# Questions tagged [formal-groups]

The tag has no usage guidance.

82 questions
Filter by
Sorted by
Tagged with
558 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 ...
1 vote
41 views

502 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 ...
296 views

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 ...
115 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 ...
356 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 ...
310 views

453 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 ...
749 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 ...
275 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 $p$-...
303 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 $K$. ...
255 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 ...
169 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),...
220 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, ...
216 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 ...
679 views

### $p$-torsion of an abelian variety of $p$-rank $0$

Let $k$ be an algebraically closed field of characteristic $p > 0$ and let $A$ be an abelian variety over $k$ such that $A[p](k) = 0$, i.e., such that $A$ has $p$-rank $0$. If I am not mistaken, ...
606 views

### Infinitesimal deformations of the formal group of $\mathbb{G}_m$

For a commutative ring $R$, consider the formal group $\widehat{\mathbb{G}}_m$ over $R$ that is the completion of $\mathbb{G}_{m, R}$ along its identity section (naively, $\widehat{\mathbb{G}}_m$ is ...
197 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 ...
254 views

### Formal group law over $\mathbb{F}_p$
Let $p$ be a prime. For each $n > 0$ there is a unique 1-dimensional commutative formal group law $F$ over $\mathbf{Z}$, $F(X, Y) = X + Y + \dots \in \mathbf{Z}[[X, Y]]$, whose logarithm function ...