# Questions tagged [formal-groups]

The formal-groups tag has no usage guidance.

95
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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 ...

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### With 6 inverted, is the ring of Weierstrass curves a quotient of the Lazard ring by a regular sequence?

Let $L$ be the Lazard ring, i.e., the underlying ring of the universal one-dimensional formal group law. Let $M$ be the ring $\mathbb{Z}[c_4, c_6, 1/6]$ of Weierstrass curves over $\mathbb{Z}[1/6]$. ...

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### 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 ...

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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 $\mathbb{F}_{p^n}$ ...

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### 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 ...

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### 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 ...

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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)...

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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 ...

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### 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 $...

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### 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( \...

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### 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^...

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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 $p$ ...

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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 ...

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### 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 ...

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### 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 $\...

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### 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 ...

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### 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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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### 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 ...

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### 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 ...

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### 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 $...

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### 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 - ...

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### 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 ...

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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 ...

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### 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 ...

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### 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{...

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### 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 ...

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### 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 ...

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### 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 ...

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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 ...

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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[...

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### 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(...

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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 ...

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### 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 ...

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### 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 $\mu_{p^\infty}$ ...

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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}...

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### 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 ...

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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 ...

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### 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 ...

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### (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 \...

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### 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 ...

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### 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....

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### 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 $\...

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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 ...

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### 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 ...

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### 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 $p$-...

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### 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 $K$. ...