# Questions tagged [formal-groups]

The formal-groups tag has no usage guidance.

62
questions

**5**

votes

**1**answer

124 views

### 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(...

**2**

votes

**0**answers

75 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 ...

**2**

votes

**1**answer

352 views

### 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 ...

**4**

votes

**0**answers

99 views

### Computation of the Newton polygon of formal modules under unramfied condition

When reading a very short paper "A generalization of the Chowla-Selberg formula" which is avaliable at http://www.math.titech.ac.jp/~taguchi/bib/cs.pdf by Yukiyoshi Nakkajima and Yuichiro Taguchi, ...

**9**

votes

**1**answer

461 views

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

**2**

votes

**0**answers

90 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}...

**40**

votes

**3**answers

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 ...

**6**

votes

**0**answers

98 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 ...

**8**

votes

**1**answer

284 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 ...

**9**

votes

**1**answer

235 views

### (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 \...

**4**

votes

**3**answers

207 views

### 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 ...

**7**

votes

**1**answer

503 views

### 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....

**3**

votes

**0**answers

215 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 $\...

**12**

votes

**0**answers

359 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 ...

**10**

votes

**2**answers

666 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 ...