Organized with melody chan, antonella grassi, rohini ramadas, and julie rana. That space time geometry could be based on a padic or even a. David lubicz, an introduction to the algorithmic of p padic numbers. Now i find real numbers much, much more confusing than padic numbers.
Chiarellotto differential modules of bounded spectral norm by g. We give a padic proof of hodge symmetry for smooth and projective varieties of dimension three over the field of complex numbers. Algebraic padic expansions david lampert department of mathematics, universiry of michigan, ann arbor. Padic numbers are far removed from our everyday intuitions, scholze said. Western algebraic geometry online, april 1819, 2020.
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. We study the p adic deformation properties of algebraic cycle classes modulo rational equivalence. 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. Padic methods in number theory and algebraic geometry. The arizona winter school 2007, on which the current book is based, was a unique opportunity to introduce graduate students to this subject. Zeta functions in algebraic geometry mircea mustat. 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. Such methods were used to compute composed products of polynomi. A formalization in homotopy type theory and there in coq is discussed in. 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. Existence of invariant norms in padic representations of. 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. Do they have any important applications outside of algebra that i could easily ex. In recent decades, \p\adic geometry and \p\adic cohomology theories have become indispensable tools in number theory, algebraic geometry, and the theory of automorphic representations. Over the years, though, they have come to feel natural to him. Motivation congruences between modular forms and padic zeta functions. Introduction to p adic analytic number theory download. The goal here is to use padic representations to study the geometry of algebraic varieties xover k. 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. Weil cohomology in practice this page is due to be taken down in summer 2020. 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 studies the arithmetic of algebraic number. On christols theorem a generalization to systems of pdes with logarithmic singularities depending upon parameters by f. 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. Women in algebraic geometry, july 2731, 2019, icerm.
We study the padic deformation properties of algebraic cycle classes modulo rational equivalence. One could hope for a padic analytic proof of the hodgetate decomposition. The articles presented here are drawn from a wide area of mathematics. The homogeneous coordinate ring of a projective variety, 5. 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. 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. Padicnumbersanintroductionuniversitext download free. Quotients of locally closed subschemes108 appendix b. 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. 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. Is there anything resembling a manifold on the field of padic or complex padic fields.
The theory has its beginnings in jeanpierre serre and john tates study of tate modules of abelian varieties and the notion of hodgetate representation. Convergent power series over complete nonarchimedean elds118 b. We give a formalism of methods of emerton, which would permit to. In fact, the aim of this course is to describe such an analytic approach. Review of the use of p padic numbers in arithmetic geometry includes. The padic numbers were invented and introduced to number theory by kurthensel1 aroundtheyear1900motivatedbytheideaofbringingthe powerful tool of power series to this area of mathematics. However, here we wish to relay another, deeper, relation between the cohomology of algebraic varieties and the structures underlying the langlands corresondence, a. Unrami ed extensions of q pand teichmuller lifts1 b. Each of the technical terms, however, can be seen as the. 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. This is the first book to deal with applications of. 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. Enrico beltrametti and his collaborators cassinelli and blasi in the late 1960s and early 1970s. An elementary construction of shimura varieties as moduli of abelian schemes.
In lectures 1215, we study shtukas with one leg, and their connection to pdivisible groups and padic hodge theory. Pdf padic local langlands correspondence and geometry. This is the first book to deal with applications of padic numbers in theoretical and mathematical physics. Msri derived algebraic geometry and its applications. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. 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. 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. Multiplicative seminorms and berkovich spaces 19 5. This thesis concerns the geometry behind the padic local langlands correspondence. A profound synthesis of this topic was made by weil.
Minerva school on padic methods in arithmetic algebraic. If so is there a connection to algebraic geometry as rich as in the reals. Introduction for about twenty years, the use of padic methods in symbolic computation has been gaining popularity. For those unfamiliar with padic fields, i will give a lightning introduction below. 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.
Moreover, mandell has used this observation to develop a \ p adic counterpart of rational homotopy theory. Rational and padic homotopy theory december 15, 2011. I dont expect readers to possess any of the technical vocabulary introduced in the previous paragraph. In lectures 1215, we study shtukas with one leg, and their connection to p.
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