This assumption is not necessary in padic geometry. The fundamental theorem of arithmetic for integers. More precisely, the goal is to bring the reader to the frontier of current developments in arithmetic geometry and number theory through the works of deningerwerner in vector bundles on curves over p adic fields. Arithmetic geometry is centered around diophantine geometry, the study of rational points of algebraic varieties. Arithmetic and geometry of the open padic disc penn math. We start off with kummers work and iwasawas treatment of cyclotomic fields, which eventually paved the way to the modern padic variational techniques. Dino lorenzini, an invitation to arithmetic geometry graduate studies in mathematics, vol 9 gsm9. This book is an introduction to number theory and arithmetic geometry, and the goal of the text is to use geometry as the. For transforming arithmetic algebraic geometry over padic fields through. Arithmetic and geometry of the open padic disc scott corry a. Roughly speaking, our result says that the special fiber of a galois cover of the open padic disc wants to be the field of norms of the characteristic zero fibers, and. The field of p adic numbers, absolute values, ostrowskis theorem for q pdf 6. The theory has its beginnings in jeanpierre serre and john tate s study of tate modules of abelian varieties and the notion. Heegner points and sylvesters conjecture, with john voight, arithmetic geometry, clay mathematics proceedings, 8.
At a few points, we have expanded slightly on the material, in particular so as to provide a full construction of local shimura varieties and general moduli spaces of shtukas, along with some. Peter scholze and the future of arithmetic geometry. Completing q with respect to p leads to the field qp of padic numbers. Dec 12, 2012 the first essential difference between padic numbers and the reals comes when you try to do arithmetic. A profound synthesis of this topic was made by weil. In this talk, we will explain a result on the comparison between hidas canonical period of.
Jun 28, 2016 arithmetic geometry elliptic curves geometry hyperbolic geometry langlands program mathematics modular forms number theory p adic numbers podcast polynomials profile topology in 2010, a startling rumor filtered through the number theory community and reached jared weinstein. Ostrowskis theorem for number fields no lecture notes but see ostrowskis theorem for number fields pdf by keith conrad 7. An ordinary integer n also called a rational integer in this context is a special case of a padic integer, whose ordern residue is simply n mod p n the sum or the product of two padic integers is defined as the padic integer whose ordern residue is the sum or the product of the ordern residues of both operands. Workshop padic arithmetic geometry abstracts john coates, ramdorai sujatha title. Of course, this approach is not meant to replace more systematic expositions of padic analysis or geometry.
American mathematical society, providence, ri, 2009. In mathematics, arithmetic geometry is roughly the application of techniques from algebraic. Weil cohomology in practice this page is due to be taken down in summer 2020. Review of the use of p padic numbers in arithmetic geometry includes. This course introduces some of the basic results and techniques in arithmetic geometry, with emphasis on diophantine applications. A formalization in homotopy type theory and there in coq is discussed in. However, if the characteristic of the base field p is positive, no direct. Bhargav bhatt perfectoid spaces are a class of algebrogeometric objects living in the realm of padic geometry that were introduced by peter scholze sch12 in his. Motivation congruences between modular forms and padic zeta functions. Peter scholze and the future of arithmetic geometry quanta. This is a revised version of the lecture notes for the course on padic geometry given by p. Arithmetic and geometry of the open padic disc scott corry florian pop, advisor motivated by the local lifting problem for galois covers of curves, this thesis investigates galois branched covers of the open padic disc.
The goal here is to use p adic representations to study the geometry of algebraic varieties xover k. In mathematics, the p adic 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. The field of padic numbers, absolute values, ostrowskis theorem for q pdf 6. This book is an introduction to number theory and arithmetic geometry, and the goal of the text is to use geometry as the motivation to prove the main theorems in the book. 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. Nowadays, the field of mathematics that studies the interactions between number theory and algebraic geometry is known as arithmetic geometry. Its main purpose is to introduce the reader to padic analytic geometry and to the theory of padic analytic functions and di. In complex geometry, a basic consequence of hodge theory is that this spectral sequence degenerates at e. Alvaro pelayo, vladimir voevodsky, michael warren, a preliminary univalent formalization of the padic numbers arxiv. One interesting aspect of noncommutative iwasawa theory is that nonabelian padic lie extensions of number. Im a student interested in arithmetic geometry, and this implies i use etale cohomology a lot. Aug 01, 2018 mathematicians had noticed in the 1970s that many problems concerning padic numbers become easier if you expand the padic numbers by creating an infinite tower of number systems in which each one wraps around the one below it p times, with the padic numbers at the bottom of the tower. Algebraic curves an algebraic variety is the zero locus of nite number of polynomial equations in nite number of variables. Regarding its definition, etale cohomology is a purely algebrogeometric object.
