Automorphic galois representations and the inverse galois. Compatible systems of symplectic galois representations. Then there exists a bound b a such that for all primes b a the representation is surjective. Milestones in inverse galois theory the inverse galois problem was perhaps known to galois. The general expectation is that for each g6 s1 the list of occurring dis in nite. Motives with exceptional galois groups and the inverse. Inverse galois problem for ordinary curves international. Thisgeneralizesaresultofwiesewiese, which inspired this paper. Functoriality and the inverse galois problem the work in kot92, clo91andht01 attaches galois representations to many such more precisely, if there is a. This article is the last part of a series of three on compatible systems of symplectic galois representations and applications to the inverse galois problem cf. The inverse galois problem, hilbertian fields, and hilberts irreducibility theorem logan chariker contents 1. Inverse galois problem for small simple groups department of. The inverse galois problem is to find a field extension with a given galois group as long as one does not also specify the ground field, the problem is not very difficult, and all finite groups do occur as galois groups.
The inverse galois problem concerns whether every finite group appears as the galois group of. Motives with exceptional galois groups and the inverse galois. May 08, 2009 pdf in this article new cases of the inverse galois problem are established. The inverse problem of galois theory, as formulated for the pair g,k, consists of two parts.
This problem is one of the greatest open problems in group theory. This gives a realization gl 2f as galois group for all prime. Dieulefaity, sug woo shin z, gabor wiese x 21st july 2014 abstract this article is the third and last part of a series of three articles about compatible systems of. Knapp, basic algebra, digital second edition east setauket, ny. Few special problems inform us beyond their computational consequences.
Automorphic galois representations and the inverse galois problem. This second edition addresses the question of which finite groups occur as galois groups over a given field. In the early nineteenth century, the following result was known as folklore. Our main theorem is the following new result for the inverse galois problem over q for symplectic groups. In particular, this includes the question of the structure and the representations of the absolute galois group of k, as well as its finite epimorphic images, generally referred to as the inverse problem of galois theory. The inverse galois problem galois theory is named after the famous 19thcentury mathematician evariste galois. Functoriality and the inverse galois problem chandrashekhar khare and michael larsen abstract. Introduction inverse galois problem cornell university. There has been much work on using modular forms to realize explicit simple groups of the form psl 2f r as galois groups of extensions of q, cf. We do this by constructing galois covers of ordinary semistable curves, and then deforming them. This problem, first posed in the early 19th century, is unsolved. This paper by no means proves the inverse galois problem to hold or not to hold for all. Consequences of the inverse galois problem mathoverflow.
Riemanns generalization of the rst features his famous. The inverse galois problem university of minnesota, morris. Answering the inverse galois problem for solvable extensions required class field theory one of the pinnacles of early 20th century mathematics. Inverse galois problem, elliptic curve over a finite field, hilbert irreducibility theorem, noethers problem, and rigidity method. Automorphic construction of compatible systems with suitable local properties.
Outline 1 the problem 2 compact riemann surfaces 3 covering spaces 4 connection to eld theory 5 proof of the main theorem 6 references. Groups of type b n and g 2 chandrashekhar khare, michael larsen, and gordan savin 1. Recall that the inverse galois problem igp asks whether every finite group g occurs as the galois group of some galois extension of q. Inverse galois theory is concerned with the question of which finite groups occur as galois groups over a given field. Andrea ferraguti joint mathematics meeting 2020 the inverse problem for arboreal galois representations of index two 1812020111. In fact, there are many groups gfor which it is not even known if such a eld extension exists. This is not a question one usually expects a student to solve.
We consider the inverse galois problem over function fields of positive characteristic p, for example, over the projective line. The inverse galois problem student theses faculty of science and. We describe a method to construct certain galois covers of the projective line and other curves, which are ordinary in the sense that their jacobian has maximal ptorsion. The inverse galois problem for symplectic groups valentijn karemaker university of pennsylvania joint with s.
The goal of this project, is to study the inverse galois problem. In x3 we begin to solve the inverse galois problem for special classes of groups. In other words, determine whether there exists a galois exten. In our previous work kls, which generalised a result of wiese w, the langlands functoriality principle was used to show that for every positive. The group extension problem and the embedding problem, which are both explored in x2. The book will serve as a guide to progress on the inverse galois problem and as an aid in using this work in other areas of mathematics. Using galois theory, certain problems in eld theory can be reduced to group theory, which is, in some sense, simpler and better understood. His irreducibility theorem established a connection between galois groups over and galois groups over. The other bundles those complex tori, with their prime degree isogenies. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. In the study of galois theory, after computing a few galois groups of a given field, it is very natural to ask the question of whether or not every finite. Yun, motives with exceptional galois groups and the inverse galois problem, invent. If this ptorsion group has maximum size, the curve is called ordinary.
The inverse problem of galois theory was developed in the early 1800 s as an approach to understand polynomials and their roots. Pdf in this article new cases of the inverse galois problem are established. There has been considerable progress in this as yet unsolved problem. This includes coding theory and other finite field applications. Statement questiondoes every nite group occur as the galois group of some nite galois extension l of q. Genus 3 curves and the inverse galois problem icerm. In particular, this includes the question of the structure and the representations of the absolute galois group of k, as well as its finite epimorphic images.
