Questions tagged [formal-groups]

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Kummer map on the formal group of elliptic curve

In section 3(Page 50) of this paper, it is mentioned that the Kummer map, $\hat{E}(\mathfrak{m}_n)\rightarrow H^{1}(k_n,T)$ along with the Weil pairing induces a cup product of Galois cohomology ...
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The relation between formal group $\frac{X+Y}{1-XY}$ and algebraic group $x^2+y^2=1$ with group law $*$

$T(X,Y)=\frac{X+Y}{1-XY}$ is a power series which satisfies axiom of formal group. My book reads this formal group comes from algebraic group $S: x^2+y^2=1$, with group law $*$:$(x_1,y_1)*(x_2,y_2)=(...
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Basic question on the connection between Complex-oriented cohomology theories and Formal Group Law

In wikipedia is stated that a complex-orientable cohomology theory is a multiplicative cohomology theory $E$ such that the restriction map $E^2(\mathbb{C}\mathbf{P}^\infty) \to E^2(\mathbb{C}\mathbf{...
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Formal group of Abelian Varieties

I am reading Barry Mazur and Tom Weston's note Euler Systems and Arithmetic Geometry ([here][1] is the link). I have a question about the following fact in section 2.2 page 87: Let $K$ be a local ...
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A tricky Witt vector computation

Let $K$ be a local field of characteristic $p$, and $K^{sep}$ be its separable closure. Let $F_q : W(\mathcal K) \longrightarrow W(\mathcal K)$ be the homomorphism of Witt vectors $$ (w_0, w_1, \ldots ...
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All formal group has Frobenius power series as homomorphisms?(Lubin Tate theory)

Let $L/ \Bbb{Q}_p$ be finite extension, $o$ be it's ring of integers. Frobenius power series is defined as $Φ(X)∈o[[X]]$ s.t.$Φ(X)=πX+$(higher term) and $Φ(X)≡X^qmodπo[X]$. It is well known that For ...
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'Additive Inverse' in Formal Group

The definition of a formal group is given here. Apologies in advance if this question sounds trivial/obvious. I am still trying to wrap my head around the idea of formal group. Suppose $F$ is a ...
UnsinkableSam's user avatar
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Why is multiplication by $m$ map a homomorphism for formal groups?

I have started learning about formal groups defined over a R1NG, $R$. Then, I studied multiplication by $m$ map on a formal group. The definition is as follows: Def: Let $F$ be a formal group over $R$....
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Silverman AEC: Lemma IV.6.3 [on the convergence of formal logarithm]

In Silverman's Arithmetic of Elliptic Curves, Lemma IV.6.3(a) states that Let $R$ be a ring of characteristic $0$, complete with respect to a discrete valuation $v$, and let $p\in\mathbb{Z}$ be a ...
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Silverman AEC: Corollary IV.4.4.

In Silverman's Arithmetic of Elliptic Curves, Corollary IV.4.4 states that (for an arbitrary ring $R$), Let $\mathcal{F}/R$ be a formal group and let $p\in \mathbb{Z}$ be a prime. There are power ...
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How to confirm $\phi(F_1(x,y))=F_2(\phi(x),\phi(y))$,where $F_1$ and $F_2$ are formal group law of elliptic curve $E_1$, $E_2$.

This question is from Silverman's 'the arithmetic of elliptic curves',$p134$. Let $K$ be a field of characteristic $p > 0$, let $E_1/ K$ and $ E_2/K$ be elliptic curves, and let $\phi : E_1 \to E_2$...
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Inverse of formal multiplicative group

I am reading about the formal multiplicative group, with addition given by $F(x,y)=x+y+xy$, and I am wondering if there is a nice way to describe the inverse of an element. So if I let $x+y+xy=0$, ...
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elliptic curve $E:y^2=x^3-x$ can be transformed to move the identity element to the origin

For example, elliptic curve $E:y^2=x^3-x$ can be transformed to move the identity element to the origin, $(z,w)=(0,0)$, we do the change of coordinates $z=-x/y$, $w=-1/y$. The equation of this curve ...
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Group object in the category of formal schemes vs. formal group law

I've been trying to see how both definitions of a formal group coincide: Take a ring $ R $, a group object in the category of formal schemes over $ R $ is then given by two morphims $ e : R \...
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torsion element of group associated to formal group

Let $R$ be complete local ring $M$ be the maximal ideal of $R$ $F$ be a formal group defined over $R$, with group law $F(X,Y)$. According Silverman's book 'the arithmetic of elliptic curves', example ...
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Completed tensor product

I want to know if this notion of completed tensor product is the one that yields $$ k[\![ x ]\!] \hat{\otimes} k[\![ y ]\!] \cong k[\![ x,y ]\!]. $$ Here I should be considering the inverse limit ...
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Ghost Component of Big Witt Vectors

Given any $\mathbb{Z}_{(p)}$-algebra $R$, we can form the big Witt vectors $(a_n) \in \prod_{n=1}^\infty R$. Then there is an isomorphism from $$BigWitt(R) \to 1+xR[[x]].$$ I believe one can define ...
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What does "completing an elliptic curve $E$ along its identity section $\sigma_0$" mean?

