Search

We use a rigidity argument to prove the existence of two related degree 28 covers of the projective plane with Galois group \(SU_{3}(3).2\mathop{\cong}G_{2}(2)\). Constructing corresponding two-parameter polynomials directly from the defining group-theoretic data seems beyond feasibility. Instead we provide two independent constructions of these polynomials, one from 3-division points on covers of the projective line studied by Deligne and Mostow, and one from 2-division points of genus three curves studied by Shioda. We explain how one of the covers also arises as a 2-division polynomial for a family of G2 motives in the classification of Dettweiler and Reiter. We conclude by specializing our two covers to get interesting three-point covers and number fields which would be hard to construct directly.

Let D be a Dedekind domain with quotient field K. The ring of integer-valued polynomials on D is the subring Int(D) = { f ∈ K[X]: f(D) ⊆ D} of the polynomial ring K[X]. The Polya-Ostrowski group PO(D) of D is a subgroup of the class group of D generated by the well-known factorial ideals n! D of D. A regular basis of Int(D) is a D-module basis consisting of one polynomial of each degree. It is well known that Int(D) has a regular basis if and only if the group PO(D) is trivial, if and only if the D-module Int n (D) = { f ∈ Int(D): degf ≤ n} is free for all n. In this paper we provide evidence for and prove special cases of the conjecture that, if PO(D) is finite, then the natural density of the set of nonnegative integers n such that Int n (D) is free exists, is rational, and is at least 1∕ | PO(D) | . Moreover, we compute this density or determine a conjectural value for several examples of Galois number fields of degrees 2, 3, 4, 5, and 6 over \(\mathbb{Q}\).

Para>In 1882, Lindemann wrote a paper in which he sketched a general result, special cases of which imply the transcendence of e and π. This general result was later proved rigorously by K. Weierstrass in 1885. Before we begin, we make some remarks pertaining to algebraic number theory. Let K be an algebraic number field.

This paper presents the current status on modularity of Calabi–Yau varieties since the last update in 2003. We will focus on Calabi–Yau varieties of dimension at most three. Here modularity refers to at least two different types: arithmetic modularity and geometric modularity. These will include: (1) the modularity (automorphy) of Galois representations of Calabi–Yau varieties (or motives) defined over \(\mathbb{Q}\) or number fields, (2) the modularity of solutions of Picard–Fuchs differential equations of families of Calabi–Yau varieties, and mirror maps (mirror moonshine), (3) the modularity of generating functions of invariants counting certain quantities on Calabi–Yau varieties, and (4) the modularity of moduli for families of Calabi–Yau varieties. The topic (4) is commonly known as geometric modularity.

This chapter is an introductory and historical discussion of problems concerning arithmetic invariants and L-values. Often a number-theoretic object has an associated L-function. A number field has its Dedekind zeta function, an algebraic variety has its Hasse–Weil zeta function, an automorphic form/representation has its Langlands L-functions, and we have L-functions associated with Galois representations, compatible systems of Galois representations and motives. Number theorists all agree that L-functions and L-values are useful invariants for studying the object with number-theoretic goals in mind. From L-functions, number theorists have created more invariants, for example, λ- and μ-invariant from p-adic L-functions and the ℒ-invariant from exceptional zeros of p-adic L-functions.

We establish a spectral identity for moments of Rankin–Selberg L-functions on GL r ×GL r − 1 over arbitrary number fields, generalizing our previous results for r = 2.

6.1. Let F be a field. A map \(\,\nu : F\to {\bold{R}}\cup\{\infty\}\) is called an order function of F if it satisfies the following conditions

Let F be either an algebraic number field or its completion at a prime \(\,v\in{\bold{v}},\) and \((V,\,\varphi)\) a nondegenerate quadratic space over F of dimension \(\,n.\) We then define the characteristic (quaternion) algebra of \((V,\,\varphi)\) as follows.

The description of arithmetic Kleinian groups and arithmetic Fuchsian groups via quaternion algebras and their orders is convenient, as it links up with the earlier use of quaternion algebras and number fields as commensurability invariants of general Kleinian groups. This description also clarifies the connections between the arithmetic, on the one hand, and the topological, geometric and group-theoretic properties of the groups, on the other. This has been illustrated in Chapter 9 and further aspects will be pursued in the remaining chapters.

