TAILIEUCHUNG - Introduction to arithmetic geometry

To help you have more documents to serve the needs of learning and research, invite you to consult the "Introduction to arithmetic geometry" below. Hope content useful document serves the academic needs and research. | Fall 2013 09 10 2013 Introduction to Arithmetic Geometry Lecture 2 Plane conics A conic is a plane projective curve of degree 2. Such a curve has the form C k ax2 by2 cz2 dxy exz fyz with a b c d e f 2 k. Assuming the characteristic of k is not 2 we can make d e f 0 via an invertible linear transformation. First if a b c 0 we can make one of them nonzero by replacing a variable by its sum with another in this case one of d e f must be nonzero say d and then replacing y with x y yields an equation with a 0. So assume without loss of generality that a 0. Replacing x with x 2ay kills the xy term and we can similarly kill the xz term by replacing x with x 2az we are just completing the square . Finally if f 0 we can make b nonzero and then replace y with y 2bz to eliminate the yz term. Each of these substitutions corresponds to an invertible linear transformation of the projective plane as does their composition. So we now assume char k 2 and that C has the diagonal form ax2 by2 cz2 0. 1 If any of the coefficients a b c are zero then this curve is not 2 For example if the coefficient c is zero we can factor the LHS of 1 over k ax2 by2 ựãx y byXx ax V by 0. In this case C k is the union of two projective lines that intersect at 0 0 1 but C k might contain only one point as when k Q and a b 0 for example . We now summarize this discussion with the following theorem. Theorem . Over a field whose characteristic is not 2 every geometrically irreducible conic is isomorphic to a diagonal curve ax2 by2 cz2 0 with abc 0. Remark . This does not hold in characteristic 2. Parameterization of rational points on a conic Suppose xo yo zo is a rational point on the diagonal conic C ax2 by2 cz2 0. Without loss of generality we assume z0 0 and consider the substitution x xoW U y yoW V z zoW 2 1In Lecture 1 we defined a plane projective curve f x y z 0 to be reducible if f gh for some g h 2 k x y z where k is the algebraic closure of k. Some authors .

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