By Neal I. Koblitz
The idea of elliptic curves and modular types offers a fruitful assembly floor for such diversified parts as quantity thought, complicated research, algebraic geometry, and illustration concept. This publication starts off out with an issue from straight forward quantity thought and proceeds to guide its reader into the fashionable conception, overlaying such themes because the Hasse-Weil L-function and the conjecture of Birch and Swinnerton-Dyer. This new version info the present country of information of elliptic curves.
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Extra info for Introduction to Elliptic Curves and Modular Forms
In what follows, we shall assume that neither PI nor P2 is the point at infinity 0, and that PI # - P2. Then the line through PI and P2 (the tangent line at PI if PI = P2) can be written in the form Y = mx + [3, where m = (Y2 - YI)/(X 2 - Xl) if PI # P2 and m = dy/dxl(xl'Yl if PI = P2. '(x l )/2YI. In both cases the y-intercept is [3 = Yl - mx l · Then x 3 , the x-coordinate of the sum, is the third root of the cubic f(x) - (mx + [3)2, two of whose roots are Xl' x 2. 4). In principle, we could have simply defined the group law by means of these formulas, and then verified algebraically that the axioms of a commutative group are satisfied.
5), we obtain successively: a=4; b= 0; Thus, c = -60G4 , d = -140G 6 . It is traditional to denote 24 I. 1): tJ'(Z)2 = f(tJ(z)), where f(x) = 4x 3 - g2X - g3 E IC[x]. 9), we would obtain relations between the various Gk (see Problems 4-5 below). 9) has an elegant and basic geometric interpretation. , the fundamental parallelogram II with opposite sides glued) to Ifl>l defined by zH(tJ(z),tJ'(z),I) OH(O, 1,0). 9). Here f(x) E IC[x] is a cubic polynomial with distinct roots. Thus, every point z in CjL maps to a point on the elliptic curve y2 = f(x) in Ifl>~.
To carry out this plan, we must expand ,fO(z) and ,fO'(Z)2 near the origin. Since both are even functions, only even powers of z will appear. Let e be the minimum absolute value of nonzero lattice points I. We shall take r < 1, and assume that z is in the disc of radius re about the origin. 1) of ,fO(z). We do this by differentiating the geometric series 1/(1 - x) = 1 + x + x 2 + ... 1), we obtain I ,fO(z) = Z2 z Z2 Z3 + 1Ie L 2+ 3+ 4+ ... + (k [3 14 15 1*0 Zk-2 1)- + ... [k' 23 §6. 2)). To check the claim of absolute convergence of the double series, we write the sum of the absolute values of the terms in the inner sum in the form (recall: Izl < rill): 3 + _r2 4 + _r3 5 + ...
Introduction to Elliptic Curves and Modular Forms by Neal I. Koblitz
Categories: Algebraic Geometry