One of the main aims of this chapter is to complete the classification theorem for quaternion algebras over a number field by establishing the existence part of that theorem. This theorem, together with other results in this chapter, make use of the rings of adeles and groups of ideles associated to number fields and quaternion algebras. These rings and groups and their component parts are locally compact groups so that some aspects of their Haar measures, duality and abstract harmonic analysis go into this study. The results on adeles and ideles which are discussed here are aimed towards their application, in the next chapter, of producing discrete arithmetic subgroups of finite covolume. They will also enable us to make volume calculations on arithmetic Kleinian and Fuchsian groups in subsequent chapters. For these purposes and other applications subsequently, there are two crucial results here. One is the Strong Approximation Theorem, which is proved in the last section of this chapter. The other, which is central in subsequent results giving the covolume of arithmetic Fuchsian and Kleinian groups in terms of the arithmetic data, is that the Tamagawa number is 1. The Tamagawa number is the volume of a certain quotient of an idele group measured with respect to its Tamagawa measure. The Tamagawa measures can be invariantly defined on the local components of the rings of adeles and groups of ideles and these are fully discussed here. The relevant quotients are shown to be compact and so will have finite volume. The proof that the Tamagawa volume, which is, by definition, the Tamagawa number, is precisely 1, is not included.

In this chapter, arithmetic Kleinian groups are described in terms of quaternion algebras. An almost identical description leads to arithmetic Fuchsian groups. Both of these are special cases of discrete groups which arise from the group of elements of norm 1 in an order in a quaternion algebra over a number field. Such groups are discrete subgroups of a finite product of locally compact groups, which will be shown, using the results of the preceding chapter, to give quotient spaces of finite volume. Suitable arithmetic restrictions on the quaternion algebras then yield discrete subgroups of SL(2, ℂ) and SL(2, ℝ) of finite covolume and in this way, the existence of arithmetic Kleinian and arithmetic Fuchsian groups is obtained.

The basic algebraic theory of quaternion algebras was given in Chapter 2. That sufficed for the results obtained so far on deducing information on a Kleinian group Γ from its invariant trace field kΓ and invariant quaternion algebra AΓ. We have yet to expound on the arithmetic theory of quaternion algebras over number fields. This will be essential in extracting more information on the quaternion algebras and, more importantly, in deducing the existence of discrete Kleinian groups of finite covolume. These will be arithmetic Kleinian groups about which a great deal of the remainder of the book will be concerned. All of this is based around the structure of orders in quaternion algebras which encapsulate the arithmetic theory of quaternion algebras. These were introduced in Chapter 2, but we now give a more systematic study, particularly from a local-global viewpoint.

In this chapter we shall cover some of the basics needed to start using Maple, to access the on-line help, and to compute with numbers. A discussion at length of how to interact with Maple will follow in Chapter 4. During the overview of the number fields that are supported by Maple, we shall also introduce you gently to the way Maple stores data internally.

The main algebraic invariants associated to a Kleinian group are its invariant trace field and invariant quaternion algebra. For a finite-covolume Kleinian group, its invariant trace field is shown in this chapter to be a number field (i.e., a finite extension of the rationals). This allows the invariants and the algebraic number-theoretic structure of such fields to be used in the study of these groups. This will be carried out in subsequent chapters. The invariant trace field is not, in general, the trace field itself but the trace field of a suitable subgroup of finite index. It is an invariant of the commensurability class of the group and that is established in this chapter. This invariance applies more generally to any finitely generated non-elementary subgroup of PSL(2, C). Likewise, the invariance, with respect to commensurability, of the associated quaternion algebra is also established. Given generators for the group, these invariants, the trace field and the quaternion algebra, can be readily computed and techniques are developed here to simplify these computations.

