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In this note, we study conjectures of Artin and Langlands and derive the automorphy of all solvable groups of order at most 200, three groups excepted.

In this note, we study Artin's conjecture via group theory and derive Langlands reciprocity for certain solvable Galois extensions of number fields, which extends the previous work of Arthur and Clozel. In particular, we show that all nearly nilpotent groups and all groups of order less than 60 are of automorphic type.

We compute Artin L-functions of certain central extensions over the field of rational numbers and give an algorithm to compute a set of quadratic forms that determines the splitting of prime numbers.

Let G be a finite group containing a non-abelian Sylow 2-subgroup. We elementarily show that every G-Galois field extension L/K has a hyperbolic trace form in the presence of root of unity.

We know explicit Picard modular functions, corresponding to a family of hyperelliptic curves, with the property that their values in CM points generate abelian extensions of the associated reflex fields (Matsumoto, Ann Sc Norm Sup Pisa 16(4):557–578, 1989, Riedel, In: Arithmetic and geometry around hypergeometric functions. Birkhauser, Basel, 2007, pp 273–285). In this note we study the number fields and their extensions occuring in this way. We show that every sextic CM field containing the fourth roots of unity is projectively generated by a singular modulus and appears as reflex field. In order to investigate the abelian extensions, we use the class field theoretic description of the field of moduli. In the unramified case we develop conditions that assure that the Picard-Shimura class field is equal to the reflex field or to the Hilbert class field. Finally, we determine these class fields for odd class numbers up to 11.

We establish several close links between the Galois structures of a range of arithmetic modules including certain natural families of ray class groups, the values at strictly positive integers of p-adic Artin L-series, the Shafarevich–Weil Theorem and the conjectural surjectivity of certain norm maps in cyclotomic ℤ p {\mathbb{Z}_{p}} -extensions. Non-commutative Iwasawa theory and the theory of organising matrices play a key role in our approach.

This paper contains a refined account of nonlinear number fields which includes new results not found in the paper: Geometric Galois theory, nonlinear number fields and a Galois group interpretation of the idele class group [Int J Math 16(6):567–593, 2005 (Errata Int J Math 21(10):1383–1385, 2010)].

Given a number field \(k\) and an isoclinism class of stem groups (in the sense of P. Hall) it essentially suffices to realize one of these groups as a Galois group over \(k\). Indeed the other groups then appear as Galois groups of subfields of the composita with certain abelian extensions of \(k\) whose existence can be decided by class field theory.

We classify the Bieri–Neumann–Strebel–Renz invariant Σ1(G) for a class of Artin groups based on the full graph with 4 vertices.

This paper justifies an assertion in [Eld09] that Galois scaffolds make the questions of Galois module structure tractable. Let k be a perfect field of characteristic p and let K = k((T )). For the class of characteristic p elementary abelian p-extensions L/K with Galois scaffolds described in [Eld09], we give a necessary and sufficient condition for the valuation ring OL to be free over its associated order AL/K in K[Gal(L/K)]. Interestingly, this condition agrees with the condition found by Y. Miyata, concerning a class of cyclic Kummer extensions in characteristic zero.

We construct unconditionally several families of number fields with the largest possible class numbers. They are number fields of degree 4 and 5 whose Galois closures have the Galois group A4; S4, and S5. We first construct families of number fields with smallest regulators, and by using the strong Artin conjecture and applying the zero density result of Kowalski–Michel, we choose subfamilies of L-functions that are zero-free close to 1. For these subfamilies, the L-functions have the extremal value at s = 1, and by the class number formula, we obtain the extreme class numbers.

We provide a characterization of integers of the form described in the title in the context of class field theory. A more down-to-earth characterization is given when the extended class number is 1, 2, or 3. For completeness we present a formula for the extended class number. Some examples are also given.

Let F be a global function field and let F^ab be its maximal abelian extension. Following an approach of D.Hayes, we shall construct a continuous homomorphism \rho: Gal(F^ab/F) \to C_F, where C_F is the idele class group of F. Using class field theory, we shall show that our \rho is an isomorphism of topological groups whose inverse is the Artin map of F. As a consequence of the construction of \rho, we obtain an explicit description of F^ab. Fix a place \infty of F, and let A be the subring of F consisting of those elements which are regular away from \infty. We construct \rho by combining the Galois action on the torsion points of a suitable Drinfeld A-module with an associated \infty-adic representation studied by J.-K. Yu.

