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, assorted, and amazingly vast," and hence, "many vital subject matters needed to be omitted." I incorporated a quick creation to 10 extra subject matters as an appendix to the 1st quantity, with the tacit realizing that finally there can be a moment quantity containing the main points. you're now conserving that moment quantity. it became out that even these ten themes wouldn't healthy regrettably, right into a unmarried booklet, so i used to be pressured to make a few offerings. the next fabric is roofed during this e-book: I. Elliptic and modular capabilities for the entire modular staff. II. Elliptic curves with complicated multiplication. III. Elliptic surfaces and specialization theorems. IV. Neron versions, Kodaira-Neron class of detailed fibers, Tate's set of rules, and Ogg's conductor-discriminant formulation. V. Tate's idea of q-curves over p-adic fields. VI. Neron's concept of canonical neighborhood peak functions.
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Additional info for Advanced Topics in the Arithmetic of Elliptic Curves
Elliptic and Modular Functions Next we look at the behavior of G 2k (T) as T ----- ioo. Since the series for G 2k converges uniformly, we can take the limit term-by-term. Terms of the form (mT + n)-2k with m i- 0 will tend to zero, whereas the others give n-2k. Hence 00 lim G 2k (T) = T~'lOO n=-oo 1 ----u; n = 2((2k). n#D o This shows that G 2k is holomorphic at 00 and gives its value. 3. 2) we know that G 4(T) and G 6(T) are modular forms of weights 4 and 6 respectively. 2)) 4 and ((4) = ;0 we find that 92(00) = 120((4) = 47r4 3' ~(oo) = O.
It induces a (complex analytic) isomorphism PROOF. 3), both ~(7) and 92(7)3 = 26 33 53 G 4 (7)3 are modular forms, and both have weight 12, so their quotient is a modular function of weight o. 7a) with k = 0, j defines a meromorphic function on X(l). B. This means that j is meromorphic relative to 35 §4. , JP'1(C). 3). ord oo ~ = 1. Thus j has a simple pole at the cusp 00 E X(l) and no other poles on X(l), so the map j : X(l) ----+ JP'1(C) is an analytic map of degree 1 between compact Riemann surfaces.
Let X/C be a smooth projective curve of genus g, let k 2:: I be an integer, and let w E O'X. (a) Let Kx be a canonical divisor on X [AEC II §4]. Then div(w) is linearly equivalent to kKx . (b) deg(divw) = k(2g - 2). PROOF. (a) Let r] E 01- be a non-zero I-form with divisor diver]) = Kx. Then F = w/r]k E 01- = C(X) 28 I. Elliptic and Modular Functions is a function on X, so div(w) = k div(7]) + div(w/7]k) = kKx + div(F) is linearly equivalent to kKx . (b) From (a), deg(divw} = kdeg(Kx }. 4bJ, which says that deg(Kx) = 2g - 2.
Advanced Topics in the Arithmetic of Elliptic Curves by Joseph H. Silverman