By Neal I. Koblitz

The concept of elliptic curves and modular varieties presents a fruitful assembly flooring for such different components as quantity idea, advanced research, algebraic geometry, and illustration concept. This publication starts off out with an issue from trouble-free quantity concept and proceeds to steer its reader into the trendy concept, masking such issues because the Hasse-Weil L-function and the conjecture of Birch and Swinnerton-Dyer. This re-creation info the present nation of information of elliptic curves.

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B) In the case of the elliptic curve y2 = x 3 - n2 x find an explicit formula for these x-coordinates. Show that they are never rational (for any n). 5. Given a point Q on an elliptic curve, how many points P are there such that 2P = Q? Describe geometrically how to find them. 6. Show that if K is any subfield of IC containing g2 and g3, then the points on the elliptic curve y2 = 4x 3 - g2X - g3 whose coordinates are in K form a subgroup 36 I. From Congruent Numbers to Elliptic Curves of the group of all points.

C) If the function v = e Z is defined by the usual series, use part (b) to show that e is the inverse function of z = log v. (d) Show that the map e Z gives a one-to-one correspondence between C/L and IC - {O}. Under this one-to-one correspondence, the additive group law in IC/L becomes what group law in IC - {O}? Z 10. J(z; L). J(z + a) for some constant a. 11. J(z; L). Let RI be an unbounded simply connected open region in the complex plane which does not contain the roots e l , e z , e 3 of the cubic 4x 3 - g2 X - g3' 28 I.

Setfl (z) = 1. Prove that for N = 2, 3, 4, ... we have: p(Nz) = p(z) - fN-l (Z)fN+l (Z)/fN(Z)z. 4. In the notation of Proposition 14, suppose that (JEGal(KN/K) fixes all x-coordinates of points of order N. That is, (JI K/i = identity. Show that the image of (J in GL z (lL/ N lL) is ± 1. Conclude that Gal(KN/K;) = {± I} n G, where G is the image of Gal (KN/K) in GLz(lL/NlL). What is the analogous situation for cyclotomic fields? 5. Let L = {mw l + nw z }, and let E be the elliptic curve yZ = 4x 3 - g2(L)x - g3(L).