In this paper, we describe a number of interesting l-adic representations V of the Galois group of the rational function field with trivial L-function: L(V,s)=1.

This article comes from notes written for my lectures at the summer school “Abecedarian of SIDE” held at the CRM (Montreal) in June 2016. They are intended to give a short introduction to difference Galois theory, leaving aside the technicalities.

A class invariant is the value of a modular function that generates a ring class field of an imaginary quadratic number field such as the singular moduli of level 1. In this paper, using Shimura Reciprocity Law, we compute the Galois actions of a class invariant from a generalized Weber function over quadratic number fields with discriminant .

We study the problem of finding the Artin L-functions with the smallest conductor for a given Galois type. We adapt standard analytic techniques to our novel situation of fixed Galois type and get much improved lower bounds on the smallest conductor. For small Galois types we use complete tables of number fields to determine the actual smallest conductor., 32 pages, 3 figures

Among abelian extensions of a congruence function field, an asymptotic relation of class number and genus is established. The proof is classical, employing well-known results from congruence function field theory., 11 pages

We establish some new cases of Artin’s conjecture. Our results apply to Galois representations over $$\mathbf{Q }$$ with image $$S_5$$ satisfying certain local hypotheses, the most important of which is that complex conjugation is conjugate to $$(12)(34)$$ . In fact, we prove the stronger claim conjectured by Langlands that these representations are automorphic. For the irreducible representations of dimensions 4 and 6, our result follows from known 2-dimensional cases of Artin’s conjecture (proved by Sasaki) as well as the functorial properties of the Asai transfer proved by Ramakrishnan. For the irreducible representations of dimension 5, we encounter the problem of descending an automorphic form from a quadratic extension compatibly with the Galois representation. This problem is partly solved by working instead with a four dimensional representation of some central extension of $$S_5$$ . Our modularity results in this case are contingent on the non-vanishing of a certain Dedekind zeta function on the real line in the critical strip. A result of Booker show that one can (in principle) explicitly verify this non-vanishing, and with Booker’s help we give an example, verifying Artin’s conjecture for representations coming from the (Galois closure) of the quintic field $$K$$ of smallest discriminant (1609).

In this paper, we show how to construct the first layer of anti-cyclotomic -extension of imaginary quadratic fields when the Sylow subgroup of class group of k is 3-elementary, and give an example. This example is different from the one we obtained before in the sense that when we write is obtained from non-units of .

Let G be a finite nilpotent group and K a number field with torsion relatively prime to the order of G. By a sequence of central group extensions with cyclic kernel we obtain an upper bound for the minimum number of prime ideals of K ramified in a Galois extension of K with Galois group isomorphic to G. This sharpens and extends results of Geyer and Jarden and of Plans. Alternatively, we show how to use Frohlich’s result on realizing the Schur multiplicator in order to realize a family of groups given by central extensions with minimal ramification.

For every integer l bigger than one, we find elements x and y in the mapping class group of an appropriate orientable surface S, satisfying the Artin relation of length l. That is, xyx... = yxy..., where each side of the equality contains l terms. By direct computations, we first find elements x and y in Mod(S) satisfying Artin relations of every even length bigger than 6, and every odd length bigger than 1. Then using the theory of Artin groups, we give two more alternative ways for finding Artin relations in Mod(S). The first provides Artin relations of every length greater than 3, while the second produces Artin relations of every even length greater than 4., 18 pages, 3 figures

We study the relation between the minus part of the p -class subgroup of a dihedral extension over an imaginary quadratic field and the special value of the Artin L -function at 0.

In the present paper it is shown that one can recover much of the inertia structure of (quasi) divisors of a function field K|k over an algebraically closed base field k from the maximal pro-l abelian-by-central Galois theory of K. The results play a central role in the birational anabelian geometry and related questions.

In this paper, we determine completely the initial Fitting ideal of the minus part of the ideal class group of an abelian number field over Q up to the 2-component. This answers an open question of Mazur and Wiles (Invent Math 76:179–330, 1984) up to the 2-component, and proves Conjecture 0.1 in Kurihara (J Reine Angew Math 561:39–86, 2003). We also study Brumer’s conjecture and prove a stronger version for a CM-field, assuming certain conditions, in particular on the Galois group.

