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Analysis on Arithmetic Schemes. II

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Analysis on Arithmetic Schemes. II Analysis on arithmetic schemes. II Ivan Fesenko Abstract. We construct adelic objects for rank two integral structures on arithmetic surfaces and develop measure and integration theory, as well as elements of harmonic analysis. Using the topological Milnor K2-delic and K1 × K1-delic objects associated to an arithmetic surface, an adelic zeta integral is defined. Its unramified version is closely related to the square of the zeta function of the surface. For a proper regular model of an elliptic curve over a global field, a two-dimensional version of the theory of Tate and Iwasawa is derived. Using adelic analytic duality and a two-dimensional theta formula, the study of the zeta integral is reduced to the study of a boundary integral term. The work includes first applications to three fundamental properties of the zeta function: its meromorphic continuation and functional equation and a hypothesis on its mean periodicity; the location of its poles and a hypothesis on the permanence of the sign of the fourth logarithmic derivative of a boundary function; and its pole at the central point where the boundary integral explicitly relates the analytic and arithmetic ranks. Contents Introduction ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 2 1. A-delic objects with respect to two-dimensional structure :::::::::::::::::::::::::::::: 9 1.1. Basic objects ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 10 1.2. Adelic dualities ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 16 1.3. Adelic measure and integration ::::::::::::::::::::::::::::::::::::::::::::::::: 22 2. Two-dimensional K-delic objects :::::::::::::::::::::::::::::::::::::::::::::::::: 33 3. Two-dimensional zeta integral and its first properties ::::::::::::::::::::::::::::::::: 37 3.1. Group T and integrals over it :::::::::::::::::::::::::::::::::::::::::::::::::::39 3.2. Zeta function ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 46 3.3. Definition of the zeta integral :::::::::::::::::::::::::::::::::::::::::::::::::: 47 3.4. First calculation of the zeta integral ::::::::::::::::::::::::::::::::::::::::::::: 49 3.5. Subgroups of T and associated integrals ::::::::::::::::::::::::::::::::::::::::: 55 3.6. Two-dimensional theta-formula :::::::::::::::::::::::::::::::::::::::::::::::: 61 3.7. Second calculation of the zeta integral :::::::::::::::::::::::::::::::::::::::::: 63 4. The boundary term and first applications :::::::::::::::::::::::::::::::::::::::::::: 67 4.1. The boundary function :::::::::::::::::::::::::::::::::::::::::::::::::::::::: 68 4.2. Mean-periodicity and meromorphic continuation of the zeta integral ::::::::::::::::70 4.3. Monotone behaviour of log derivatives of the boundary function and poles of the zeta integral ::::::::::::::::::::::::::::::::::::::::::::::::::: 74 4.4. Supplementary comments ::::::::::::::::::::::::::::::::::::::::::::::::::::: 86 4.5. Behaviour at the central point :::::::::::::::::::::::::::::::::::::::::::::::::: 88 Bibliography ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::90 List of notation ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::93 a slightly updated version (see footnotes) of the original published in J. K-theory, 5(2010), 437–557 2 I. Fesenko Analysis on arithmetic schemes. II Introduction 0. A conceptual way to understand the meromorphic continuation and functional equation of zeta functions of global fields and their twists by Dirichlet characters is to lift them to zeta integrals on appropriate adelic objects and then calculate the integrals in two different ways, using a powerful adelic duality. For number fields this first appeared in [24], [57], [25]; for the general case see e.g. [60]. For a continuous homomorphism χ from the idele class group to the multiplicative group of complex numbers, and an appropriate function f on the adeles, one derives the following formula concerning the zeta integral ζ(f; χ): ζ(f; χ) = ξ(f; χ) + ξ(f;b χb) + !(f; χ); <(s(χ)) > 1 in which the first two terms are absolutely convergent integrals on the plane. The boundary term !(f; χ) involves an integral of f and its transform fb over the set f0g, which is the (weak) boundary of the multiplicative group of the global field. This boundary term is a rational function either of s or of q−s. The meromorphic continuation, functional equation, and the location of the poles of the zeta integral and the determination of their residues then follow from analytic properties of the boundary term. This K1-adelic method was extended to algebraic groups over global fields in [18], where one works with zeta integrals of an automorphic representation. The appropriate analogue of the boundary term has finitely many poles. In particular, in the cuspidal case the corresponding boundary term vanishes. 