Katz mazur arithmetic moduli of elliptic curves pdf

We are constantly discovering new ways of understanding algebraic curves and their arithmetic properties. Elliptic and modular curves over rings springerlink. Mazurs first proof of this theorem depended upon a complete analysis of the rational points on certain modular curves. Available formats pdf please select a format to send. Depending on the restrictions applied to the classes of algebraic curves considered, the corresponding moduli problem and the moduli space is different. Ellr will denote the category whose objects are elliptic curves e s. 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. Arithmetic moduli of elliptic curves by nicholas m. Katzmazur chapter 1 ag questions mathematics stack exchange. Katz, barry mazur, arithmetic moduli of elliptic curves, annals of mathematics studies, vol. Richard hain, lectures on moduli spaces of elliptic curves arxiv.

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. Math 679 is a graduate level mathematics course whose purpose is to prove mazurs theorem. The arithmetic study of the moduli spaces began with jacobis fundamenta nova in 1829, and the modern theory was erected. I managed to read on ignoring them but they are still bugging me.

Height and arithmetic intersection for a family of semistable curves kawaguchi, shu, journal of mathematics of kyoto university, 1999. Denote the moduli stack over specz of smooth elliptic curves with n marked points and r nonzero tangent vectors by m1. Elliptic curves, lfunctions, and cmpoints shouwu zhang department of mathematics columbia university new york, ny 10027 july 11, 2002 contents 1 elliptic curves. This process is experimental and the keywords may be updated as the learning algorithm improves. I think most of them are due to not being comfortable enough with the algebraic geometry notions involved. Chales rezk on the book by nicholas katz and barry mazur entitled arithemtic moduli of elliptic curves.

Here the n marked points and the anchor points of the r tangent vectors are distinct. Other readers will always be interested in your opinion of the books youve read. 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. Barry mazur and publisher princeton university press. I will give a survey of current approaches, results, and conjectures in this vibrant subject. This is the first lecture on the arithmetic moduli theory of elliptic curves. Lfunctions of universal elliptic curves over igusa curves. In km, katz and mazur developed the theory of drinfeld level structures on elliptic curves over arbitrary.

When there is enough level to remove nontrivial isotropy groups, the work in km constructs a. Arithmetic moduli of elliptic curves or to study duality of abelian varieties. Save up to 80% by choosing the etextbook option for isbn. In the past decade mathematicians have made further substantial progress in the field.

Questions about rational pointsthe interplay of arithmetic and algebrahave fascinated mathematicians from diophantus to the present. 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. 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. Arithmetic moduli of elliptic curves annals of mathematics studies 108 john coates. Mazurs theorem is a wellknown and important result, however it is not often taught in classroom settings. Rubin arithmetic theory of elliptic urveslectures given at the 3rd session of th. I then proved in a fair amount of detail that the moduli problem of elliptic curves with. 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. Arithmetic moduli of elliptic curves katz, nicholas m. I began by explaining why the natural moduli problem for elliptic curves is not representable by a scheme.

Global results buium, alexandru and poonen, bjorn, duke mathematical journal, 2009. There are some applications of compactifications of the 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. We study the multiplicationbyp map on an elliptic curve, which gives a strati. These keywords were added by machine and not by the authors. This breakthrough has farreaching consequences for the arithmetic of elliptic curves. Good introductory references on moduli stacks, for. This book is devoted to giving an account of the arithmetic theory of the moduli spaces of elliptic curves.

In km, katz and mazur developed the theory of drinfeld level structures on elliptic curves over arbitrary schemes, thereby removing the. We illustrate the theory of log abelian varieties and their moduli in the case of log elliptic curves. Wiles theorem and the arithmetic of elliptic curves. View the article pdf and any associated supplements and figures for a period of 48 hours. 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. Arithmetically cohenmacaulay reducible curves in projective spaces. Independence of points on elliptic curves arising from special points on modular and shimura curves, i. He is currently a professor of mathematics at princeton university and. Arithmetic moduli of elliptic curves pdf free download epdf.

These informal notes are an expanded version of lectures on the moduli space of elliptic curves given at zhejiang university in july, 2008. 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. The arithmetic of elliptic curves is a graduatelevel textbook designed to introduce the reader to an important topic in modern mathematics. State of the art in arithmetic moduli of elliptic curves. 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. In what follows, constant use will be made of the notations and results of katzmazur 4. 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. 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. Hence, by normalization over m 1 one gets proper normal. In chapter 6 a topic from the arithmetic of elliptic curves is discussed. Since elliptic curves are front and center of this story, it is appropriate to. Oberseminar modular curves and the eisenstein ideal.

Browse other questions tagged ellipticcurves arithmeticgeometry or. There are two things that make it difficult to read. 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. Pdf integral points on moduli schemes of elliptic curves.