Questions tagged [formal-groups]

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44 votes
2 answers
7k views

What is known about the sum x^{n^2}/n?

It follows from a general theorem of Honda that the formal group with the logarithm $$ x+x^{2^s}/2+x^{3^s}/3+x^{4^s}/4+\cdots $$ has integer coefficients. I became interested in it because its $p$-...
მამუკა ჯიბლაძე's user avatar
74 votes
15 answers
17k views

$f(f(x))=\exp(x)-1$ and other functions "just in the middle" between linear and exponential

The question is about the function $f(x)$ so that $f(f(x))=\exp (x)-1$. The question is open ended and it was discussed quite recently in the comment thread in Aaronson's blog here http://...
Gil Kalai's user avatar
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46 votes
2 answers
5k views

Formal group laws and L-series

Let E be an elliptic curve, let $L(s) = \sum a_n n^{-s}$ denote its L-function, and set $$ f(x) = \sum a_n \frac{x^n}{n}. $$ Then Honda has observed that $$ F(X,Y) = f^{-1}(f(X) + f(Y)) $$ defines ...
Franz Lemmermeyer's user avatar
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 ...
Jair Taylor's user avatar
22 votes
3 answers
2k views

Is there a better proof of this fact in number theory/formal group theory?

Let $\Phi_n$ be the $n$'th cyclotomic polynomial, and put \begin{align*} a_n &= \Phi_n(1) \\ b_n &= \gcd\left(\left(\begin{array}{c} n \\ 1\end{array}\right),\dotsc,\left(\begin{array}{c} n ...
Neil Strickland's user avatar
19 votes
1 answer
2k views

What do formal group laws of height $\geq 3$ look like?

By the classification of formal groups in characteristic $p$, we know that isomorphism classes of connected smooth $1$-dimensional formal groups, equivalently group scheme structures on $\operatorname{...
Will Sawin's user avatar
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16 votes
3 answers
1k views

Isomorphism between two universal p-typical formal group laws

EDIT: I've tried to alter the question so that its basic nature is clearer, as it's been unclear to a number of people now. At any prime p, there is a graded polynomial ring $V \cong {\mathbb Z}_{(p)}...
Tyler Lawson's user avatar
  • 51.1k
15 votes
2 answers
1k 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 ...
Daniel Hoffmann's user avatar
9 votes
1 answer
726 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 $\mu_{p^\infty}$ ...
sawdada's user avatar
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9 votes
1 answer
687 views

Generalizing detropicalization

Given an identity in max,plus arithmetic, are there ways to turn it into an ordinary algebraic identity it other than by replacing addition by multiplication and replacing max by series-plus or by ...
James Propp's user avatar
  • 19.4k
7 votes
2 answers
692 views

Reference request: Spec A_* is the automorphism group of the additive formal group law

Dear all, I'm seeking a reference for a claim made in lecture 8 of Jacob Lurie's chromatic homotopy theory notes (http://www.math.harvard.edu/~lurie/252xnotes/Lecture8.pdf). More particularly, ...
Saul Glasman's user avatar
  • 2,148
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
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