Advanced Topics in the Arithmetic of Elliptic Curves - download pdf or read online

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.

Show description

Read or Download Advanced Topics in the Arithmetic of Elliptic Curves PDF

Best algebraic geometry books

Read e-book online Complex functions: An algebraic and geometric viewpoint PDF

Elliptic capabilities and Riemann surfaces performed a massive function in nineteenth-century arithmetic. this day there's a nice revival of curiosity in those subject matters not just for his or her personal sake but additionally due to their functions to such a lot of components of mathematical examine from staff concept and quantity concept to topology and differential equations.

Download e-book for kindle: Concise course in algebraic topology by J. P. May

Algebraic topology is a easy a part of sleek arithmetic, and a few wisdom of this zone is vital for any complicated paintings when it comes to geometry, together with topology itself, differential geometry, algebraic geometry, and Lie teams. This booklet presents a close therapy of algebraic topology either for lecturers of the topic and for complex graduate scholars in arithmetic both focusing on this region or carrying on with directly to different fields.

Extra info for Advanced Topics in the Arithmetic of Elliptic Curves

Sample text

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.

Download PDF sample

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


by Kevin
4.3

Rated 4.09 of 5 – based on 35 votes