| Abstract:
Let k be an infinite field.
A group law over k is an element f(x,y)
in the field k(x,y) of rational functions over k
such that the binary
algebra (k-S; f(x,y)) is a group for
some 'forbidden' subset S of k.
Here are some well-known examples of such group laws:
- the additive group where S = the empty set and and
f(x,y) = x+y
- the multiplicative group where S = {0}
and f(x,y) = xy
- the circle-group where S = {-1} and
f(x,y) = x+y+xy
- the geometric group law induced by a nodal cubic
over k: here S = {-1,1} and
f(x,y) = (1+xy)/(x+y),
These are special cases of the so-called one-dimensional affine
groups (e.g. 2.6.6, page 66 in [1]). In all the above examples, the
cardinality of the forbidden set S is less than 3. These are
precisely the 'singular' elements where the cancellation law for the
corresponding f(x,y) fails. Is it possible to have a group
law on k-S
for some set S with, say |S|= 3?
For example, is there a polynomially
defined group law on the set R - {0,1,2} where
R is the field of real
numbers? This is not possible. By sharp contrast, if the
field k is
finite then we do have rational group laws on the set
k-S for every
subset S of the field k.
- T. A. Springer, Linear Algebraic Groups, Birkhauser Verlag,
Stuttgart, 1981.
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