P-adic methods in number theory and algebraic geometry pdf

Padic methods in number theory and algebraic geometry. A conference inspired by the mathematics of robert coleman. Journal of number theory 23, 279284 1986 algebraic p adic expansions david lampert department of mathematics, universiry of michigan, ann arbor. The padic numbers and more generally local fields have become increasingly important in a wide range of mathematical disciplines. Introduction to p adic analytic number theory download.

Click download or read online button to get introduction to p adic analytic number theory book now. They are also relevant in certain parts of algebraic geometry, they are one of. The following view how to do science with numbers has been put forward by volovich in 2, 3. In algebra and number theory, one quite often deals with valuations instead of absolute values. Weil cohomology in practice this page is due to be taken down in summer 2020. He gave the first definition of the field of padic numbers as the set. Arithmetic geometry and number theory series on number. Browse other questions tagged algebraicgeometry biglist schemes padicnumbertheory bigpicture or ask your own question. And finally, a brand new concept by weng called the geometric arithmetic program that uses algebraic and or analytic methods, based on geometric considerations, to develop the promising and yet to be cultivated land of global arithmetic that includes nonabelian class field theory, riemann hypothesis and nonabelian zeta and l functions, etc. Welcome,you are looking at books for reading, the number theory in function fields, you will able to read or download in pdf or epub books and notice some of author may have lock the live reading for some of country.

Padic methods in number theory and algebraic geometry by alan c adolphson, steven 1 sperber and marvin d 0 tretkoff topics. We will further develop some properties of the padic numbers in later sections. There is an article by brian conrad comparing the various foundations for padic geometry and an introduction to berkovichs version of the theory by matthew. These special sessions are the starting point for this volume. The field q p is also given a topology derived from a metric, which is itself derived from the p adic order, an alternative valuation on the rational numbers. Two years later, a weeklong meeting in 2014 on perfectoid spaces and their applications at the. Several other such lists algtopconf, sarah whitehouses conference page have been merged into mathmeetings. The articles presented here are drawn from a wide area of mathematics. Together with eric larson and ravi vakil, i organize the stanford algebraic geometry seminar. Algebraic geometry into a number theory approach to finite string theory. Adolphson, alan, sperber, steven, tretkoff, marvin. Using padic analysis, dwork was the first to prove the rationality of the zeta function of a variety over a finite field. Algebraic geometry an overview sciencedirect topics.

It contains written versions of short courses taught at the arizona winter school, which is aimed at equipping graduate students for research. Given p, the corresponding field is called the field of padic numbers. My research is in the intersection of algebraic geometry and number theory. It also uses standard constructions of modern algebraic geometry, such as the category of schemes and iwasawa theory, and other 20thcentury techniques not available to fermat. This course is an introduction to p adic hodge theory, the study of p adic galois representations of p adic fields. The field q p is also given a topology derived from a metric, which is itself derived from the padic order, an alternative valuation on the rational numbers. All contributions combine main concepts and techniques with motivating examples and. More formally, for a given prime p, the field q p of p adic numbers is a completion of the rational numbers.

The extension is achieved by an alternative interpretation of the concept of closeness or absolute value. In x2 we will turn our attention to the case of padic homotopy theory. Dworks use of padic analysis in algebraic geometry. The main objects that we study in algebraic number theory are number. For now, we will focus on the problem of classifying all possible completions of an algebraic.

Algebraic geometry has an original and interesting point of view on. This thesis lies in the framework of the padic langlands programme, whose goal is to deepen our understanding about fundamental problems in number theory, such as the solution of diophantine equations, the study of elliptic curves, and the study of shimura varieties. Schedule for the talks the talks will take place in the room sala di rappresentanza, dipartimento di matematica federigo en. I am currently particularly interested in the geometry of algebraic curves and applications of geometric methods to rational points. We will discuss how hensels lemma from number theory can be used to find geometric approximations to solutions of the equations describing the essential. A study of the padic frobenius lifts and padic periods. In mathematics, the padic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. Combinatorial and additive number theory cant 2016, may 2427, 2016, cuny graduate center, new york, usa. Completions padic elds adeles and ideles note that we will not do anything nontrivial with zeta functions or lfunctions. Padic methods in number theory and algebraic geometry book. The ams bookstore is open, but rapid changes related to the spread of covid 19 may cause delays in delivery services for print products. Recently, important new results have arisen from analytic, geometric, and p adic methods. The padic hodge decomposition according to beilinson jt.

