Real algebraic tori
Last updated: April 10, 2005
The following data are equivalent:
- a complex algebraic torus T with involution θ
- a real algebraic torus (see here)
- a lattice X with involution θ
It will be convenient to use 3. As explained in Fokko du Cloux's 2004
notes, it is always possible
to find a basis
e1, ... ,ep,
ep+1, ... ,ep+q,
ep+q+1, ... , ep+q+2r
of X such that the vectors e1, ... ,ep are eigenvectors
of θ for the eigenvalue +1, ep+1, ... ,ep+q are
eigenvectors of theta for the eigenvalue -1, and the vectors ej and
ej+r are interchanged by θ for p+q < j ≤ p+q+r.
Here the numbers p, q and r are entirely determined:
-
p+r is the number of +1 eigenvalues of θ (say over the rational numbers);
- q+r is the number of -1 eigenvalues of θ
- and r is computed as follows: θ induces a unipotent automorphism
of the Z/2-vector space X/2X; then r is the dimension of the image of
θ-Id in this vector space.
It follows that the group of real points of T is isomorphic to the product
of p copies of the compact torus U(1), q copies of the multiplicative
group of the reals, and r copies of the multiplicative group of the complex
numbers. In particular, the component group of T(R) is isomorphic to
the product of q copies of Z/2.
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