There is a strong analogy between number fields and function fields, hence between the theories of the two kinds of global fields. For instance, one may consider analogies between the \(p\)-adic numbers and Laurent series over \(\mathbb{F}_p\), or, much deeper, one has the notion of the genus of a number field, which is analogous to the genus of an algebraic curve on which a given function field is the field of rational functions (see [A. Weil, Rev. Sci. Paris 77, 104–106 (1939; JFM 65.1140.01)]). These analogies have brought important number-theoretic applications. For example the genus notion makes the statement of the Riemann-Roch theorem for algebraic curves extend to arithmetic geometry. It is thus expected that the geometry of arithmetic schemes, i.e. schemes of finite type over \(\mathbb{Z}\), should behave similar to the algebraic geometry of schemes of finite type over a regular projective curve.
However, this is just an expectation and it seems that the tools that have been developed up to the present point make it difficult to approach both settings in an uniform way. For example, the core problem of arithmetic schemes is that they are not compact. The main idea of the Arakelov intersection theory of arithmetic schemes is to “compactify” the scheme by considering the associated complex analytic variety at infinity, endowed it with analytic data such as hermitian metrics, Green currents, etc. This approach has led to rich number theoretical applications such as the proof of Mordell’s conjecture by G. Faltings [Invent. Math. 73, No. 3, 349–366; erratum ibid. 75, 381 (1983; Zbl 0588.14026)] and to the Bogomolov conjecture (see e.g. [S. W. Zhang, Ann. of Math. (2) 147, No. 1, 159–165 (1998; Zbl 0991.11034)]). Even though this approach has led to such important results, the proofs mostly involve subtle tools in complex analysis and the transition between algebraic and analytic techniques is sometimes hard to track. This makes that statements in the number field case do not necessarily lead to analogous statements in the function field case and viceversa.
The present manuscript provides a uniform fundament for Arakelov geometry for both settings. This is done by introducing the notion of an adelic curve. This is a field equipped with a family of absolute values parametrized by a measure space such that the logarithmic absolute value of each non-zero element of the field is an integrable function on the measure space with \(0\) as its integral. The latter is called the product formula. With this notion, the authors develop an Arakelov theory over adelic curves. As the authors point out, the notion of an adelic curve has already been present in the literature, although in somewhat less general settings (see e.g. [W. Gubler, Symp. Math. 37, 190–227 (1997; Zbl 0991.11034)]).
In the present book, the authors first formalize the notion of adelic curves and discuss algebraic covers thereof, which are fundamental for the notion of heights in this setting. Then they set up the theory of adelic vector bundles over adelic curves and discuss adelic analogues of stability and slope theory. They also discuss arithmetic invariants of adelic vector bundles such as the Arakelov degree. Then they study metrized line bundles on arithmetic varieties over adelic curves. These differ from the classical setting of adelic metrics in [S. W. Zhang, J. Algebr. Geom. 4, No. 2, 281–300 (1995; Zbl 0861.14019)] and [A. Morowaki, Mem. Am. Math. Soc. 242, No. 1144, 122 p. (2016; Zbl 1388.14076)] which are based on the notion of model metric, and this notion is no longer adequate in this general setting. Positivity properties of these metrized line bundles are also discussed. In the last chapter the authors give a generalization of the Nakai-Moishezon’s criterion settled in the setting of Arakelov geometry over an adelic curve.
Assuming basic knowledge of algebraic geometry and algebraic number theory, this manuscript is almost self-contained. It is suitable for researchers in arithmetic geometry as well as graduate students focusing on these topics.