Let K be a field of characteristic 0, which is algebraically closed to radicals. Let F = K((X)) be the valued field of Laurent power series and let G = Aut(F/K). We prove that if L is a subfield of F, K ≠ L, such that L/K is a sub-extension of F/K and F/L is a Galois algebraic extension (L/K is Galois coalgebraic in F/K), then L is closed in F, F/L is a finite extension and Gal(F/L) is a finite cyclic group of G. We also prove that there is a one-to-one and onto correspondence between the set of all finite subgroups of G and the set of all Galois coalgebraic sub-extensions of F/K. Some other auxiliary results which are useful by their own are given.

Let q be a power of a prime number p. Let k=Fq(t) be the rational function field with constant field Fq. Let K=k(α) be an Artin–Schreier extension of k. In this paper, we explicitly describe the ambiguous ideal classes and the genus field of K. Using these results, we study the p-part of the ideal class group of the integral closure of Fq[t] in K. We also give an analogue of the Rédei–Reichardt formula for K.

Öz bulunamadı.

In §1, we introduce the notion of potential stable automorphy of modular galois representations, and state a general result on the ubiquity of such representations. In §2 we state some rather precise grouptheoretic results on the monodromy of the Dwork family, and use them to prove the general result of §1. In §3 we discuss variants and possible future applications of the general result. In §4 we prove the grouptheoretic results stated in §2, as well as some supplements to those results.

Text All current techniques for showing that a number field has an infinite p -class field tower depend on one of various forms of the Golod–Shafarevich inequality. Such techniques can also be used to restrict the types of p -groups which can occur as Galois groups of finite p -class field towers. In the case that the base field is a quadratic imaginary number field, the theory culminates in showing that a finite such group must be of one of three possible presentation types. By keeping track of the error terms arising in standard proofs of Golod–Shafarevich type inequalities, we prove a Golod–Shafarevich equality for analytic pro- p -groups. As an application, we further work of Skopin [V.A. Skopin, Certain finite groups. Modules and homology in group theory and Galois theory, Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 31 (1973) 115–139 (in Russian)], showing that groups of the third of the three types mentioned above are necessarily tremendously large. Video For a video summary of this paper, please visit http://www.youtube.com/watch?v=13GudVNQUUI .

Given a finite group G, what are the necessary and sufficient conditions on a field K for it to have an extension field with Galois group G over K? A large body of research has accumulated concerning this question, particularly in regard to small 2-groups. Specifically, conditions have been determined for all but one of the groups of order 32. In this work, we solve the problem for this last group. With these new results, complete conditions are now known for all 2-groups of order less than 64.

We introduce the idele class group for quasi-projective curves over p-adic fields and show that the kernel of the reciprocity map is divisible. This extends Saito’s class field theory for projective curves (Saito in J Number Theory 21:44–80, 1985).

In recent papers, Fesenko has defined the non-abelian local reciprocity map for every totally-ramified arithmetically profinite ($APF$) Galois extension of a given local field $K$ by extending the works of Hazewinkel and Neukirch-Iwasawa. The theory of Fesenko extends the previous non-abelian generalizations of local class field theory given by Koch-de Shalit and by A. Gurevich. In this paper, which is research-expository in nature, we give a detailed account of Fesenko's work including all the skipped proofs.

Let K / Q be a finite Galois extension, and let s 0 ≠ 1 be a complex number. We prove that the multiplicative semigroup of Artin L-functions in K / Q which are holomorphic at s 0 is finitely generated. We obtain a criterion for Artin's conjecture and we discuss the case of icosahedral extensions.

Let K ( 1 ) K^{(1)} be the Hilbert class field of the imaginary quadratic field K = Q ( d ) , d > 0. K=Q(\sqrt {d}),d>0. We derive the product representations of a class of Dirichlet L-series associated with the character group constructed from the Artin map between the ideal class group of K K and the Galois group G a l ( K ( 1 ) / K ) Gal(K^{(1)}/K) . The application of the Mellin transform to the product representations of these Dirichlet series yields a family of generating functions for representations of positive integers by the subgroups of the quadratic forms.

We prove that the the kernel of the reciprocity map for a product of curves over a $p$-adic field with split semi-stable reduction is divisible. We also consider the $K_1$ of a product of curves over a number field., Comment: 14 pages

Let A be a central simple algebra over a field F. Denote the reduced norm of A over F by Nrd: A* → F* and its kernel by SL1(A). For a field extension K of F, we study the first Galois Cohomology group H 1(K,SL1(A)) by two methods, valuation theory for division algebras and K-theory. We shall show that this group fails to be stable under purely transcendental extension and formal Laurent series.

