By Joseph H. Silverman

ISBN-10: 0387943285

ISBN-13: 9780387943282

ISBN-10: 1461208513

ISBN-13: 9781461208518

In the creation to the 1st quantity of The mathematics of Elliptic Curves (Springer-Verlag, 1986), I saw that "the concept of elliptic curves is wealthy, diversified, and amazingly vast," and subsequently, "many very important themes needed to be omitted." I incorporated a quick advent to 10 extra themes as an appendix to the 1st quantity, with the tacit realizing that at last there could 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 publication, so i used to be pressured to make a few offerings. the next fabric is roofed during this e-book: I. Elliptic and modular services for the total modular team. II. Elliptic curves with complicated multiplication. III. Elliptic surfaces and specialization theorems. IV. Neron types, Kodaira-Neron type of precise fibers, Tate's set of rules, and Ogg's conductor-discriminant formulation. V. Tate's thought of q-curves over p-adic fields. VI. Neron's idea of canonical neighborhood top functions.

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We set the notation M2k = {modular forms Mgk = of weight 2k for r(1)}, {cusp forms of weight 2k for reI)}. Note that both M2k and Mgk are C-vector spaces. 9. For all k ~ 2, the Eisenstein series G 2k(r) is in M2k but is not in Mgk • The modular discriminant 6(r) is in MP2. 3). 10. (a) For all integers k ~ 2, M2k ~ Mgk +CG 2k· (b) For all integers k, the map is an isomorphism of C- vector spaces. (c) The dimension of M2k as a C-vector space is given by dim M2k ={ [~/6J [k/6 + IJ if k < 0; if k ~ 0, k == 1 (mod 6); if k ~ 0, k =t 1 (mod 6).

1. If J is a modular function, then it is easy to see that the order of vanishing of J at 7 E H depends only on the r(l)-equivalence class of 7. The point is that since J(7) = (cr + d)2k J(7) and cr + die 0, we have ordr(J) = ord r (J 0 ')'-1) = ord"Yr(J). 7b) really does not depend on the choice of the representative 7 x . PROOF. (a) As we have seen, the k-form J(7) (d7)k is invariant for the action of r(l) on H. We must show that for each x = ¢(7x ) E X(l), the k-form J(7) (d7}k descends locally around x to a meromorphic k-form on X(l), and that it vanishes to the indicated order.

It follows that G(z) = cz for some constant c E IC. ) Since the r-fold composition Go··· 0 G(z) = z and r is chosen minimally, we conclude that c is a primitive rth-root of unity. 5. Suppose first that x =I 00. 5), I(Tx) is cyclic, say generated by R. 6) implies that gX(RT) = (g(T) for all T E H, where ( is a primitive rth-root of unity. Hence so 'l/Jx is well defined on the quotient I(Tx )\Ux ' Next we check that 'l/Jx is injective. Let T1, T2 E Ux. Then 'l/Jx(¢(Td) = 'l/Jx(¢(T2)) ~ gx(Td T = gx(T2)'" ~ :s: i < r, gx(Td = gx(RiT2) for some 0 :s: i < r, T1 = RiT2 for some 0 :s: i < r, ~ ¢(T1) = ¢(T2)' ~ gx(T1) = (igx(T2) ~ for some 0 Hence 'l/Jx is injective.

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

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