  • 1,444
8 votes
1 answer
401 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 ...
Tim Campion's user avatar
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1 vote
0 answers
140 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 ...
ali's user avatar
  • 1,016
7 votes
0 answers
171 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 ...
S. carmeli's user avatar
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6 votes
0 answers
205 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[...
sawdada's user avatar
  • 6,100
7 votes
1 answer
539 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(...
xir's user avatar
  • 1,712
2 votes
0 answers
114 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 ...
user avatar
1 vote
1 answer
433 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 ...
mick's user avatar
  • 597
9 votes
1 answer
654 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 $\mu_{p^\infty}$ ...
sawdada's user avatar
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1 vote
0 answers
136 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}...
wuzx's user avatar
  • 517
40 votes
3 answers
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 ...
Jair Taylor's user avatar
7 votes
0 answers
119 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 ...
James Francese's user avatar
9 votes
1 answer
365 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 ...
Catherine Ray's user avatar
11 votes
1 answer
323 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 \...
A Rock and a Hard Place's user avatar
5 votes
3 answers
279 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 ...
José Navarro's user avatar
9 votes
1 answer
863 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....
guest's user avatar
  • 528
3 votes
0 answers
301 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 $\...
John Pardon's user avatar
  • 17.9k
14 votes
0 answers
492 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 ...
Eric Peterson's user avatar
10 votes
2 answers
784 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 ...
Catherine Ray's user avatar
5 votes
0 answers
301 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 $p$-...
SomeGuy's user avatar
  • 813
2 votes
0 answers
319 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 $K$. ...
Hugo Chapdelaine's user avatar
2 votes
0 answers
275 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 ...
Bear's user avatar
  • 855
3 votes
1 answer
184 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),...
Alexey Ustinov's user avatar
2 votes
0 answers
229 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, ...
Jeremy Hahn's user avatar