We will further develop some properties of the p adic numbers in later sections. Number theory and representation theory seminar analytic number theory, algebraic number theory, arithmetic geometry, automorphic forms, and even some things not beginning with the letter a. This project consists of using or techniques to improve current. This is the fourteenth in a series of annual workshops sponsored by the new york number theory seminar on problems and results in combinatorial and additive number theory and related parts of mathematics. Minerva school on padic methods in arithmetic algebraic geometry. However, formatting rules can vary widely between applications and fields of interest or study. They are now seen as essential tools in many areas, including number theory, algebraic geometry, group representation theory, the modern theory of automorphic forms, and algebraic topology. Using embedded resolution of singularities and the change of variable formula for padic integrals, igusa showed that z fs is a rational function of 1ps, see igu. Special stanford number theory algebraic geometry seminar.

The only proofs that are included are those coming from algebraic number theory. Important applications of padic numbers outside of algebra. Journal of number theory 23, 279284 1986 algebraic padic expansions david lampert department of mathematics, university of michigan, ann arbor, michigan 48109 communicated by h. In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory.

In x2 we will turn our attention to the case of p adic homotopy theory. More formally, for a given prime p, the field q p of padic numbers is a completion of the rational numbers. P adic methods in number theory and algebraic geometry by alan c adolphson, steven 1 sperber and marvin d 0 tretkoff topics. This course is an introduction to padic hodge theory, the study of padic galois representations of padic fields. This site is like a library, use search box in the widget to get ebook that you want.

Slide 11 the padics as a foundational tool hensel showed, in several papers and in his books, that one could give a simpler account of the theory of divisibility in algebraic number. In the past few decades, research in number theory has progressed at a rapid rate on many fronts. The langlands programme aims at creating a correspondence. And finally, a brand new concept by weng called the geometric arithmetic program that uses algebraic andor analytic methods, based on geometric considerations, to develop the promising and yet to be cultivated land of global arithmetic that includes nonabelian class field theory, riemann hypothesis and nonabelian zeta and l functions, etc. We will say that a space x is p nite if it has nitely many connected components and nitely many nonzero homotopy groups, each of which is. Number theory is one of the oldest branches of mathematics and is concerned with the properties of numbers in general. Browse other questions tagged algebraic geometry biglist schemes p adic number theory bigpicture or ask your own question. Salt lake city 2015, part 2, proceedings of symposia in pure mathematics 97, ams, providence 2018, 495572, arxiv. The aim of this book is to familiarize the reader with fundamental topics in number theory.

Important applications of padic numbers outside of algebra and number theory. Newest padicnumbertheory questions mathematics stack. One approach to constructing the field qp of padic numbers is to copy. Number theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. The padic numbers are first reformulated as formal series, in the sense of deformation theory, together with basic facts, including how a canonical lift of frobenius, as a companion deformation of their symmetries 15. Chapter 15, written by samir siksek, deals with the use. Special stanford number theory algebraic geometry seminar revisiting the padic approach to zeta functions. Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. In particular, this proved borevichs conjecture about the rationality of p f. Well develop methods for explaining things like this. Counting the number of points on an algebraic variety over a finite field is one of the oldest topics in algebraic geometry, dating back to the work of gauss. Algebraic number theory involves using techniques from mostly commutative algebra and. Over the last decades, padic methods have taken some importance in symbolic computa.

Padic methods in number theory and algebraic geometry contemporary mathematics 9780821851456. Bachelor projects for mathematics and mathematicseconomics. Number theory in function fields download pdfepub ebook. Existence of invariant norms in padic representations of. Zeta functions in algebraic geometry mircea mustat. It uses both commutative algebra the theory of commutative rings. Padic methods in number theory and algebraic geometry core. From what i have seen, in algebraic geometry, this method is not used much. We will further develop some properties of the padic numbers in later. Journal of number theory 23, 279284 1986 algebraic p adic expansions david lampert department of mathematics, university of michigan, ann arbor, michigan 48109 communicated by h. Journal of number theory 23, 279284 1986 algebraic padic expansions david lampert department of mathematics, universiry of michigan, ann arbor.