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I was looking to classify non trivial, commutative, associative, group structures on $\mathbb{R}$ (minus a countable number of point) starting with the trivial ones

$$ (\mathbb{R} , + )$$ $$ (\mathbb{R} - \lbrace 0 \rbrace, \times) $$ and then came across the following operation

$$ \mu (a,b ) = a + b + ab $$

$\mu$ is commutative, associative, has identity $0$, and every element is invertible with $a^{-1_\mu} = - \frac{a}{1+a}$

$$ (\mathbb{R} - \lbrace -1 \rbrace , \mu) $$

Then also forms a group.

I then noticed the following rather odd observation:

$$ x +y = 0 $$ $$ xy = 1$$ $$ x + y + xy = 0$$

All form hyperbolas (the first being a degenerate case) when graphed as implicit relations.

And that got me wondering, does every hyperbola come equipped with a natural arithmetic (and given that all ellipses are complex hyperbolas) do all non-parabolic conic sections come equipped with a natural arithmetic?

Sidharth Ghoshal
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  • It's unclear what you're asking. Also note that $\mu$ is just multiplication shifted by 1. – jgon Jan 05 '18 at 08:31
  • By it's unclear what you're asking, I mean are you asking whether $F(x,y) = ax^2+bxy+cy^2 + dx+ey+f$ defines a group structure? For $b^2-4ac \ne 0$ that is. Or hm. I'm very confused as to what you're asking. – jgon Jan 05 '18 at 08:35
  • yes, what i'm asking is a bit subjective, but it can be restated as does $F(x,y) = ax^2 +bx + cy^2 + dx + ey + f $ define a group structure, where $x+y$ yields the $(a,b) \rightarrow a+b$ operation, $xy-1$ yields the $(a,b ) \rightarrow ab$ operation and $x+y+xy$ yields the $(a,b) \rightarrow a+b+ab$ operation. – Sidharth Ghoshal Jan 05 '18 at 08:41
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    You shouldn't consider transferring properties from hyperbolas to ellipses by using complexification. The good approach is to use projective transformations. In this way you remain all the time with real coefficients. – Jean Marie Jan 05 '18 at 08:55

1 Answers1

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There are natural geometric group laws on (projective) conics and cubics (additional bonus is that the operation preserves the subsets of rational points on them, and it is the restriction to those that are most commonly studied). For conics fix any point $O$ (neutral element) on it and given two points $A$, $B$ on the conic draw a line parallel to $AB$ through $O$. In the projective plane it will have exactly one other point of intersection with the conic, which is defined to be the sum. For hyperbolas the operation preserves the subset of real points (including at infinity, which can also be made into $O$). Addition on conics is discussed in Lemmermeyer's Conics - a Poor Man's Elliptic Curves.

The specific algebraic expression for the operation is highly indeterminate, however. Given some self-bijection $f$ of $\mathbb{R}$ you can get a new addition by simply setting $a\oplus b=f(f^{-1}(a)+f^{-1}(b))$, and similarly with multiplication, except $f$ need not be defined at $0$. A (real part of) hyperbola can be mapped bijectively onto $\mathbb{R} - \textrm{pt}$. You can play around with choices of $O$ and such a map (projection to one of the axes, say) that converts geometric addition on the hyperbola into your specific formulas. However, your multiplication examples will not be uniform with the addition one because $x+y=0$ is not a hyperbola, or even a conic. Of course one can trivially define geometric addition on a line too by picking the origin $O$ and splicing segments accordingly, and map it into the standard one on $\mathbb{R}$ by scaled projection.

Conifold
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