From here, I got to know the method of getting formal group/formal group law from Elliptic from. It says: Given an elliptic curve $E$ over $R$, $E\to\text{Spec}(R)$, we get a formal group $\hat{E}$ ...
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Logarithm and Lubin-Tate formal group

Let $K$ be a finite extension, by Milne's online note "class field theory", $m_{\mathbb{C}_p}$ has a natural $O_K$ module structure where the action is given by $[a]_f$. For such a $f$, there exists a ...
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Definition formal group law in the book complex cobordism.... of Doug Ravenel

I have read the chapter from the Complex Cobordism.... book of Doug Ravenel. It is here https://web.math.rochester.edu/people/faculty/doug/mybooks/ravenelA2.pdf. In the definitions $(A2.1.19)$, page ...
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What book does this appendix on formal group laws belong to?

Can you please identify the book that the appendix https://web.math.rochester.edu/people/faculty/doug/mybooks/ravenelA2.pdf belongs to? This chapter is on universal formal group laws and strict ...
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Formal group laws VS Formal Lie groups

Warning: this is a soft question about usual terminology, to make sure I understand things correctly. Let $R$ be any commutative ring and $n\geq 1$. Consider the $R$-algebra $\mathcal A = R[[X_1,\...
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Why a particular choice of power series $f$ working for Lubin-Tate theory?

This is related to Iwasawa's Local Class Field Theory Chpt 3 and 4. Let $k$ be a local field and $K$ be maximal unramified algebraic extension of $k$. Set $\Omega$ the algebraic closure of $k$. Take $...
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What role does ring $R$ play in formal group?

This is related to Iwasawa's Local Class Field Theory, Chpt 4, sec 1's Lemma 4.2 proof. Let $R$ be a commutative $k-$algebra s.t. $char(k)=0$. Let $G_a(X,Y)=X+Y$ be a commutative formal group over $...
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Can we consider formal power series $P(t)=\sum_{m=0}^{\infty} \binom{2m}{m} \left[m^l(4t-1)^l+F_l(t) \right]t^m$ as formal group law?

Can we consider the formal power series $P(t)=\sum_{m=0}^{\infty} \binom{2m}{m} \left[m^l(4t-1)^l+F_l(t) \right]t^m$ as a formal group law in 1 variable ? Where $l \geq 0$ are integers and $F_l(t)$ ...
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What is the relation between Lie algebra/ group and formal group laws

$\text{Relation between Lie algebra/ group and formal group laws:}$ I have found a sentence within a short article but could not grasp its meaning fully. The sentence (exactly copied) is the ...
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Are formal groups associated to various groups different?

Let $G$ be an affine algebraic group and $X$ be an abelian variety over $Spec(R)$, where $R$ is a commutative noetherian ring. Let $\widehat{G}$ and $\widehat{X}$ be the completion along the identity ...
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Why do we say always "$1$-dimensional formal group law"?

Why do we say always "$1$-dimensional formal group law"? What is the meaning of $1$-dimensional in this phrase?
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Definition of an invariant differential form on a formal group

These lecture notes define an invariant differential form on a formal group as follows: My question concerns the statement Equivalently, this can be restated as $$ P(F(T,S))F_X(T,S) = P(T). $$ ...
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The formal group law of height $ \ 2$ or $ \ 3 $ over $\mathbb{F}_p$

I need to calculate formal group law over $\mathbb{F}_p$ of height $ \ 2$ or $ \ 3$ ? I need the arithmetic calculation. Please help me with a method
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Height of a formal group law homomorphism

I am reading about formal group homomorphisms defined over a ring $R$ of characteristic $p > 0$ from Silverman's Arithmetic of Elliptic Curves. Given a homomorphism $f$, he shows that if $f'(0)=0$ ...
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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 $$G(x + y) = f(G(x),G(y)). $$ Consider the equation $$ h(2x) = f(h(x),h(x)) = A(h(x)). $$ This equation has many solutions. ...
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Is $ \frac{x+y}{x^3 y^3 - x^3 y - x y^3 + 2 xy + 1} $ a formal group law?

Is $ \frac{x+y}{x^3 y^3 - x^3 y - x y^3 + 2 xy + 1} $ a formal group law on the interval $[-1,1]$ ? It is a lot of work to check on associativity imo. Maybe there is a shortcut around checking ...
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Is $\Bbb Z[x_1,x_2,\Bbb…]/(x_ix_j- {i+j \choose i}x_{i+j})$ a noetherian ring?

I computed homomorphism from formal additive group to formal multiplicative group, realizing that it's represented by $R=\Bbb Z[x_1,x_2,\Bbb…]/(x_ix_j- {i+j \choose i}x_{i+j}) $. Here $x_i\ (i \in \...
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What is the classification of 1-dimensional commutative formal group laws over $\mathbb{Z}$ up to isomorphism?

