By Joseph H. Silverman
In the advent to the 1st quantity of The mathematics of Elliptic Curves (Springer-Verlag, 1986), I saw that "the conception of elliptic curves is wealthy, diversified, and amazingly vast," and thus, "many very important issues needed to be omitted." I integrated a short creation to 10 extra issues as an appendix to the 1st quantity, with the tacit realizing that at last there may be a moment quantity containing the main points. you're now retaining that moment quantity. it grew to become out that even these ten subject matters wouldn't healthy regrettably, right into a unmarried e-book, so i used to be pressured to make a few offerings. the next fabric is roofed during this booklet: I. Elliptic and modular capabilities for the whole modular staff. II. Elliptic curves with advanced multiplication. III. Elliptic surfaces and specialization theorems. IV. Neron types, Kodaira-Neron class of unique fibers, Tate's set of rules, and Ogg's conductor-discriminant formulation. V. Tate's concept of q-curves over p-adic fields. VI. Neron's concept of canonical neighborhood peak functions.
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Extra resources for Advanced Topics in the Arithmetic of Elliptic Curves
Note further that since gx is a local isomorphism, we have We must show that F(z) (dz)k is a meromorphic function of w = ZT. 5) that J(T) (dT)k is r(l)-invariant. This implies F(z) (dz)k = J(T) (dT)k = J(g;;I((Z)) (dg;;I((z))k = F((z) (d(z)k = J(RT) (dRT)k = F((z)(k (dz)k. In particular, the function zk F(z) is invariant under the substitution z f-+ (z. Since ( is a primitive rth_root of unity, it follows that for some meromorphic function FI (w). Hence F(z) (dz)k = r-kzk(l-T)F(z) (d(ZT))k = r- k z-1'k FI(zT) (d(ZT))k = r-kw-kFI(w) (dW)k, which proves that J(T) (dT)k descends to a meromorphic k-form wf in a neighborhood of x.
Hence in this case we find that o which completes the proof of (c). Any elliptic function can be factored as a product of Weierstrass ufunctions reflecting its zeros and poles. We give a general result and two important examples. To ease notation, since the lattice A is fixed, we will write u(z) and p(z) instead of u(z; A) and p(z; A). 5. Let fez) be a non-zero elliptic function for the lattice A. Write the divisor of f as r div(f) = L ni(ai) i=1 for some ai E C, and let r b = Lniai. ) Then there is a constant c E C* so that u(z) rrr n· fez) = c u(z _ b).
However, by subtracting an appropriate constant from each term, we can create a series which does converge and has the desired properties. 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.
Advanced Topics in the Arithmetic of Elliptic Curves by Joseph H. Silverman