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Galois Theory of Rigid Correspondences.

In this section we set up the Galois theory and birational geometry that will be used to describe those field extensions that can be reached by a tower of generally convergent algorithms.

All varieties will be irreducible and complex projective. Let V be a variety, k = K(V) its function field.

An irreducible polynomial p in k[z] determines a finite field extension tex2html_wrap_inline1761 , where tex2html_wrap_inline1763 is a root of p; the extension is unique up to isomorphism over k.

To obtain a geometric picture for the field extension, consider p(z) as a family of polynomials tex2html_wrap_inline1771 whose coefficients are rational functions of v. The polynomial p determines a subvariety tex2html_wrap_inline1777 which is the closure of the set of (v,z) such that tex2html_wrap_inline1781 . The function field tex2html_wrap_inline1783 where tex2html_wrap_inline1785 denotes the rational function obtained by projecting W to tex2html_wrap_inline1789 .

W may be thought of as the graph of a multi-valued function W(v) which sends v to the roots of tex2html_wrap_inline1797 . We call such a multi-valued map a rational correspondence.

We say W is a rigid correspondence if its set of values assumes only one conformal configuration on the Riemann sphere: i.e. there exists a finite set tex2html_wrap_inline1801 such that the set W(v) is equal to tex2html_wrap_inline1805 for some Möbius transformation tex2html_wrap_inline1807 depending on v. In this case we say the field extension tex2html_wrap_inline1811 is a rigid extension.

Now let k' denote a finite Galois extension of k with Galois group G.

T 2.1. The field extension k'/k is the splitting field of a rigid extension if and only if there exists:

a faithful homomorphism tex2html_wrap_inline1821 and
an element tex2html_wrap_inline1823 in tex2html_wrap_inline1825 such that
tex2html_wrap_inline1827 for all g in G.

Proof. Let k' be the splitting field of a rigid correspondence tex2html_wrap_inline1835 . For simplicity, assume tex2html_wrap_inline1837 is at least 3. Let tex2html_wrap_inline1839 , i = 1,2,3 denote three distinct conjugates of tex2html_wrap_inline1841 under G. tex2html_wrap_inline1845 acts triply transitively on the projective line tex2html_wrap_inline1847 ; take tex2html_wrap_inline1849 to be the unique group element which moves tex2html_wrap_inline1851 to tex2html_wrap_inline1853 .

We claim that tex2html_wrap_inline1855 is in tex2html_wrap_inline1857 for all g in G. Indeed, tex2html_wrap_inline1863 is just the cross-ratio of tex2html_wrap_inline1865 and tex2html_wrap_inline1867 , which is constant by rigidity. Let tex2html_wrap_inline1869 be the image under tex2html_wrap_inline1871 of the conjugates of tex2html_wrap_inline1873 .

Define tex2html_wrap_inline1875 . Then tex2html_wrap_inline1877 permutes A, so it is an element of tex2html_wrap_inline1881 . Because G acts trivially on tex2html_wrap_inline1885 , tex2html_wrap_inline1887 is a homomorphism; e.g.


and since tex2html_wrap_inline1889 fixes A pointwise only if g fixes the conjugates of tex2html_wrap_inline1895 , it is faithful; thus we have verified (a-c).

Conversely, given the data (a-c), set tex2html_wrap_inline1897 for any x in tex2html_wrap_inline1901 with trivial stabilizer in tex2html_wrap_inline1903 ; then tex2html_wrap_inline1905 is rigid over k and tex2html_wrap_inline1909 .

Cohomological Interpretation. The map tex2html_wrap_inline1911 determines an element tex2html_wrap_inline1913 of the Galois cohomology group tex2html_wrap_inline1915 , which is naturally a subgroup of the Brauer group of k; condition (c) simply says tex2html_wrap_inline1919 is the coboundary of tex2html_wrap_inline1921 , so tex2html_wrap_inline1923 .

A geometric formulation of the vanishing of this class is the following. Let tex2html_wrap_inline1925 denote the rational map of varieties corresponding to the field extension tex2html_wrap_inline1927 . Form the Severi-Brauer variety tex2html_wrap_inline1929 , where G acts on W by birational transformations and on tex2html_wrap_inline1935 via the representation tex2html_wrap_inline1937 . Then tex2html_wrap_inline1939 is a flat tex2html_wrap_inline1941 bundle outside the branch locus of the map tex2html_wrap_inline1943 . We can factor tex2html_wrap_inline1945 through the inclusion tex2html_wrap_inline1947 for any x in tex2html_wrap_inline1951 with trivial stabilizer.

The cohomology class of tex2html_wrap_inline1953 vanishes if and only if tex2html_wrap_inline1955 is birational to tex2html_wrap_inline1957 ; in which case tex2html_wrap_inline1959 presents W as a rigid correspondence.

More on Galois cohomology and interpretations of the Brauer group can be found in [6], [7] and [14].

next up previous
Next: Purely Iterative Algorithms. Up: Solving the quintic by Previous: Introduction.

Peter Doyle