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Advanced Topics in the Arithmetic of Elliptic Curves by Joseph H. Silverman

By Joseph H. Silverman

In the advent to the 1st quantity of The mathematics of Elliptic Curves (Springer-Verlag, 1986), I saw that "the idea of elliptic curves is wealthy, different, and amazingly vast," and hence, "many vital themes needed to be omitted." I integrated a quick creation to 10 extra themes as an appendix to the 1st quantity, with the tacit figuring out that at last there should be a moment quantity containing the main points. you're now protecting that moment quantity. it grew to become out that even these ten issues wouldn't healthy regrettably, right into a unmarried booklet, so i used to be compelled to make a few offerings. the next fabric is roofed during this ebook: I. Elliptic and modular features for the entire modular staff. II. Elliptic curves with advanced multiplication. III. Elliptic surfaces and specialization theorems. IV. Neron types, Kodaira-Neron category of targeted 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 conception of canonical neighborhood top functions.

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3aJ. (b) Taking the derivative of PROOF. 2a) we see that the second derivative is -po Note that the logarithms are locally well defined up to the addition of a constant which disappears when we differentiate and also that we must take the principal branch of log ~) for almost all w in order to ensure the convergence of the series. 2b), (1 - d a(z+w;A) dz log a(z; A) = ((z so a(z + Wi A) = + w; A) - ((z; A) = T/(w), Ce'7(w)za(zi A) for some constant C not depending on z. Note also that a is an odd function, a fact that is clear from the product defining a.

This is how we "discovered" p(z; A) in [AEC VI §3]. We apply the same principle to express p(z; r) as a function of u and q. Exponentiating the conditions (i) and (ii), we look for a function F(u; q) satisfying (iii) F( qku; q) = F( u; q) for all u E C*, k E 2; (iv) F(u; q) has a double pole at each u E qZ and no other poles. As above, we look for F to be an average F(u; q) = 2: J(qn u ) nEZ for some elementary function J. Such an F will clearly satisfy the periodicity condition (iii). To obtain (iv), we need J(T) to have a double pole at T = 1.

In order to have a richer source of functions, we will study functions on H that have "nice" transformation properties relative to the action of r(l) on H. Although these transformation properties may look somewhat artificial at first, the corresponding functions actually define differential forms on X(l), so they are in fact natural objects to study. ) 24 I. Elliptic and Modular Functions Definition. Let k E Z, and let f(7) be a function on H. We say that f is weakly modular of weight 2k (for r(l)) if the following two conditions are satisfied: (i) f is meromorphic on H; (ii) f(-y7) = (C7 for all)' = (~ ~) E r(l), + d)2k f(7) 7 E H.

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