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Analysis on arithmetic schemes. II. (English) Zbl 1225.14019

From the introduction: “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 ways, using a powerful adelic duality.” This approach to the Dedekind zeta function of a global field and its twists was introduced by J. T. Tate in this thesis [reprint in: Algebraic number theory. Proceedings of an instructional conference organized by the London Mathematical Society (a NATO Advanced Study Institute) with the support of the International Mathematical Union. London and New York: Academic Press (1967; Zbl 0153.07403)], and K. Iwasawa [Letter to J. Dieudonné. Zeta functions in geometry. Tokyo: Kinokuniya Company Ltd. Adv. Stud. Pure Math. 21, 445–450 (1992; Zbl 0835.11002)]. The aim of the present article may be neatly summarised as follows: Extend the Tate-Iwasawa method to dimension two, to study the zeta function \(\zeta_S(s)\) of a regular arithmetic surface \(S\). This approach is also novel in that it studies the zeta function of \(S\) directly, whereas most approaches examine the \(L\)-function of the generic fibre. There are numerous technical obstacles to this programme, many of which are overcome in the present article and the author’s previous articles.
The two-dimensional local fields and groups of adeles (\(=\) certain restricted products of two-dimensional local fields or their various rings of integers) associated to a two dimensional, Noetherian scheme were first introduced by A. Parshin [Math. USSR, Izv. 10(1976), 695–729 (1977; Zbl 0366.14003)] (the theory was extended to arbitrary dimension by A. Beilinson [Funct. Anal. Appl. 14, 34–35 (1980); translation from Funkts. Anal. Prilozh. 14, No. 1, 44–45 (1980; Zbl 0509.14018)]). As opposed to familiar local fields, two-dimensional local fields are not locally compact, therefore do not posses a Haar measure, and new theories of measure and harmonic analysis are required if the techniques of Tate and Iwasawa are to be carried out. These were developed in part I of the author’s article [Doc. Math., J. DMV Extra Vol., 261–284 (2003; Zbl 1130.11335)] (there is a different approach by the reviewer [Tokyo J. Math. 33, No. 1, 235–281 (2010; Zbl 1203.11081)]) and the required results are summarised at the beginning of §1.
§1.1 then introduces the local, global, and adelic objects associated to \(S\); the various groups of adeles are typically are not the same as Parshin’s but are smaller, with stricter conditions defining the restricted products, so that they admit restricted product measures from their two-dimensional local field components. This extension of the measure theory to the groups of adeles is carried out in §1.3, where topologies on the adelic groups are also described.
§2 is a summary, without proofs, of the idelic approach to two-dimensional global class field theory. The results are not required for the remainder of the paper, but are included to motivate the appearance of \(K_2\) groups, especially \(K_2\)-idele groups (\(=\)restricted products of \(K_2\) of two-dimensional local fields or their various rings of integers). In the case of an algebraic surface, this idelic approach to global class field theory was described by A. Parshin [Sov. Math., Dokl. 19, 1438-1442 (1978); translation from Dokl. Akad. Nauk SSSR 243, 855–858 (1978; Zbl 0443.12006)].
By analogy with Tate’s thesis, one would expect the zeta function of \(S\) to be defined as an integral, over a suitable idele group, of a well-behaved function on an adele group, twisted by the absolute value. This is the most challenging problem in the theory: the \(K_2\)-idele group does not sit inside any adele group and, in any case, integrating on it remains an open problem. This is circumnavigated in §3.1 by introducing a “\(K_1\times K_1\)-idele group” (\(=\) restricted product of \(\mathbb{G}_m\times\mathbb{G}_m\) of rings of integers of two-dimensional local fields) \(T\), equipped with a surjective homomorphism to the \(K_2\)-idele group; thus \(T\) is thought of as a covering of the \(K_2\)-idele group which forgets most information about ramification. In §3.4 it is shown from elementary calculations of integrals that \(\{\text{conductor factor}\}\times\zeta_S(s)^2\) can be realised as an integral over \(T\), and this provides an adelic interpretation of the conductor of \(S\). The appearance of the squaring factor is a natural side-effect of integrating on \(T\) and means that the difficulty of determining the sign of the functional equation is avoided.
The deepest result of the paper is the “two-dimensional theta formula” of §3.6, which provides the analogue for \(S\) of the Poisson summation formula for the ring of integers of a number field. (It is worth noting that in §1.2 the usual Poisson summation formula, or equivalently Riemann-Roch, is easily lifted to the adele groups of \(S\), and this should not be confused with the two-dimensional theta formula). It is a Stokes-type formula, stating that integrals over a certain global subgroup \(T_0\subset T\) (defined in §3.5) can be calculated as integrals over the topological boundary \(\partial T_0\). This is used in §3.6 to show that \(\zeta_S(s)^2\) differs, by entire functions which satisfy all expected properties, from an integral over \(\partial T_0\), which the author calls “the boundary term”.
So, finally, meromorphic continuation, functional equation, Riemann hypothesis, and Birch and Swinnerton-Dyer are all reduced to analogous problems for the boundary term, and thus one obtains new adelic interpretations of these problems. For example, working with the zeta function rather than the \(L\)-function means that automorphic properties are lost, but the author shows that the the notion of mean-periodicity from functional analysis now plays the equivalent role: in §4.2 it is shown that if \(H(t)\), a certain \(\mathbb{R}\)-valued function on \(\mathbb{R}\) of which the boundary term is an integral transform, is mean-periodic then the zeta function and \(L\)-function admit meromorphic continuation to the complex plane and satisfy the expected functional equation.
Moreover, in §4.3, it is shown that \(H(t)\) can be expressed classically, i.e. without using two-dimensional adeles, as an infinite sum of integral transforms of theta functions. This opened the door to the possibility of a Langlands type correspondence between mean-periodic functions and zeta functions (in place of modular forms and \(L\)-functions), a line of thought which has been developed by the author, M. Suzuki, and G. Ricotta [Mean periodicity and zeta functions, awaiting publication in Annales de l’Institut Fourier].
Similarly, in §4.3 it is shown that if the fourth derivative of \(H(t)\) doesn’t have infinitely many zeros as \(t\to\infty\), and assuming the real poles of \(\zeta_S\) in the critical strip are on the critical line, then \(\zeta_S\) satisfies the generalized Riemann hypothesis. See M. Suzuki [J. Number Theory 131, No. 10, 1770–1796 (2011; Zbl 1237.11028)] for further related research.
The programme leads in many further directions and offers a variety of open problems, some of which are included as remarks throughout the paper.

MSC:

14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
19F05 Generalized class field theory (\(K\)-theoretic aspects)
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11G05 Elliptic curves over global fields
11G99 Arithmetic algebraic geometry (Diophantine geometry)
19M05 Miscellaneous applications of \(K\)-theory
14C99 Cycles and subschemes
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