P adic algebraic geometry pdf

The homogeneous coordinate ring of a projective variety, 5. Although we begin the oral lectures with a crash course on some basic highlights from rigidanalytic geometry in the sense of tate, some awareness of those ideas is taken as. That space time geometry could be based on a padic or even a. The padic numbers were invented and introduced to number theory by kurthensel1 aroundtheyear1900motivatedbytheideaofbringingthe powerful tool of power series to this area of mathematics. Multiplicative seminorms and berkovich spaces 19 5. Two meetings of the ams in the fall of 1989one at the stevens institute of technology and the other at ball state universityincluded special sessions on the role of \ p \ adic methods in number theory and algebraic geometry. The mathematical sciences research institute msri, founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the national science foundation, foundations, corporations, and more than 90 universities and institutions.

The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. Algebraic number theory studies the arithmetic of algebraic number. We give a padic proof of hodge symmetry for smooth and projective varieties of dimension three over the field of complex numbers. Over the years, though, they have come to feel natural to him. Moreover, mandell has used this observation to develop a \ p adic counterpart of rational homotopy theory. Is there anything resembling a manifold on the field of padic or complex padic fields. Zeta functions in algebraic geometry mircea mustat.

Now i find real numbers much, much more confusing than padic numbers. Organized with melody chan, antonella grassi, rohini ramadas, and julie rana. On christols theorem a generalization to systems of pdes with logarithmic singularities depending upon parameters by f. If so is there a connection to algebraic geometry as rich as in the reals. Review of the use of p padic numbers in arithmetic geometry includes. Women in algebraic geometry, july 2731, 2019, icerm.

This book covers the following three topics in a manner accessible to graduate students who have an understanding of algebraic number theory and scheme theoretic algebraic geometry. Chiarellotto differential modules of bounded spectral norm by g. We study the p adic deformation properties of algebraic cycle classes modulo rational equivalence. Im trying to understand katzdeligne theory of false modular forms as exposed in the third appendix of the katzs paper higher congruences between modular forms. The purpose of the padic langlands programme is to create a similar correspondence between some of the padic representations of the reductive group, and some of the padic representations of the galois group. Weil cohomology in practice this page is due to be taken down in summer 2020. Unrami ed extensions of q pand teichmuller lifts1 b.

Introduction to p adic analytic number theory download. Do they have any important applications outside of algebra that i could easily ex. The articles presented here are drawn from a wide area of mathematics. Padic numbers are far removed from our everyday intuitions, scholze said. Each of the technical terms, however, can be seen as the.

This thesis concerns the geometry behind the padic local langlands correspondence. In fact, the aim of this course is to describe such an analytic approach. 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. Perfectoid geometry doesnt transcend these previous attempts but rather extracts those of their features that are relevant to the problems scholze set out to solve. We show that the crystalline chern character of a vector bundle lies in a certain part of the hodge filtration if and only if, rationally, the class of the vector bundle lifts to a formal proclass in ktheory on the padic scheme. 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. I dont expect readers to possess any of the technical vocabulary introduced in the previous paragraph. We give a formalism of methods of emerton, which would permit to. This paper presents some basic theorems giving the structure of cyclic codes of lengthn over the ring of integers modulop a and over thepadic numbers, wherep is a prime not dividingn. The goal here is to use padic representations to study the geometry of algebraic varieties xover k. Kedlaya, fall 2004 a little p adic functional analysis part 1 of 2 im going to start with a little bit of terminology and notation about nonarchimedean banach spaces trusting that you can. David lubicz, an introduction to the algorithmic of p padic numbers. Padic methods in number theory and algebraic geometry. This is the first book to deal with applications of padic numbers in theoretical and mathematical physics.

Pdf padic local langlands correspondence and geometry. An elementary construction of shimura varieties as moduli of abelian schemes. Two meetings of the ams in the fall of 1989one at the stevens institute of technology and the other at ball state universityincluded special sessions on the role of \p\adic methods in number theory and algebraic geometry. This is the first book to deal with applications of. However, here we wish to relay another, deeper, relation between the cohomology of algebraic varieties and the structures underlying the langlands corresondence, a. A formalization in homotopy type theory and there in coq is discussed in. The theory has its beginnings in jeanpierre serre and john tates study of tate modules of abelian varieties and the notion of hodgetate representation. We show that the crystalline chern character of a vector bundle lies in a certain part of the hodge filtration if and only if, rationally, the class of the vector bundle lifts to a formal proclass in ktheory on the p adic scheme.

For example, he has shown that if xis a simply connected space whose homotopy groups are nitely generated modules over z p, then xcan be functorially recovered from c x. Quotients of locally closed subschemes108 appendix b. Motivation congruences between modular forms and padic zeta functions. Such methods were used to compute composed products of polynomi. For those unfamiliar with padic fields, i will give a lightning introduction below. Such a correspondence has been shown for g gl2 when the. Please join the simons foundation and our generous member organizations in supporting arxiv during our giving campaign september 2327. Algebraic padic expansions david lampert department of mathematics, universiry of michigan, ann arbor. 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 q p. In lectures 1215, we study shtukas with one leg, and their connection to p.

The 3adic numbers, for example, provide a natural way to study equations like x 2 3y 2, in which factors of three are key. Enrico beltrametti and his collaborators cassinelli and blasi in the late 1960s and early 1970s. Padicnumbersanintroductionuniversitext download free. These special sessions are the starting point for this volume. In lectures 1215, we study shtukas with one leg, and their connection to pdivisible groups and padic hodge theory. The idea of uniformizing algebraic varieties by padic analytic spaces is due to tate, who showed how an elliptic curve eover a padic field fcan be.

Convergent power series over complete nonarchimedean elds118 b. A profound synthesis of this topic was made by weil. Minerva school on padic methods in arithmetic algebraic. Rational and padic homotopy theory december 15, 2011. Western algebraic geometry online, april 1819, 2020.