By J-P. Serre

This ebook is split into components. the 1st one is only algebraic. Its aim is the class of quadratic kinds over the sector of rational numbers (Hasse-Minkowski theorem). it really is accomplished in bankruptcy IV. the 1st 3 chapters comprise a few preliminaries: quadratic reciprocity legislations, p-adic fields, Hilbert symbols. bankruptcy V applies the previous effects to essential quadratic kinds of discriminant ± I. those kinds happen in numerous questions: modular capabilities, differential topology, finite teams. the second one half (Chapters VI and VII) makes use of "analytic" equipment (holomor phic functions). bankruptcy VI provides the evidence of the "theorem on mathematics progressions" because of Dirichlet; this theorem is used at a serious aspect within the first half (Chapter ailing, no. 2.2). bankruptcy VII bargains with modular types, and specifically, with theta features. a few of the quadratic varieties of bankruptcy V reappear right here. the 2 components correspond to lectures given in 1962 and 1964 to moment yr scholars on the Ecole Normale Superieure. A redaction of those lectures within the type of duplicated notes, used to be made via J.-J. Sansuc (Chapters I-IV) and J.-P. Ramis and G. Ruget (Chapters VI-VII). They have been very valuable to me; I expand the following my gratitude to their authors.

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If the figures P, A, B, C, Ax and A* are considered as point sets and the following statements are valid: P = AuBuC, AnAx = AiaAl9 A n B = B n C = C n A = 0; BnA1 = B29 CnAx = C^. 29 CLOSED SUBPLANE If A is rotated through angles of 2π/21 and 4π/21, it is carried into B and C respectively; moreover the triangles representing second order lines of A are carried into triangles representing second order lines of B and C, respectively. This example suggests the following definition: A subplane of a finite projective plane is a finite projective plane every point of which is a point of the original plane but not conversely; and every line of which is a subset of a line of the original plane.

These form a proper triangle P5PeP7. — We have already seen an example like this on a finite plane, too. Indeed, let us take the quadrangle ΡχΡ^ΡιζΡ9 of Fig. 21. Its diagonal points are the points P5 = P x P 6 n P 1 3 P 9 , P12 = PiPiznP6P9, P8 = P i P 9 n P e P 1 3 and the triangle P5P12PS is a proper triangle. But there exist finite planes in which complete quadrangles may have collinear diagonal points. Let us consider the Fano plane shown in Fig. 22(b). 4* 54 BASIC CONCEPTS In this plane the diagonal points of the quadrangle Ρ4 = Ρ 1 ρ 2 η Ρ 3 Ρ β , P7 = P1P3nP2P6, P^PSPQ Ρδ = are: P1P6nP2Pi.

Our example is a central-axial collineation. 6) il 2 3 4 5 6 7 8 9 10 11 12 13 2 3 4 7 5 6 10 8 9 13 11 12 π= ^ λ= 1 2 3 4 5 6 7 8 9 10 11 12 13 1 2 3 4 6 7 5 9 10 8 12 13 11 induced by π. Thefixedpoint P1 of the collineation (π, λ) is a centre and the invariant line lx is an axis, but now Ρι€/χ. If the centre lies on the axis, the collineation is said to be an elation if the centre and the axis are nonincident, the collineation is said to be an homology. Thus the central-axial collineation given in the first axample is an homology whilst the second example is an elation.

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