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Additional resources for Automorphic Forms on GL(3,R)
Exercises 1. Show that, if three pairwise skew lines in PG 3 ✁ F ✂ are given, then it is possible to choose coordinates so that the lines have equations x1 ☎ x2 ☎ 0; x3 ☎ x4 ☎ 0; x3 ☎ x1 ✁ x4 x2 . ☎ Find the common transversals to these three lines. 2. Now let F be commutative. Show that the common transversals to any three of the lines found in the last question are the original three lines and the lines with equations x1 ☎ x3 α, x2 ☎ x4 α for α ✟ F, α ✦☎ 0 ✁ 1. Deduce that the Desarguesian spread defined by a quadratic extension of F is regular.
3. The general case 37 a subspace. Any subspace, equipped with the lines it contains, is a linear space in its own right. A linear space is called thick if it satisfies: (LS1+) Any line contains at least three points. Finally, we will impose Veblen’s Axiom: (V) A line meeting two sides of a triangle, not at a vertex, meets the third side also. 4 (Veblen–Young Theorem) Let ✂ X ✄✪✆✫☎ be a linear space, which is thick and satisfies Veblen’s Axiom (V). Then one of the following holds: (a) X / ✎✷✆✦✎ 0; / (b) ✸ X ✸✹✎ 1, ✆✦✎ 0; (c) ✆✦✎✺✑ X ✓ , ✸ X ✸✼✻ 3; (d) ✂ X ✄✪✆✫☎ is a projective plane; (e) ✂ X ✄✪✆✫☎ is a projective space over a skew field, not necessarily of finite dimension.
The POINTS are the points of X and the parallel classes of lines of ◆ . The LINES are the lines of ✆ and the parallel classes of planes in ◆ . Incidence is hopefully obvious: as in the old space, together with incidence between any line and its parallel class, as well as between a parallel class C of lines and a parallel class ◗ of planes if a plane in ◗ contains a line in C. By Step 2, this is a linear space; and clearly every LINE contains at least three POINTS. , the parallel classes) “ideal”.