This chapter gathers together number-theoretic concepts and results which will be used at various stages throughout the book. There are few proofs in this chapter and it should be regarded as a synopsis of some of the main results in algebraic number theory, the proofs and details of which can be found in one of the many excellent texts on algebraic number theory. Being labelled Chapter 0, the implication is that this is a reference section, and key results given in this chapter will be referred back to subsequently as required in the book. It is certainly not necessary for the reader to absorb all the material here before proceeding further. The basic ideas in Sections 0.1,0.2 and 0.3 will arise frequently in the succeeding chapters. However, until Chapter 6, only these basic ideas together with, in a couple of sections, some ideas from Sections 0.6,0.7 and 0.9 are required to understand the proofs and examples. Thus we suggest that the readers with a passing familiarity with basic notions in algebraic number theory should return to this chapter only when they encounter a concept with which they are unfamiliar.

In this chapter, we return to considering arithmetic Kleinian and Fuchsian groups and the related quaternion algebras. Recall that the wide commensurability classes of arithmetic Kleinian groups are in one-to-one correspondence with the isomorphism classes of quaternion algebras over a number field with one complex place which are ramified at all the real places. There is a similar one-to-one correspondence for arithmetic Fuchsian groups. Thus, for a suitable quaternion algebra A, let C(A) denote the (narrow) commensurability class of associated arithmetic Kleinian or Fuchsian groups. In this chapter, we investigate how the elements of C(A)are distributed and, in particular, determine the maximal elements of C(A) of which there are infinitely many. Since these groups are all of finite covolume, their volumes are, of course, commensurable. As a starting point to determining these volumes, a formula for the groups Pρ(O 1), where O is a maximal order in A, is obtained in terms of the number-theoretic data defining the number field and the quaternion algebra. This relies critically on the fact that the Tamagawa number of the quotient A A 1 /A k 1 of the idele group A A 1, is 1, as discussed in Chapter 7. From this formula, one can determine the covolumes of the maximal elements of C(A)and show that all of these volumes are integral multiples of a single number. Much of this chapter is based on work of Borel.

In this chapter, we will discuss the structure and properties of the set of closed geodesics, particularly in arithmetic hyperbolic 2- and 3-orbifolds. As discussed briefly in §5.3.4, this is closely connected to properties of loxodromic elements in Kleinian or Fuchsian groups, and in the case of arithmetic groups, the traces and eigenvalues of these loxodromic elements carry extra arithmetic data that can be used to help understand the set of geodesics in arithmetic hyperbolic 3-manifolds. We also consider torsion that arises in arithmetic Fuchsian and Kleinian groups. Although, on the face of things, this appears to have little to do with lengths, the existence of torsion and eigenvalues of loxodromic elements in arithmetic groups is closely tied to the algebra and number theory of the invariant trace field and quaternion algebra. In particular, their existence depends on the existence of embeddings into the quaternion algebra of suitable quadratic extensions of the defining field. Such embeddings were characterised in Chapter 7 and these results are refined in this chapter to consider embeddings of orders inside these quadratic extensions into orders in the quaternion algebras.

Let K/F be an algebraic function field over the field of constants F. Throughout this book we have been emphasizing the analogy between the arithmetic of K and that of an algebraic number field. This analogy is particulary clear when we choose an element x ∈ K which is not a constant. The ring A = F[x] ⊂ k = F(x) then plays the role of the pair ℤ ⊂ ℚ in number theory. K is an algebraic extension of F(x) and the analogue of the ring of integers in an algebraic number field is the integral closure of A in K. Let’s call this ring B. We will show that B is a Dedekind domain. We will investigate the unit group and the class group of B. We will also associate zeta and L-functions to B.

In the last chapter we explored the arithmetic of constant field extensions and noted (as was pointed out by Iwasawa) that these extensions can be thought of as function field analogues of cyclotomic extensions of number fields. This analogy led to various conjectures about the behavior of class groups in number fields which have proved very fruitful for the development of algebraic number theory and arithmetic geometry. There is another function field analogy to cyclotomic number fields which was first discovered by L. Carlitz [3] in the late 1930s. This ingenious analogy was not well known until D. Hayes, in 1973, published an exposition of Carlitz’s idea and showed that it provided an explicit class field theory for the rational function field (see Hayes [1]). Later developments, due independently to Hayes and V. Drinfeld, showed that Carlitz’s ideas can be generalized to provide an explicit class field theory for any global function field, i.e., an explicit construction of all abelian extensions of such a field (see Drinfeld [1] and Hayes [2]). This is a complete solution to Hilbert’s 9-th problem in the function field case. Nothing remotely so satisfying is known for number fields except for the field of rational numbers (cyclotomic theory) and imaginary quadratic number fields (the theory of complex multiplication).