All 1-dimensional commutative formal group laws over a field $k$ of characteristic $\geq$ 0 are classified up to isomorphism by the characteristic of $k$ and their height. This is result of Michel ...
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What is an example of a formal group law that does not come from an abelian variety?

I am curious to find an example of a 1-d formal group law that does not come from a splitting of the formal group law of an abelian variety. I am aware that we can craft logarithms from the formal ...
Catherine Ray's user avatar
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1 answer
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What does it mean to decompose a formal group law into a formal summand of formal group laws?

Background: Associated to an abelian variety of dimension $n$ is a formal group law of dimension $n$, that is, a formal power series, $F(\bar{x}, \bar{y}) = \sum_{i, j} c_{ij} \bar{x}^i \bar{y}^j$, ...
Catherine Ray's user avatar
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2 answers
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Show that $G$ is a group under $*$

Let $G$ be the set of rational numbers $x$ with $x \neq\frac{-1}{2}.$ For $x, y ∈ G$ define $$x ∗ y = 2xy + x + y.$$ Show that $G$ is a group under $*$. I know how to show that associativity holds ...
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do formal group laws induce group structures on schemes (as opposed to formal schemes)

Let $R$ be a ring and $f \in R[[x]]$ a commutative formal group law over $R$, meaning $f(f(x, y), z)=f(x, f(y, z))$, $\ f(x, y)=f(y, x)$ and $f(x, y)=x+y + \text{higher order terms}$. Let $G=\...
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Flat quotients of power series rings

I read the following statement in some algebraic topology notes and I want to know if it is true and, if so, why. Let $R$ be a ring and $f(x)$ a power series in $R[[x]]$. Suppose that $R[[x]]/(f)$ ...
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Reconstructing formal groups from the p-map, realizing p-maps from formal group

Suppose $F$ is a formal group over $\mathbb{Z}_p$. There are few trivial condition that $f \in \mathbb{Z}_p[[t]]$, the power series representing the $p$-map should satisfy: 1)$f \equiv g(t^p) mod p$ ...
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How can we continuously deform a height 1 formal group law into a height 2 formal group law?

A Quick Review: The complex elliptic curve $\mathbb{C}/(\mathbb{Z} + \tau \mathbb{Z})$ may be rewritten using the exponential, $\text{exp(}{2 \pi i \tau}) =: q$ as $\mathbb{C}^\times/q^\mathbb{Z}$ . ...
Catherine Ray's user avatar
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0 answers
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Can the torsion of a formal group have rank going to infinity?

Suppose $F(x,y) \in \mathbb{Z}_p[[x,y]]$ is a formal group over $\mathbb{Z}_p$. I denote with $G(-)$ the corresponding functor, so that $G(K)$ will denote for me the maximal ideal of $O_K$ equipped ...
Localarithmetic's user avatar
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1 answer
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Is a formal deformation of a Lie algebra an example of a formal group law?

I stumbled across the following definition of the formal deformation of a Lie algebra, and it looks like a group object in the category of formal schemes (not necessarily commutative or 1-dimensional)....
Catherine Ray's user avatar
7 votes
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316 views

What is the group structure on the ring of power series around a point that makes it "the completion of an elliptic curve" along that point?

I've been struggling to understand the explicit details of the completion of an elliptic curve about the origin, and am desperately confused by the explicit details of the resulting group operation. ...
Catherine Ray's user avatar
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0 answers
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Galois action on torsion points of formal group

My question is about a statement in Lang's Cyclotomic Fields, Ch. 8, $\S$2, although I've modified the notation a little. Let $R$ be a complete discrete valuation ring with fraction field $K$, ...
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Are there invariants of formal group laws other than height?

By a theorem of Lazard, 1-d formal group laws over separably closed fields of char $p$ are classified up to isomorphism by their height. Are there invariants of formal group laws other than height (...
Catherine Ray's user avatar
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188 views

Uncertainties on the details of the Connor-Floyd isomorphism and Formal Group Laws

Let $\Omega^{\bullet}(-)$ be the complex cobordism cohomology. $\Omega^n(X) = \{ (M, f) \mid f: M \to X \}$ where cobordant maps are identified, $M$ and $X$ are smooth manifolds, and $\dim(M ) = n$. ...
Catherine Ray's user avatar
2 votes
1 answer
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Showing the center of an endomorphism ring is a direct summand

I am reading A. Fröhlich's Formal Groups, and I am working on the proof that if $F$ is a formal group defined over a separably closed field $k$ of characteristic $p$, then the endomorphism ring $E$ of ...
Annie Carter's user avatar
11 votes
1 answer
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Cogroup structures on the profinite completion of the integers

Let $\mathsf{ProFinGrp}$ be the category of profinite groups (with continuous homomorphisms). This is equivalent to the Pro-category of $\mathsf{FinGrp}$. Notice that $\widehat{\mathbb{Z}} = \lim_{n&...
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