By Joseph H. Silverman

In the creation to the 1st quantity of The mathematics of Elliptic Curves (Springer-Verlag, 1986), I saw that "the idea of elliptic curves is wealthy, diverse, and amazingly vast," and subsequently, "many very important issues needed to be omitted." I integrated a short advent to 10 extra issues as an appendix to the 1st quantity, with the tacit knowing that at last there can be a moment quantity containing the main points. you're now conserving that moment quantity. it became out that even these ten subject matters wouldn't healthy regrettably, right into a unmarried booklet, so i used to be compelled to make a few offerings. the subsequent fabric is roofed during this e-book: I. Elliptic and modular features for the total modular workforce. II. Elliptic curves with advanced multiplication. III. Elliptic surfaces and specialization theorems. IV. Neron types, Kodaira-Neron category of unique fibers, Tate's set of rules, and Ogg's conductor-discriminant formulation. V. Tate's conception of q-curves over p-adic fields. VI. Neron's thought of canonical neighborhood top functions.

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**Example text**

G = {lattices in C} from §1. Much of our preceding discussion is summarized in the following proposition. 4. GjC* {A} = {AT} +- r(1)\H +- T 37 §4. Uniformization and Fields of Moduli Here AT = ZT + Z, {EA} denotes the C-isomorphism class of the elliptic curve EA : y2 = 4x 3 - g2(A)x - g3(A), and {A} is the homothety class of the lattice A. PROOF. 1). 2bc). 3). 4. Let {E} E e££c be an isomorphism class of elliptic curves, and choose a Weierstrass equation E : y2 = x 3 + Ax + B for some curve E in this class.

1 - qn u)2 27riu 27ri (1 - qnu) Unfortunately, the series is clearly divergent, the nth term goes to 1 as n ---t 00. But just as in the original definition of ~, we can improve the convergence by adding a constant onto each term. 2a) has the form 1 ""' (27ri)2~(Z;T) = ~ n2:0 where qnu (1- qnu)2 +L n2:1 qnu-l (1- qnu-I)2 + GI , 52 1. ) Now integrate: for some constant of integration C 2 = C 2 (q). Note that the last series is absolutely and uniformly convergent on compact subsets of C " AT, so it defines a meromorphic function on C.

4. GjC* {A} = {AT} +- r(1)\H +- T 37 §4. Uniformization and Fields of Moduli Here AT = ZT + Z, {EA} denotes the C-isomorphism class of the elliptic curve EA : y2 = 4x 3 - g2(A)x - g3(A), and {A} is the homothety class of the lattice A. PROOF. 1). 2bc). 3). 4. Let {E} E e££c be an isomorphism class of elliptic curves, and choose a Weierstrass equation E : y2 = x 3 + Ax + B for some curve E in this class. ) Switching WI TE and W2 W2 WI =- W2 = 1'1,)'2 1,2 for the homology dx Y if necessary, we may assume that E H.