After that, we will give an elementary proof for the inverse galois problem when. Let k be a field, f a finite subfield and g a connected solvable algebraic matric group defined over f. Any finite abelian group g occurs as a galois group over q. This project explores rigidity, a powerful method used to show that a given group goccurs as a galois group over q. The inverse galois problem concerns whether every nite group appears as the galois group of some galois extension of the eld of rational numbers q. Namely, given a finite group g, the goal is to find a totally real extension kq, neces.
In galois theory, the inverse galois problem concerns whether or not every finite group appears as the galois group of some galois extension of the rational numbers q. We give an introduction to the inverse galois problem and compare some radically different approaches to. Moduli spaces and the inverse galois problem for cubic surfaces. In addition to simply asking whether every finite group is the galois group of a galois extension of the rationals, it is of interest to study the. The regular inverse galois problem uci mathematics. The main result is that for a fixed integer n, there is a positive. Pdf for a given finite group g, the inverse galois problem consists of determining whether g occurs as a galois group over a base field k. Pdf for a given finite group g, the inverse galois problem consists of determining whether g occurs as a galois group over a base field k, or in.
When no marking is chosen, the subgroup gis determined only up to conjugation. Andrea ferraguti joint mathematics meeting 2020 the inverse problem for arboreal galois representations of index two 181202091 examples over the rationals let f. This can be seen as evidence that the ability to solve the inverse galois problem will entail a deeper understanding of a variety of mathematical things. The inverse galois problem is galois theory \in reverse. Introduction in the study of galois theory, after computing a few galois groups of a given. Conditions on g and k are given which ensure the existence of a galois extension of k with group isomorphic to the frational points of g. Given a nite group g, nd a nite, normal extension lq with gallq g. The more wellknown, describes the \abelian analytic functions on a one dimensional compact complex torus. There are some permutation groups for which generic polynomials are known, which define all algebraic extensions of q having a particular group as galois group. This book describes a constructive approach to the inverse galois. Pdf on modular forms and the inverse galois problem. Vila joint mathematics meetings, atlanta january 7, 2017.
The inverse galois problem and explicit computation of. A reconstruction theorem finally, one can ask if any two maximal subgroups m a and m b are isomorphic whenever a 6. We provethat for any prime and any even integer n, there are in nitely many exponents k for which pspn fkappears asagaloisgroupoverq. Here, we shall discuss some of the most significant results on this. Inverse galois problem and significant methods arxiv. Inverse galois problem for totally real number fields cornell. In particular, this includes the question of the structure and the representations of the absolute galois group of k and also the question about its finite epimorphic images, the socalled inverse problem of galois theory. Constructive aspects of the inverse galois problem by c. Group theory and a first course in algebraic curves are sufficient for understanding many papers in the volume. We obtain our galois extensions by studying the galois action on the second etale cohomology groups of a specific elliptic surface. Namely, given a nite group g, the question is whether goccurs as a galois group of some nite extension of q.
Inverse problem of galois theory has been a difficult problem. The inverse galois problem is a major open problem in abstract algebra and has been extensive studied. Outline 1 the problem 2 compact riemann surfaces 3 covering spaces 4 connection to eld. This problem, rst posed in the 19th century, is in general unsolved 3. Moduli spaces and the inverse galois problem for cubic. In this thesis we investigate a variant of the inverse galois problem. In the form of modular towers, the rigp generalizes many of the general conjectures of arithmetic geometry, especially those involving properties of modular. Compatible systems of symplectic galois representations and. The inverse galois problem states whether any finite group can be realized as a galois group over q field of rational numbers. Galois theory answers that question by establishing a connection between eld and group theory. Determine whether goccurs as a galois group over k. Inverse galois problem for totally real number fields. Elementary formulations of the r egular inverse galois problem have several varients, of which the most precise and attackable is the nielsen version of the rigp, and the most general is the regular split embedding problem. The inverse galois problem with restricted ramification.
The inverse galois problem asks whether every nite group is isomorphic to the galois group of some extension of q. As an application of the considerations on moduli schemes, we obtain the following a. The inverse problem for arboreal galois representations of. Assume that d 2, 6 or d is odd and, furthermore, assume that end q a z. The inverse galois problem asks which nite groups occur as galois groups of extensions of q, and is still an open problem. Constructive aspects of the inverse galois problem. Sorry, we are unable to provide the full text but you may find it at the following locations. He studied wether it was possible to express roots of polynomials using radicals.
An additional useful reference is the book generic polynomials. For curves over q with good reduction at p, one can ask about the size of the ptorsion of the reduction of its jacobian. The regular inverse galois problem and hurwitz spaces we begin with an outline of theoretical prerequisites for the later computations. I once saw an application of a solved case of the inverse galois problem. If q is not required to be the base eld, then it is a theorem that every nite group occurs as the galois group of some nite extension of elds.