Scholzes initial motivation, very roughly speaking, was to study equations over certain kinds of fields of characteristic zero by reinterpreting them as equations over fields of prime characteristic p, where other methods are available. Tohoku university, katahira campus, sendai, japan access program and abstract pdf program. Peter scholzes lectures on padic geometry, fall 2014 last updated on 29 jan. The quantity a n is called the ordern residue of such a padic integer. Alexander paulin math berkeley university of california, berkeley. This is just a very partial answer that is based on my experience trying to do some work with extension of padic numbers. New developments of arithmetic geometry, motive, galois theory, and their practical applications, we have a workshop on padic geometry at komaba, university of tokyo. He provided 2 sets of notes, on explicit reciprocity law for lubintate groups dvi, ps, pdf and on crystalline sheaves, syntomic cohomology, and padic polylogarithms dvi, ps, pdf. Ostrowskis theorem for number fields no lecture notes but see ostrowskis theorem for number fields pdf. Multiplicative seminorms and berkovich spaces 19 5. Ostrowskis classi cation of absolute values on q 5 5. In mathematics, padic hodge theory is a theory that provides a way to classify and study padic galois representations of characteristic 0 local fields with residual characteristic p such as qp. My work predominantly explores the padic and geometric aspects of the langlands program and is highly interdisciplinary, involving number theory, arithmetic geometry, algebraic geometry, padic analytic geometry, dmodule. More precisely, the goal is to bring the reader to the frontier of current developments in arithmetic geometry and number theory through the works of deningerwerner in vector bundles on curves over padic fields.
We discuss recent developments in padic geometry, ranging from foundational results. Therefore, as quotients of padic integers, all padic numbers are simply of the following form, where m may be negative. At a few points, we have expanded slightly on the material, in particular so as to provide a full construction of local shimura varieties and. Find materials for this course in the pages linked along the left. The extension is achieved by an alternative interpretation of the concept of closeness or absolute value. Comparison of periods and r t theorems the comparison of several periods of automorphic forms is important in iwasawa theory. Today we will give a crash course in nonarchmidean geometry, since it plays such a vital role.
This is a revised version of the lecture notes for the course on p adic geometry given by p. This is just a very partial answer that is based on my experience trying to do some work with extension of p adic numbers. When the base field is a padic field, they are also directly connected by padic hodge the ory. He has developed powerful methods in algebraic geometry over padic fields, and has proved striking theorems in this area. Introduction to padic hodge theory lectures by serin hong. In fact, if youre working with padic numbers, there are no negative numbers at all. Notes by aleksander horawa these are notes from math 679 taught by serin hong in winter 2020, latexed by aleksander horawa who is the only person responsible for any mistakes that may be found in them. Padic method and its applications in arithmetic geometry, 2007 as an internaional workshop of jsps coretocore program no. Date october 31 mon november 2 wed, 2016 location tohoku university, katahira campus, tokyo electron house of creativity, sendai, japan lecture room in 3rd floor organizers. Algebraic geometry and number theory that took place in june 2014 in istan. Thanks to arthur ogus and jay pottharst for helpful comments.
The inverse of a noninvertible padic integer is thus the product of an invertible padic integer by a negative power of p. Tsuji gave a series of 4 lectures at usc and caltech in february, 2001 on various topics related to padic hodge theory. These notes grew out of a mini lecture series i gave in the cimpa summer school. Minerva school on padic methods in arithmetic algebraic. Arithmetic geometry and number theory series on number. The padic upper half plane, with jeremy teitelbaum, padic geometry. There is padic arithmetic in the free software programs sage, pari and gap, but their main limitation i dont know if things have changed recently was their inability to with relative extension, that is extensions of another field which is itself a proper extension of. Peter scholzes lectures on padic geometry, fall 2014. Galois representations in arithmetic geometry takeshi saito when. As an application, i will show that the dmodule theoretic pushforward of an isocrystal along a proper smooth morphism is an isocrystal. Lately padic numbers have attracted a great deal of attention in modern theoretical physics as a promising new approach for describing the nonarchimedean geometry of spacetime at small distances. The first essential difference between padic numbers and the reals comes when you try to do arithmetic.
There is p adic arithmetic in the free software programs sage, pari and gap, but their main limitation i dont know if things have changed recently was their inability to with relative extension, that is extensions of. Padic method and its applications in arithmetic geometry. Pdf arithmetic geometry download full pdf book download. Introduction to arithmetic geometry mit mathematics. Modular elliptic curves and fermats last theorem pdf.
The main goal of this survey is to provide a general overview of the theme of padic variation, both from a historical and technical view point. These methods have proved extremely powerful and enabled us to gain access. David lubicz, an introduction to the algorithmic of p padic numbers. Minerva school on padic methods in arithmetic algebraic geometry. In more abstract terms, arithmetic geometry can be defined as the study of schemes of finite type over the spectrum of the ring of integers. As i said above, for integers, if you dont do anything with negative numbers, padic arithmetic is the same as real number integer arithmetic. This is the first book to deal with applications of padic numbers in theoretical and mathematical physics.
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