Katz mazur arithmetic moduli of elliptic curves pdf

The notion of elliptic curves over arbitrary schemes is indispensable for the. The arithmetic study of the moduli spaces began with jacobis fundamenta nova in 1829, and the modern theory was erected. Lfunctions of universal elliptic curves over igusa curves. Good introductory references on moduli stacks, for. Rubin arithmetic theory of elliptic urveslectures given at the 3rd session of th. These keywords were added by machine and not by the authors. Global results buium, alexandru and poonen, bjorn, duke mathematical journal, 2009. I was reading through katz mazur arithmetic moduli of elliptic curves, chapter 1, and ran into some small issues which might have a lot to do actually with notation. In km, katz and mazur developed the theory of drinfeld level structures on elliptic curves over arbitrary. Browse other questions tagged ellipticcurves arithmeticgeometry or. The arithmetic of elliptic curves is a graduatelevel textbook designed to introduce the reader to an important topic in modern mathematics. Pdf integral points on moduli schemes of elliptic curves. Arithmetic moduli of elliptic curves or to study duality of abelian varieties. In what follows, constant use will be made of the notations and results of katzmazur 4.

The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. Elliptic curves, lfunctions, and cmpoints shouwu zhang department of mathematics columbia university new york, ny 10027 july 11, 2002 contents 1 elliptic curves. Mazurs first proof of this theorem depended upon a complete analysis of the rational points on certain modular curves. Since elliptic curves are front and center of this story, it is appropriate to. I then proved in a fair amount of detail that the moduli problem of elliptic curves with. There are some applications of compactifications of the moduli spaces. Katz, barry mazur, arithmetic moduli of elliptic curves, annals of mathematics studies, vol. Independence of points on elliptic curves arising from special points on modular and shimura curves, i. Nakamura 12 constructed two compactifi cations of the moduli space of abelian varieties with noncommutative levels and proved that the first compactified moduli consists of all the. Ellr will denote the category whose objects are elliptic curves e s. There are two things that make it difficult to read.

This work is a comprehensive treatment of recent developments in the study of elliptic curves and their moduli spaces. I was reading through katzmazur arithmetic moduli of elliptic curves, chapter 1, and ran into some small issues which might have a lot to do actually with notation. This book is devoted to giving an account of the arithmetic theory of the moduli spaces of elliptic curves. This is the first lecture on the arithmetic moduli theory of elliptic curves. I was reading through katzmazur arithmetic moduli of elliptic curves, chapter 1, and ran into some small issues which might have a lot to do. Mazurs theorem is a wellknown and important result, however it is not often taught in classroom settings.

The arithmetic study of the moduli spaces began with jacobis fundamenta nova in 1829, and the modern theory was erected by eichlershimura, igusa, and delignerapoport. Nicholas michael katz born december 7, 1943 is an american mathematician, working in arithmetic geometry, particularly on padic methods, monodromy and moduli problems, and number theory. Whether youve loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. Chales rezk on the book by nicholas katz and barry mazur entitled arithemtic moduli of elliptic curves. In any case, the theory of picard schemes is indispensable in arithmetic geometry, e. In the past decade mathematicians have made further substantial progress in the field. Here the n marked points and the anchor points of the r tangent vectors are distinct. Arithmetically cohenmacaulay reducible curves in projective spaces. View the article pdf and any associated supplements and figures for a period of 48 hours. We are constantly discovering new ways of understanding algebraic curves and their arithmetic properties. Katz and barry mazur princeton university press princeton, n. The essential fact about elliptic curves with a 5torsion point sketched in the above paragraphs is that for any scheme s, there is a bijection between, on the one hand, isomorphism classes of pairs e,p of an elliptic curve esand a section p of exact order 5 in all geometric.

Arithmetic moduli of elliptic curves by nicholas m. Save up to 80% by choosing the etextbook option for isbn. In km, katz and mazur developed the theory of drinfeld level structures on elliptic curves over arbitrary schemes, thereby removing the. Elliptic and modular curves over rings springerlink. He is currently a professor of mathematics at princeton university and.

Arithmetic moduli of elliptic curves katz, nicholas m. Pdf we combine the method of faltings arakelov, parsin, szpiro with the shimurataniyama conjecture to prove effective finiteness results for integral points on moduli schemes of elliptic. Questions about rational pointsthe interplay of arithmetic and algebrahave fascinated mathematicians from diophantus to the present. Modular curves at infinite level 3, katz and mazur pose a moduli problem for elliptic curves and level structures which turns out to be representable much of the time. These informal notes are an expanded version of lectures on the moduli space of elliptic curves given at zhejiang university in july, 2008. Other readers will always be interested in your opinion of the books youve read. Their goal is to introduce and motivate basic concepts and constructions such as orbifolds and stacks important in the study of moduli spaces of curves and abelian varieties through the example of elliptic curves. This breakthrough has farreaching consequences for the arithmetic of elliptic curves. Height and arithmetic intersection for a family of semistable curves kawaguchi, shu, journal of mathematics of kyoto university, 1999. Mazurs torsion theorem, which gives a complete list of the possible torsion subgroups of elliptic curves over the rational numbers, is a deep and important result in the arithmetic of elliptic curves. I began by explaining why the natural moduli problem for elliptic curves is not representable by a scheme. When there is enough level to remove nontrivial isotropy groups, the work in km constructs a.

Arithmetic moduli of elliptic curves pdf free download epdf. We illustrate the theory of log abelian varieties and their moduli in the case of log elliptic curves. We study the multiplicationbyp map on an elliptic curve, which gives a strati. State of the art in arithmetic moduli of elliptic curves. Oberseminar modular curves and the eisenstein ideal. I think most of them are due to not being comfortable enough with the algebraic geometry notions involved. Barry mazur and publisher princeton university press. It is certainly true that some of the deepest results in the subject, such as mazurs theorem bounding torsion over q and faltings proof of the isogeny conjecture, re. I will give a survey of current approaches, results, and conjectures in this vibrant subject. Arithmetic moduli of elliptic curves annals of mathematics studies 108 john coates. In chapter 6 a topic from the arithmetic of elliptic curves is discussed. Depending on the restrictions applied to the classes of algebraic curves considered, the corresponding moduli problem and the moduli space is different. Math 679 is a graduate level mathematics course whose purpose is to prove mazurs theorem.

Available formats pdf please select a format to send. Wiles theorem and the arithmetic of elliptic curves. In algebraic geometry, a moduli space of algebraic curves is a geometric space typically a scheme or an algebraic stack whose points represent isomorphism classes of algebraic curves. Richard hain, lectures on moduli spaces of elliptic curves arxiv.