Galois representations in arithmetic geometry takeshi saito. Rep d, factors through rep v d if and only if the corresponding representation v b of g. We also make a conjecture about the structure of mod l. More generally, algebraic geometry over non algebraically closed fields or fields of positive characteristic is also referred to as arithmetic algebraic geometry. Algebraic cycles, modular forms and euler systems abstract. With etale cohomology, geometric problems can be studied using linear algebra as in the diagram algebraic geometry linear algebra varieties over a.
This means that algebraic varieties over q yield padic galois representations galqq. Conference symposium on galois representations and arithmetic algebraic geometry october 1416, 1985 and january 711, 1986 kyoto university and university of tokyo. The rst examples are the representations on torsion points of elliptic curves and abelian varieties. The theme of the program was arithmetic algebraic geometry. Braids, galois groups, and some arithmetic functions. Moreover, very interesting examples of padic galois representations arise from geometry. Western algebraic geometry online, april 1819, 2020.
For example, if l is a galois extension of a number field k, the ring of integers o l of l is a galois module over o k for the galois group of lk see hilbertspeiser theorem. It is known that a galois representation of q arising from an algebraic variety enjoys the following two properties. These representations are a crucial tool for understanding arithmetic within geometry see the theorems of tate and faltings for example. Questions relating to the representations of the absolute galois group gal. Arithmetic geometry and symmetries around galois and. One can then ask which padic galois representations of q arise in this way. One course will focus entirely on local problems padic representations of galois groups of padic fields, a second course will have a more global flavor galois deformation theory and global applications, and a third on lvalues will rely on the other two courses. Introductory text on galois representations mathoverflow. Basic notions of rigid analytic geometry chapter 8 galois.
Numerous and frequentlyupdated resource results are available from this search. The choice of lecture topics was heavily influenced by the recent spectacular work of wiles on modular elliptic curves and fermats last theorem. Arithmetic algebraic geometry is in a fascinating stage of growth, providing a rich variety of applications of new tools to both old and new problems. The geometric viewpoint is stressed throughout the book. Galois deformation ring and barsotti tate representations. Some examples of geometric galois representations hard. Representative of these recent developments is the notion of arakelov geometry, a way of completing a variety over the ring of integers of a number field by adding fibres over the archimedean. Galois representations in arithmetic algebraic geometry london mathematical society lecture note series book 254 kindle edition by scholl, a. To understand which galois representations come from algebraic geometry, it is essential to study the submoduli spaces which parametrize those with certain padic hodge theoretic conditions.
The main emphasis of the articles in the volume is on elliptic curves, galois representations, and modular forms. Zdzislaw wojtkowiak lie algebras of galois representations on fundamental groups 601 go yamashita padic multiple zeta values, padic multiple lvalues, and motivic galois groups 629 yuichiro hoshi and shinichi mochizuki topics surrounding the combinatorial anabelian geometry of hyperbolic curves i. Jul 10, 2009 one course will focus entirely on local problems padic representations of galois groups of padic fields, a second course will have a more global flavor galois deformation theory and global applications, and a third on lvalues will rely on the other two courses. Galois representations and arithmetic algebraic geometry. Galois representations have become increasingly important in arithmetic, algebraic geometry and the theory of automorphic forms. In this volume the author gives a unified presentation of some of the basic tools and concepts in number theory, commutative algebra, and algebraic geometry, and for the first time in a book at this level, brings out the deep analogies between them. The term galois representation is frequently used when the gmodule is a vector space over a field or a free module over a ring in representation theory, but can also be used as a synonym for gmodule. Mathematical society of japan, 1987 373 pp dates publication date. Langlandskottwitz method for gl2 from automorphic to galois very brie y deformation of galois representations taylorwileskisin method from galois to automorphic application to modularity of elliptic curves over q and fermats last theorem prerequisite reasonable knowledge of general algebra, algebraic number theory, and commutative.
Gouvea and others published deformations of galois representations find, read and cite all the research you need on researchgate. In the first part of this paper we try to explain to a general mathematical audience some of the remarkable web of conjectures linking representations of galois groups with algebraic geometry, complex analysis and discrete subgroups of lie groups. Arithmetic geometry is centered around diophantine geometry, the study of rational points of algebraic varieties. We need to consider hodge structures together with adic representations in order. The title was interpreted loosely and the symposium covered recent developments on the interface between algebraic number theory and. There are natural sources of galois representations in algebraic geometry, and the langlands program conjecturally connects them with automorphic forms. Linear representations of the galois group over local fields. We also discuss how this representation can be computed in several simple cases. In the final week of the program there will be three minicourses that. Explicit methods in arithmetic geometry in characteristic p, june 1622, 2019. Newest galoisrepresentations questions feed subscribe to rss. Cambridge university press 9780521644198 galois representations in arithmetic algebraic geometry. It uses both commutative algebra the theory of commutative rings and geometric intuition.
Reading it is a good way to get a concrete feeling of what galois representations are. In the broad sense it means the part of algebraic geometry which studies the geometry and arithmetic of algebraic varieties which are maximally nonabelian in a certain sense. An overview of a theorem of flach abstract abstract no javascript ps pdf, appendix to deformations of galois representations by fernando gouvea, in arithmetic algebraic geometry park city, ut 1999. The arithmetic of modular symbols that the geometry of modular curves has something to say about the arithmetic of cyclotomic. The fundamental conjectures of anabelian algebraic geometry the term anabelian algebraic geometry geometrie algebrique anabelienne was introduced by grothendieck. It is moreover equipped with arithmetic structure, i. A concise introduction to the theory of groups, including the representation theory of finite groups. On the satake isomorphism chapter 4 galois representations in. Symposium on galois representations and arithmetic algebraic geometry october 1416, 1985 and january 711, 1986. Out of the 2dimensional galois representation attached to a level p newform congruent to an eisenstein series modulo p. Scholl skip to main content accessibility help we use cookies to distinguish you from other users and to provide you with a better experience on our websites. Modularity of some potentially barsottitate galois. Algebraic geometry studies the set of solutions of a multivariable polynomial equation or a system of such equations, usually over r or c. Conversely, we can study galois representation, a highly arithmetic object, using geometric or representation theoretic methods.
