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 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.

Show description

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

Best algebraic geometry books

Download e-book for kindle: The legacy of Niels Henrik Abel: the Abel bicentennial, by Olav Arnfinn Laudal, Ragni Piene

This booklet encompasses a sequence of analysis papers on topics on the topic of the paintings of Niels Henrik Abel, written by means of a few of the most popular experts of their fields. many of the authors were particularly invited to provide papers, discussing the effect of Abel in a mathematical-historical context.

New PDF release: Tata Lectures on Theta III

The second one in a chain of 3 volumes surveying the speculation of theta capabilities, this quantity provides emphasis to the unique homes of the theta capabilities linked to compact Riemann surfaces and the way they result in ideas of the Korteweg-de-Vries equations in addition to different non-linear differential equations of mathematical physics.

Download e-book for iPad: Logarithmic forms and diophantine geometry by A. Baker, G. Wüstholz

There's now a lot interaction among stories on logarithmic types and deep features of mathematics algebraic geometry. New gentle has been shed, for example, at the well-known conjectures of Tate and Shafarevich on the subject of abelian kinds and the linked celebrated discoveries of Faltings setting up the Mordell conjecture.

Mostly surfaces by Richard Evan Schwartz PDF

This ebook provides a few issues on the topic of surfaces, equivalent to Euclidean, round and hyperbolic geometry, the basic staff, common protecting surfaces, Riemannian manifolds, the Gauss-Bonnet Theorem, and the Riemann mapping theorem. the most notion is to get to a couple fascinating arithmetic with out an excessive amount of formality.

Additional info for Advanced Topics in the Arithmetic of Elliptic Curves

Example text

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.

Download PDF sample

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

by William

Rated 4.84 of 5 – based on 48 votes