By Torsten Ekedahl
Those lecture notes grew out of a one semester introductory direction on elliptic curves given to an viewers of computing device technology and arithmetic scholars, and imagine in basic terms minimum historical past wisdom. After having coated uncomplicated analytic and algebraic features, placing targeted emphasis on explaining the interaction among algebraic and analytic formulation, they cross directly to a few extra really expert themes. those contain the $j$-function from an algebraic and analytic point of view, a dialogue of elliptic curves over finite fields, derivation of recursion formulation for the department polynomials, the algebraic constitution of the torsion issues of an elliptic curve, complicated multiplication, and modular types. to be able to encourage simple difficulties the publication begins very slowly yet considers a few elements corresponding to modular types of better point which aren't often taken care of. It provides greater than a hundred workouts and a Mathematica TM workstation that treats a few calculations related to elliptic curves. The booklet is geared toward scholars of arithmetic with a basic curiosity in elliptic curves but additionally at scholars of computing device technological know-how drawn to their cryptographic points. A ebook of the ecu Mathematical Society (EMS). disbursed in the Americas by way of the yankee Mathematical Society.
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Additional resources for One semester of elliptic curves
If p1 , p2 , p3 ∈ X are the three intersection points (counted with multiplicity) of a line and X, then p1 + p2 + p3 = 0. 48 5 The group structure on an elliptic curve Let us show how this determines the group structure. If p1 and p2 are two points on X, then we draw a line through them (or the tangent to X at them if they are equal) and let p3 be the third intersection point. , p3 = −(p1 + p2 ) so that in order to get the sum of p1 + p2 we need to find the inverse of p3 . , the line given by its x-coordinate being equal to that of p3 ) and let p4 be the third point of intersection.
We shall see a simpler proof of this fact later. What does this relation mean? Well, we have the following linear algebra exercise. Exercise 20. Show that the three pairs (r, s), (u, v), and (x, y) of C2 (or R2 ) lie on a line precisely when r s 1 u v 1 = 0. x y 1 Hence what the relation says is that if u + v + w = 0, then (℘ (u), ℘ (u)), (℘ (v), ℘ (v)), and (℘ (w), ℘ (w)) lie on a line. Now, u, v, w is the value of the elliptic integral and we know by the addition formula that u + v + w = 0 if (℘ (u), ℘ (u)), (℘ (v), ℘ (v)), and (℘ (w), ℘ (w)) lie on a line.
The homogeneity implies that the condition F (s, t) = 0 depends only on the homogeneous coordinate (s : t) (as it should). By construction the zeroes of F , seen as points on P1 (k) and then as points on , are the points of intersection between and X but we may also attach a multiplicity to each such point. Exercise 32. i) Show that every homogeneous polynomial F (s, t) = 0 over an algebraically closed field factors as i (bi s − ai t)ni , where the points (ai : bi ) are the distinct zeroes of F and the multiplicities ni are uniquely determined by the zero (ai : bi ).