Galois representations and arithmetic algebraic geometry, y. Deformations of galois representations iwr heidelberg. Encoded in the riemann zeta function is lots of deep arithmetic information. Organized with melody chan, antonella grassi, rohini ramadas, and julie rana. Newest galoisrepresentations questions mathematics. This book is a conference proceedings based on the 1996 durham symposium on galois representations in arithmetic algebraic geometry. Geometric properties of families of galois representations. I would guess that, in a sense the linear part of the galois group, i. L download it once and read it on your kindle device, pc, phones or tablets.
Such representations arise throughout arithmetic geometry, where typically v is a cohomology space attached to an algebraic variety. I have been interested in algebraic number theory and arithmetic geometry since 1995, when i was a graduate student in tsinghua university, china. They were all chosen for their quality and relevance to the application of algebraic geometry to arithmetic problems. Braids, galois groups, and some arithmetic functions 101 base point b, this already indicates that the use of the fundamental groupoid on a suitable sstable set b of base points would be better. Galois representations are central in modern number theory, perhaps most famously in the proof of fermat s last theorem. Number theory has applications in computer science due to connections with cryptography. Applications of anabelian geometry to galois representations. Newest galoisrepresentations questions mathematics stack. Many objects that arise in number theory are naturally galois representations. Grothendiecks padic etale cohomology attaches to such an xa collec. In mathematics, a galois module is a gmodule, with g being the galois group of some extension of fields. Galois representations in arithmetic algebraic geometry.
Galois representations in arithmetic geometry ii 3 glq vgl nq, where v is an ndimensional vector space over the adic number. Autv, where v is a nitedimensional vector space over q, which is unrami ed outside a nite set of primes of k. The existence of the galois representations is taken as a black box, but he explains the galois theoretic significance of various congruences on the coefficients of the modular forms. This book contains conference proceedings from the 1996 durham symposium on galois representations in arithmetic algebraic geometry. Rep d such that for any nite q palgebra band point. Euler systems and arithmetic geometry barry mazur and tom. Publication information advanced studies in pure mathematics, volume 12 tokyo, japan. Arithmetic geometry is a branch of algebraic geometry studying schemes usually of finite type over the spectrum specz of the commutative ring of integers.
Arithmetic geometry is centered around diophantine geometry, the study of rational points of algebraic varieties 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. In contrast to most such accounts the notes study abstract algebraic varieties, and not just subvarieties of affine and projective space. The title was interpreted loosely and the symposium covered recent developments on the interface between algebraic number theory and arithmetic algebraic geometry. Galois representations in arithmetic geometry takeshi saito when.
The main emphasis of the articles in the volume is on elliptic curves, galois representations, and. These representations are a crucial tool for understanding arithmetic within geometry see the. Algebraic families of galois representations 3 theorem b 6. Cambridge core number theory galois representations in arithmetic algebraic geometry edited by a. Proves the duality theorems in galois, etale, and flat cohomology that have come to play an increasingly important role in number theory and arithmetic geometry, 2006 second corrected texed edition paperback.
The study of galois modules for extensions of local or global fields is an important tool. In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Galois representations are central in modern number theory, perhaps most famously in the proof of fermats last theorem. If we take the fermat curve as the algebraic variety above, we obtain the example given at the beginning. The aim of the three main courses is to present an overview of many of these ideas and applications, aimed at advanced graduate students and post docs with a strong background in number theory, galois cohomology, and basic algebraic geometry. Women in algebraic geometry, july 2731, 2019, icerm. In this case a continuous representation may have in. Arithmetic geometry is the study of integer or rational solutions of systems of. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel. The research interests of our group include galois representations, shimura varieties, automorphic forms, lattices, algorithmic aspects, rational points on varieties, and the arithmetic of k3 surfaces. Galois theory is vital to algebraic number theory, and useful sometimes to algebraic geometry. By an adic galois representation of kwe mean a continuous representation g k. Geometric deformations of orthogonal and symplectic galois. Galois representations in arithmetic and geometry icts.
Automorphy for some ladic lifts of automorphic mod l. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus. Jan 26, 2015 some examples of geometric galois representations in this post we discuss the galois representation associated to a projective scheme, where is a number field. Deformations of galois representations request pdf. Conditions on galois representations 108 chapter 14. Varieties over a field k ladic representations of the absolute. Mazur showed that the moduli spaces of galois representations are represented by universal deformation rings 30. Arithmetic algebraic geometry is in a fascinating stage of growth, providing a rich. Fields and galois theory a concise treatment of galois theory and the theory of fields, including transcendence degrees and infinite galois extensions. The author considers galois group actions on the fundamental groups of. Arithmetic algebraic geometry iaspark city mathematics.
Source galois representations and arithmetic algebraic geometry, y. But i get the feeling that it like much of algebra, in fact has the status of esoterica in the larger pure mathematical world. Right now i have been focusing on padic galois representations from algebraic geometry, more specifically, integral padic hodge theory. In the book theory of padic galois representations of fontaine, i have two questions in prop3. Galois representations in arithmetic algebraic geometry edited by a. However, formatting rules can vary widely between applications and fields of interest or study. Euler systems and arithmetic geometry barry mazur and tom weston. Galois representations in arithmetic geometry takeshi saito when he formulated an analogue of the riemann hypothesis for congruence zeta functions of varieties over.
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