We have seen in the numerical examples of the preceding chapter that the ring of algebraic integers of a quadratic number field, and also of the cyclotomic field ℚ(ζ) (where ζ is a primitive pth root of unity), are free finitely generated Abelian groups.

Let K be an algebraic number field, L|K a finite extension of degree n, and, as before, let A, B be, respectively, the rings of integers of K, L. Let P be a prime ideal of A, and let \( B\underline P = \Pi _{i = 1}^gP_i^{ei} \) be the decomposition of BP into a product of prime ideals, with \( {f_i} = [B/P{}_i:A/\underline P ] \). We shall study in more detail how this decomposition takes place. This has been done by Hilbert, assuming that L|K is a Galois extension.

In this chapter, we begin the study of extensions of ideals. Let K be an algebraic number field, let LK be an extension of finite degree, and let A (respectively, B), be the rings of algebraic integers of K (respectively, L). Let I be any nonzero fractional ideal of K. The aim of this study is to relate the decomposition of I into prime ideals of A, with the decomposition into prime ideals of B, of the fractional ideal of L generated by I.

In this chapter we introduce the important Dedekind zeta-function of an algebraic number field and obtain an asymptotic expression for the class number.

Let K be a number field, and let L|K be a Galois extension of degree n. We denote by A L , (respectively, A K ), the ring of algebraic integers of L (respectively, K).

We know already that the ring A of integers of an algebraic number field K need not be a principal ideal domain. In this chapter, we associate with every field K a numerical invariant h, which measures the extent to which A deviates from being a principal ideal domain. h will be equal to 1 if and only if A is a principal ideal domain.

We have shown that the ring A of algebraic integers of an algebraic number field is Noetherian and integrally closed. However, it is not true in general that A is a principal ideal domain.

Let K be an algebraic number field, L an extension of finite degree n over K, and let A, B be, respectively, the rings of algebraic integers in K, L. In this situation, A need not be a principal ideal domain and B need not be a free A-module. We shall introduce the relative trace and norm of fractional ideals of L and, in view of characterizing ramified prime ideals, we shall consider the relative discriminant and relative different.

The easiest way to start studying number fields is to consider them per se, as absolute extensions of √; this is, for example, what we have done in [Coh0]. In practice, however, number fields are frequently not given in this way. One of the most common other ways is to give a number field as a relative extension, in other words as an algebra L/K over some base field K that is not necessarily equal to √. necessarily equal to ℚ. In this case, the basic algebraic objects such as the ring of integers ℤ L and the ideals of ℤ L , are not only ℤ-modules, but also ℤ K- modules. The ℤ K -module structure is much richer and must be preserved. No matter what means are chosen to compute ℤ L , we have the problem of representing the result. Indeed, here we have a basic stumbling block: considered as ℤ-modules, ℤ L or ideals of ℤ L are free and hence may be represented by ℤ-bases, for instance using the Hermite normal form (HNF); see, for example, [Coh0, Chapter 2]. This theory can easily be generalized by replacing ℤ with any other explicitly computable Euclidean domain and, under certain additional conditions, to a principal ideal domain (PID). In general, ℤ K is not a PID, however, and hence there is no reason for ℤ L to be a free module over ℤ K- A simple example is given by K = ℚ(√-10) and L = K(√1) (see Exercise 22 of Chapter 2).

Class field theory deals with Abelian extensions of base fields. It gives complete answers to the existence of Abelian extensions with given relative or absolute discriminants. However, the algorithmic construction of these extensions is not completely straightforward. There are several ways to do this, but at present the most efficient general method is the use of Kummer extensions. In the next chapter, we will describe two other methods using analytic techniques, one using Stark units and Stark’s conjecture, the other using complex multiplication. Both of these methods impose restrictions on the base field, but when they are applicable they are much more efficient.

Having the necessary tools for dealing theoretically and algorithmically with modules over Dedekind domains, we are now going to study in detail relative extensions of number fields. In the first section, we emphasize the field-theoretic properties, while in the rest of this chapter we study the ring-theoretic properties.