Questions tagged [formal-groups]
The formal-groups tag has no usage guidance.
52
questions with no upvoted or accepted answers
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760
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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 ...
14
votes
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524
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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 ...
12
votes
0
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454
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 ...
11
votes
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374
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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 ...
10
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295
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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 ...
9
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260
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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
$$\mathbb{Z}_{(p)}[v_1,\...
9
votes
0
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414
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Non-commutative Formal Group Laws
Does anyone know of a good, complete reference for non-commutative formal group laws (i.e. construction of a "Lazard ring," discussion of non-commutative formal groups, perhaps some discussion of ...
7
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175
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Meaning of Elliptic Irregular Primes
The Bernoulli numbers are defined by the equation
$$
\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 ...
7
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130
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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 ...
7
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181
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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 ...
6
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368
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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 ...
6
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351
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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 ...
6
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243
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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[...
6
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288
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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(...
6
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0
answers
260
views
When do equivariant sheaves on a formal neighborhood extend?
Suppose that $X$ is a variety (in char 0) with an action of an affine algebraic group $G$. Let $Y \subset X$ be a subvariety fixed by $G$--the action map agrees with projection upon restriction to $Y$...
5
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143
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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 ...
5
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290
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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 ...
5
votes
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315
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 $p$-...
5
votes
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874
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 $G$.
...
5
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679
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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 ...
4
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382
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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 $\mathbb{F}_{p^n}$ ...
4
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241
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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 ...
4
votes
0
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84
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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 ...
4
votes
0
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149
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 ...
4
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221
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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 ...
4
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0
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202
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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
0
answers
219
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 ...
4
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361
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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 ...
3
votes
0
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258
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What are the unsolved problems in Formal groups and $L$-functions?
In the 1st page of the introduction of Hazewinkel's Formal Groups and Applications book, there are two ways of constructing formal groups (law):
$\bullet$ Given a Lie group $G$, one can define a ...
3
votes
0
answers
71
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 ...
3
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0
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315
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 $\...
3
votes
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231
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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 ...
2
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70
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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)...
2
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174
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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 $p$ ...
2
votes
0
answers
139
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 - ...
2
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0
answers
266
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 ...
2
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0
answers
117
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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
0
answers
329
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
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284
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"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 ...
2
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0
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239
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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, ...
2
votes
0
answers
71
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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 ...
2
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0
answers
114
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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 ...
1
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30
views
Does there exist an upper triangular set of gens of $ k[[X]] $ for a unipotent formal group acting on $ \operatorname{Spec}(k[[X]]) $?
For this post I define a formal group to be a group object in the category of formal schemes.
Let $ G $ be a linear algebraic group with affine coordinate ring $ k[G] $. If $ \mathfrak{m}_{e} $ is ...
1
vote
0
answers
169
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( \...
1
vote
0
answers
76
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 $\...
1
vote
0
answers
310
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 ...
1
vote
0
answers
147
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 ...
1
vote
0
answers
147
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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}...
1
vote
0
answers
150
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,...
1
vote
0
answers
144
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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 $F$. ...