By Joseph H. Silverman

ISBN-10: 0387943285

ISBN-13: 9780387943282

ISBN-10: 1461208513

ISBN-13: 9781461208518

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.

Show description

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

Best algebraic geometry books

Get Computational Commutative Algebra 1 PDF

This creation to polynomial jewelry, Gröbner bases and functions bridges the distance within the literature among idea and genuine computation. It information various functions, protecting fields as disparate as algebraic geometry and fiscal markets. to assist in a whole figuring out of those purposes, greater than forty tutorials illustrate how the speculation can be utilized.

Read e-book online Novikov Conjectures, Index Theorems, and Rigidity PDF

The Novikov Conjecture is the one most vital unsolved challenge within the topology of high-dimensional non-simply attached manifolds. those volumes are the outgrowth of a convention held on the Mathematisches Forschungsinstitut Oberwolfach (Germany) in September, 1993, with reference to `Novikov Conjectures, Index Theorems and Rigidity'.

Ramanujan's Lost Notebook - download pdf or read online

Within the spring of 1976, George Andrews of Pennsylvania nation collage visited the library at Trinity collage, Cambridge, to check the papers of the overdue G. N. Watson. between those papers, Andrews stumbled on a sheaf of 138 pages within the handwriting of Srinivasa Ramanujan. This manuscript was once quickly specified, "Ramanujan's misplaced pc.

Extra resources for Advanced Topics in the Arithmetic of Elliptic Curves

Sample text

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.

Download PDF sample

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


by Paul
4.5

Rated 5.00 of 5 – based on 49 votes