Questions tagged [formal-groups]

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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 ...
Ojen's user avatar
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14 votes
0 answers
524 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
12 votes
0 answers
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 ...
Lennart Meier's user avatar
11 votes
0 answers
374 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 ...
Jared Weinstein's user avatar
10 votes
0 answers
295 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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9 votes
0 answers
260 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,\...
Lennart Meier's user avatar
9 votes
0 answers
414 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 ...
Jonathan Beardsley's user avatar
7 votes
0 answers
175 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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7 votes
0 answers
130 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
7 votes
0 answers
181 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 ...
Jonathan Beardsley's user avatar
6 votes
0 answers
368 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
6 votes
0 answers
351 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
6 votes
0 answers
243 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
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6 votes
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288 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(...
Alexey Ustinov's user avatar
6 votes
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 $Y$...
Ian Shipman's user avatar
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5 votes
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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 ...
Homotopy theorist 's user avatar
5 votes
0 answers
290 views

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
5 votes
0 answers
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 $p$-...
SomeGuy's user avatar
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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 $G$. ...
Dima Sustretov's user avatar
5 votes
0 answers
679 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 ...
Álvaro Lozano-Robledo's user avatar
4 votes
0 answers
382 views

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
4 votes
0 answers
241 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
4 votes
0 answers
84 views

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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4 votes
0 answers
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 ...
A Rock and a Hard Place's user avatar
4 votes
0 answers
221 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 ...
მამუკა ჯიბლაძე's user avatar
4 votes
0 answers
202 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 ...
George's user avatar
  • 41
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 ...
Jonathan Beardsley's user avatar
4 votes
0 answers
361 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 ...
David Corwin's user avatar
  • 15.1k
3 votes
0 answers
258 views

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 ...
MAS's user avatar
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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 ...
kiwi's user avatar
  • 41
3 votes
0 answers
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 $\...
John Pardon's user avatar
  • 18.3k
3 votes
0 answers
231 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 ...
8k14's user avatar
  • 131
2 votes
0 answers
70 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
  • 789
2 votes
0 answers
174 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 $p$ ...
Duality's user avatar
  • 1,406
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 - ...
John Rognes's user avatar
  • 8,692
2 votes
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 ...
user237334's user avatar
2 votes
0 answers
117 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
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 $K$. ...
Hugo Chapdelaine's user avatar
2 votes
0 answers
284 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
  • 845
2 votes
0 answers
239 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
2 votes
0 answers
71 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 ...
George's user avatar
  • 21
2 votes
0 answers
114 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 ...
Jonathan Beardsley's user avatar
1 vote
0 answers
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 ...
Schemer1's user avatar
  • 789
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( \...
Duality's user avatar
  • 1,406
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 $\...
kiwi's user avatar
  • 41
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 ...
k.j.'s user avatar
  • 1,352
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 ...
ali's user avatar
  • 1,043
1 vote
0 answers
147 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
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,...
user56577's user avatar
1 vote
0 answers
144 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 $F$. ...
George's user avatar
  • 11