1. Instead of passing from the one-dimensional, commutative theory, which uses GL1 or K1 of one-dimensional local, global, and adelic objects, to the one-dimensional, noncommutative theory, i.e. using algebraic groups of the objects, this work goes in the direction of a commutative, two- dimensional theory in the following sense. The analogue of a global field is a two-dimensional global field, namely the field of rational functions of an arithmetic surface, which in this text will be a proper, regular model of a curve over a (one-dimensional) global field. Abelian extensions of a two-dimensional global field are described by higher class field theory, one exposition of which t uses products, restricted in a two-dimensional sense, of topological Milnor K2-groups of completed and semi-completed objects associated to points and curves on the surface. In this work, without using higher class field theory, but working with subobjects of the objects from explicit higher class t field theory, we lift the zeta function of the arithmetic surface to a zeta integral on K2-delic and K1 × K1-delic objects, and then develop a two-dimensional, commutative generalization of Tate and Iwasawa’s one-dimensional theory. Thus, unlike almost all works in arithmetic geometry, which study the zeta function of arithmetic schemes in dimension two and higher via the study of their L-function factors, the two-dimensional adelic analysis of this paper directly operates with the zeta function. The key ingredient, which has not been available until recently, is the theory of translation invariant measure and integration on higher dimensional local fields, developed in [11], [12]. To introduce the zeta integral we first construct the theory of measure and integration on new adelic objects. The two-dimensional adelic analysis reveals various kinds of new adelic symmetries underlying the properties of the zeta function, which are not explicitly visible in the approaches which study the L-function factors. Milnor K-theory plays an important role too, both for the motivation of some of the main objects and for their use. There are numerous links between the commutative, two-dimensional adelic theory of this paper and the one-dimensional, noncommutative theories, including aspects of the arithmetic and geometric Langlands correspondences. Analysis on arithmetic schemes. II I. Fesenko 3 The ideles, introduced by Chevalley, simplified and clarified some of fundamental structures ⊕n in algebraic number theory. Similarly, for a group G (G = Ga, Kn, or K1 ), G-adeles in higher dimensions are expected to play a central role in arithmetic geometry. Deep theorems in the arithmetic of elliptic curves in characteristic zero have so far been proved only under special restrictions on the ground global field, using quite specific methods which unfortunately cannot be extended to an arbitrary global field. In contrast, adelic methods are universal and they work over an arbitrary global field without any restrictions. The adelic perspective in dimension two leads to a unified whole understanding of the connections between three fundamental aspects of zeta functions: the meromorphic continuation and functional equation, the location of poles, and the behaviour at the central point. 2. This text deals with the relative situation of a two-dimensional, arithmetic scheme S over a one-dimensional base B, the latter being the spectrum of the ring of integers of a global number field k in characteristic zero, or a proper, smooth, connected curve over a finite field. The central case on which we concentrate in this work is that of a proper, regular model of an elliptic curve over a global field, which is the first nontrivial case of an arithmetic surface. The zeta integral in this case converges in the best possible way. In the case of proper, regular models of hyperbolic curves over global fields one should renormalize the fibre products of local zeta integrals, using the projective space over B. Prerequisites for this work are the one-dimensional theory [57], [24] and the higher dimensional, local theory [11], [12]. A closely related text which explains the main ideas, methods, constructions, and directions of applications and which is relatively free from technical details is [14]. The current text simplifies some of the objects in [14]. A brief review of [11] is included at the beginning of parts 1 and 3. The numeration of the sections continues that of [11], and its sections are referred to without explicit mentioning of [11]. The sections form rather natural divisions of the material of the paper and they are of varying length; such a simple marking of the text aims to simplify its reading. The two-dimensional
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