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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 $\operatorname{Spec} \bar{\mathbb F}_p[[x]]$, are in a bijection, defined by the height invariant, with $\mathbb N \cup \infty$ .

I understand the formal groups of height $1$ and $\infty$ quite well - they are the germs of the additive and multiplicative groups. There are extremely simple explicit formulas for the group laws, or rather the group laws of some example.

I understand the formal groups of height $2$ fairly well. They are the germs of supersingular elliptic curves. This gives a procedure to compute the power series, but not a very easy one.

What about formal groups of other heights? Is it possible to give an explicit formula for the coefficients of the power series? How difficult are they to compute? Can the power series be taken to be algebraic functions?

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    $\begingroup$ The height characterizes the isomorphism class only over separably closed ground fields. Lazard's results even give a "universal" 1-dimensional formal group with a given height. This is all explained in Hazewinkel's book on formal groups (and in other references on formal groups). There are not many kinds of 1-dimensional smooth connected commutative group varieties, so what does your final question about "algebraic functions" mean? $\endgroup$
    – user30379
    Commented Mar 9, 2013 at 7:26
  • $\begingroup$ I don't want a universal formal group, I just want one that is a nice formula. For instance, if I'm working with the additive group, there are many laws that are equivalent, but I only ever have any reason to think about the law $x+y$ - it's the easiest for every explicit computation. By algebraic functions, I mean roots of algebraic equations over $F(x,y)$. I don't think see why this would necessarily make it come from a smooth connected commutative group variety. $\endgroup$
    – Will Sawin
    Commented Mar 9, 2013 at 8:13
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    $\begingroup$ Even though a formal group law is a power series in two variables, it’s not clear to me that this is the best or most informative way to think about such a thing. Up in characteristic zero, one gets much more useful information out of the logarithm than out of the two-variable series. $\endgroup$
    – Lubin
    Commented Mar 9, 2013 at 16:17
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    $\begingroup$ (1) Google "Honda formal group." (2) Category theory tells us: don't work up to isomorphism. To expand on Lubin: in char 0, FGLs in bijection with invertible formal power series, the logarithm or invt diff form (easy version of Lazard). Lift your (putative) FGL to char zero, work with its logarithm. Given height n, choose $[p]$, say, $[p]=x^{p^n}$ (Honda), see what that + reduces mod p forces about logarithm of the lift. (3) ACL: no, LT tell you how to lower height, not how to raise it. They don't create examples out of nothing. $\endgroup$
    – Ben Wieland
    Commented Mar 12, 2013 at 0:43
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    $\begingroup$ Thanks! I'll look into that. Your statement "don't work up to isomorphism" is a little confusing to me. Am I interpreting you correctly if I say that, while it's often fine to work up to isomorphism, one should be careful when one passes from a category with more isomorphisms, like formal groups over $\mathbb Q$, to a category with fewer isomorphism, like formal groups over $\mathbb Z$ or $\mathbb Z_p$, e.g. so as to study formal groups over $\mathbb F_p$? $\endgroup$
    – Will Sawin
    Commented Mar 12, 2013 at 3:02

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A few years ago I computed some formal group laws over ${\mathbb F}_2$ of heights 2, 3, and 4. I've just put the resulting pictures online:

I find the patterns fascinating because there is a fractal element to them (patterns are repeated at different scales). I also wonder if one can define a limit that captures the large scale look of these pictures.

As for the mathematical question whether & how higher height formal group laws occur in nature, you might like Jan Stienstra's "Formal group laws arising from algebraic varieties". He computes the formal Brauer group of a K3 surface. If I recall this correctly that can give you formal group laws up to height 10.

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    $\begingroup$ Neat pictures! $\endgroup$
    – Lubin
    Commented Mar 9, 2013 at 16:12
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    $\begingroup$ @ChristianNassau Are these the Morava ones? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Oct 29, 2013 at 19:49
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    $\begingroup$ If my memory is correct I was computing the FGLs with $p$-series $[2]_F(x) = x^{2^n}$. Morava usually calls these the Honda formal group laws. I think you can find a complex orientation for Morava's $K(n)$ that gives you exactly these FGLs. $\endgroup$
    – Christian Nassau
    Commented Oct 31, 2013 at 14:10
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    $\begingroup$ Sorry, I switched providers and forgot to copy the files. They're back now (and have been transformed to pdfs). $\endgroup$
    – Christian Nassau
    Commented Jul 16, 2014 at 15:12
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    $\begingroup$ The formal group is a $F(x,y) = \sum a_{i,j} x^iy^j$ with every $a_{i,j}$ either $0$ or $1$. The picture has black dots for every $(i,j)$ where $a_{i,j}=1$. The origin of the coordinate system is at the bottom left. $\endgroup$
    – Christian Nassau
    Commented Jul 19, 2015 at 5:53

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