Welcome to MathOverflow

Question

0

1

Let G be an affine algebraic group defined over k, with k a field of characteristic zero. Suppose G has only one single k-point, can we conclude that G does not have more points?

flag	asked Jul 14 2010 at 17:16 Ana 75●3

4

I can't see what you are asking here. You say $G$ has just one $k$-point and then ask if it has any more. Are you sure this is what you wanted to ask? – Robin Chapman Jul 14 2010 at 17:18

3

Here is a simple counterexample: $k=\mathbb{Q}$, and $G = \mu_3$, the group of third roots of unity. The underling scheme of $G$ has only one $\mathbb{Q}$ point, but three $\mathbb{C}$ points. – David Speyer Jul 14 2010 at 17:38

2

Do you intend to insist that $G$ is connected (which would rule out David Speyer's example), or equivalently geometrically connected? Anyway, the answer is still negative (assuming $G \ne 1$!): over fields of characteristic 0, every smooth connected affine group is unirational and hence has a Zariski-dense locus of rational points. This relies crucially on char. 0, as well as structural facts from the theory of connected reductive groups. So if $G$ is of positive dimension, the answer is "no" (by consideration of its identity component). – BCnrd Jul 14 2010 at 18:10

Maybe I'm missing something. Doesn't that mean the answer to the OP's question is yes in the connected case? – Kevin Ventullo Jul 14 2010 at 20:36

2

@Kevin: the question is asked as a negative ("can we conclude...does not..."), so it may be confusing, but I think the answer is "no"; anyway, the content of the answer is clear (there are more $k$-pts!). Here is a proof valid for non-unipotent smooth affine groups of positive dimension over any infinite field $k$: by Grothendieck, such groups always have a non-trivial $k$-torus, and those are unirational, QED. A variant works over any infinite perfect field in the unipotent case. But over imperfect fields it can fail: over $k(t)$ for $k$ of char. $p > 2$, take $G = {y^p = x - t x^p}$. – BCnrd Jul 14 2010 at 23:05

show 4 more comments

Rational points

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.