We develop the theory of Frobenioids, which may be regarded as a category-theoretic abstraction of the theory of divisors and line bundles on models of finite separable extensions of a given function field or number field. This sort of abstraction is analogous to the role of Galois categories in Galo is theory or monoids in the geometry of log schemes. This abstract category-theoretic framework preserves many o f the important features of the classical theory of divisors and line bundles on models of finite separable extensions of a function field or number field such as the global degree of an arithmetic line bundle over a number field, but also exhibits interesting new phenomena, such as a ‘Frobenius endomorphism’ of the Frobenioid associated to a number field.

Let k be an algebraic number field and let N ( k , C l ; m ) denote the number of abelian extensions K of k with G ( K / k ) ≅ C l , the cyclic group of prime order l, and the relative discriminant D ( K / k ) of norm equal to m. In this paper, we derive an asymptotic formula for ∑ m ⩽ X N ( k , C l ; m ) using the class field theory and a method, developed by Wright. We show that our result is identical to a result of Cohen, Diaz y Diaz and Olivier, obtained by methods of classical algebraic number theory, although our methods allow for a more elegant treatment and reduce a global calculation to a series of local calculations.

Let K be an imaginary quadratic field, and let F be an abelian extension of K, containing the Hilbert class field of K. We fix a rational prime p > 2 which does not divide the number of roots of unity in the Hilbert class field of K. Also, we assume that the prime p does not divide the order of the Galois group G:=Gal(F/K). Let AF be the ideal class group of F, and EF be the group of global units of F. The purpose of this paper is to study the Galois module structures of AF and EF.

Let k be a number field, O k its ring of integers and Cl ( k ) its class group. Let Γ be the symmetric (octahedral) group S 4 . Let M be a maximal O k -order in the semisimple algebra k [ Γ ] containing O k [ Γ ] , Cl ( M ) its locally free class group, and Cl ○ ( M ) the kernel of the morphism Cl ( M ) → Cl ( k ) induced by the augmentation M → O k . Let N / k be a Galois extension with Galois group isomorphic to Γ, and O N the ring of integers of N. When N / k is tame (i.e., at most tamely ramified), extension of scalars allows us to assign to O N the class of M ⊗ O k [ Γ ] O N , denoted [ M ⊗ O k [ Γ ] O N ] , in Cl ( M ) . We define the set R ( M ) of realizable classes to be the set of classes c ∈ Cl ( M ) such that there exists a Galois extension N / k which is tame, with Galois group isomorphic to Γ, and for which [ M ⊗ O k [ Γ ] O N ] = c . In the present article, we prove that R ( M ) is the subgroup Cl ○ ( M ) of Cl ( M ) provided that the class number of k is odd.

Let l be a prime number and K be a cyclic extension of degree l of the rational function field 𝔽q(T) over a finite field of characteristic ≠ = l. Using class field theory we investigate the l-part of Pic 0(K), the group of divisor classes of degree 0 of K, considered as a Galois module. In particular we give deterministic algorithms that allow the computation of the so-called (σ - 1)-rank and the (σ - 1)2-rank of Pic 0(K), where σ denotes a generator of the Galois group of K/𝔽q(T). In the case l = 2 this yields the exact structure of the 2-torsion and the 4-torsion of Pic 0(K) for a hyperelliptic function field K (and hence of the 𝔽q-rational points on the Jacobian of the corresponding hyperelliptic curve over 𝔽q). In addition we develop similar results for l-parts of S-class groups, where S is a finite set of places of K. In many cases we are able to prove that our algorithms run in polynomial time.

Let K / Q be a finite Galois extension with the Galois group G, let χ 1 , … , χ r be the irreducible non-trivial characters of G, and let A be the C -algebra generated by the Artin L-functions L ( s , χ 1 ) , … , L ( s , χ r ) . Let B be the subalgebra of A generated by the L-functions corresponding to induced characters of non-trivial one-dimensional characters of subgroups of G. We prove: (1) B is of Krull dimension r and has the same quotient field as A ; (2) B = A iff G is M-group; (3) the integral closure of B in A equals A iff G is quasi-M-group.

Let l be a prime number and K be a cyclic extension of degree l of the rational function field Fq(T) over a finite field of characteristic ≠l. We study the l-part of the ideal class group of the integral closure of Fq[T] in K, and the l-part of the group of divisor classes of degree 0 of K as Galois modules. Using class field theory, we can describe explicitly part of the structure of these l-class groups. As an application, we get (for l=2) bounds for the order of the 4-torsion on JX(Fq), the group of points defined over Fq on the Jacobian of a hyperelliptic curve X/Fq.