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2016 | Buch

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Rigid Geometry of Curves and Their Jacobians

verfasst von: Werner Lütkebohmert

Verlag: Springer International Publishing

Buchreihe : Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics

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Über dieses Buch

This book presents some of the most important aspects of rigid geometry, namely its applications to the study of smooth algebraic curves, of their Jacobians, and of abelian varieties - all of them defined over a complete non-archimedean valued field. The text starts with a survey of the foundation of rigid geometry, and then focuses on a detailed treatment of the applications. In the case of curves with split rational reduction there is a complete analogue to the fascinating theory of Riemann surfaces. In the case of proper smooth group varieties the uniformization and the construction of abelian varieties are treated in detail.

Rigid geometry was established by John Tate and was enriched by a formal algebraic approach launched by Michel Raynaud. It has proved as a means to illustrate the geometric ideas behind the abstract methods of formal algebraic geometry as used by Mumford and Faltings. This book should be of great use to students wishing to enter this field, as well as those already working in it.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Classical Rigid Geometry

Abstract

In this chapter we give a survey of rigid geometry over non-Archimedean fields. The foundation of the theory was laid by Tate in his private Harvard notes dating back to 1961, which were later published in Inventiones mathematicae (Tate in Invent. Math. 12:257–289, 1971). Here we explain the main results from the classical point of view as studied in the late sixties; for proofs we refer to Bosch (Lectures on Formal and Rigid Geometry, vol. 2105, 2014). At that time rigid geometry was mainly inspired by complex analysis. Fundamental results were achieved by Kiehl, who introduced the Grothendieck topology and proved the basic facts concerning coherent sheaves. Moreover, Kiehl makes Serre’s theory (Serre in Ann. Inst. Fourier 6:1–42, 1956) of Géométrie Algébrique et Géométrie Analytique available for rigid analytic geometry, often referred to as GAGA; cf. (Köpf in Über eigentliche Familien algebraischer Varietäten über affinoiden Räumen, vol. 7, 1974).

We present the essential results on Tate algebras and affinoid spaces which are the building blocks of rigid geometry. By means of the Grothendieck topology we define rigid analytic spaces. Kiehl’s results on coherent sheaves are stated without proofs. As a general reference we refer to Bosch et al. (Non-Archimedean Analysis, vol. 261, 1984) and the more recent account Bosch (Lectures on Formal and Rigid Geometry, vol. 2105, 2014). We always assume that \(K\) is a non-Archimedean field in the sense of Definition 1.1.1 unless otherwise stated.

Werner Lütkebohmert

Chapter 2. Mumford Curves

Abstract

The incentive to create the theory of holomorphic functions over a non-Archimedean field was Tate’s elliptic curve. By means of rigid geometry one can explain Tate’s elliptic curve from the geometric point of view, whereas Tate originally formulated it in terms of function fields; cf. Sect. 2.1.

In the following sections we study Mumford’s generalization of Tate’s curve to curves of higher genus in the context of rigid geometry. We introduce discontinuous actions of certain subgroups \(\varGamma\) of \(\operatorname{PGL}(2,K)\) on the projective line in the style of Schottky. The structure of these groups \(\varGamma\) was found by Ihara; cf. Sect. 2.2.

Mumford curves will be introduced as orbit spaces \(\varGamma\backslash \varOmega\), where \(\varOmega\subset\mathbb{P}_{K}^{1}\) is the largest subdomain of \(\mathbb{P}_{K}^{1}\) on which \(\varGamma\) acts in an ordinary way. The construction of the quotient \(\varGamma\backslash\varOmega\) can be carried out in the framework of classical rigid geometry. Note that Mumford achieves much more general results in his ground braking article (Mumford in Compos. Math. 24:129–174, 1972) which deals exclusively with formal schemes. The concept here follows geometric constructions in order to explain the ideas behind Mumford’s construction.

Chapter 2 is somehow a counterpart of Riemann surfaces and their Jacobians. We provide the full picture of Mumford curves and their Jacobians which are rigid analytic tori. We show the duality theory of rigid analytic tori, Riemann’s period relations and, moreover, Riemann’s vanishing theorem.

Our approach is a refined version of the work of Drinfeld and Manin (J. Reine Angew. Math. 262/263:239–247, 1973) and Manin (Itogi Nauki Teh., Ser. Sovrem. Probl. Mat. 3:5–92, 1974), where they work over a \(p\)-adic field; i.e., a finite extension of \(\mathbb{Q}_{p}\). Here we consider a general non-Archimedean field as defined in Definition 1.1.1; notably we mention the work of Gerritzen (Math. Ann. 210:321–337, 1974; J. Reine Angew. Math. 297:21–34, 1978; Math. Ann. 196:323–346, 1972).

Werner Lütkebohmert

Chapter 3. Formal and Rigid Geometry

Abstract

In 1974 Raynaud proposed a program (Raynaud in Mém. Soc. Math. Fr. 39–40:319–327, 1974), where he introduced groundbreaking ideas to rigid geometry by interpreting a rigid analytic space as the generic fiber of a formal schemes over \(\operatorname{Spf}R\). Here \(\operatorname{Spf}R\) is always the formal spectrum of a complete valuation ring \(R\) of height 1, where its topology is given by an ideal \((\pi)\) for some element \(\pi\in R\) with \(0<|\pi|<1\). Due to results on flat modules (Raynaud and Gruson in Invent. Math. 13:1–89, 1971) his approach also works in the non-Noetherian case of formal schemes of topologically finite presentation over \(\operatorname{Spf}(R)\).

In Sect. 3.1 we start with a mild attempt to understand formal schemes by considering formal analytic structures on rigid analytic spaces; these consist of extra data on a given space. This allows us to define a reduction of a rigid analytic space without using the abstract method of formal schemes. For the first time Bosch introduced such spaces in Bosch (Manuscr. Math. 20:1–27, 1977).

In Sect. 3.2 admissible formal \(R\)-schemes and formal blowing-ups are defined. In a canonical way the generic fiber of an admissible formal \(R\)-scheme is a formal analytic space.

In Sect. 3.3 we will discuss the important result in Theorem 3.3.3 of Raynaud about the relationship between formal schemes and rigid analytic spaces. We omit the proof of this theorem. It heavily relies on the flattening technique (Raynaud and Gruson in Invent. Math. 13:1–89, 1971); all details were worked out by Mehlmann (in Ein Beweis für einen Satz von Raynaud über flache Homomorphismen affinoider Algebren, vol. 19, 1981) and by Bosch and the author (in Math. Ann. 295:291–317, 1993; Part II). Moreover, by using the flattening technique many properties of rigid analytic morphisms can be transferred to suitable formal \(R\)-models. In particular, the notions of properness in rigid and formal geometry correspond to each other; cf. Theorem 3.3.12.

Already at the level of Sect. 3.1 a major problem shows up; namely, if the structural rings of a formal analytic space are of topologically finite type over \(R\). When the base field is algebraically closed, this question was answered by Grauert-Remmert (in Invent. Math. 2:87–133, 1966). One can also approach the problem from an opposite direction; namely, how to arrange an \(R\)-model of an affinoid algebra which is of topologically finite type over \(R\) with reduced special fiber. This is a deep problem which was settled by Epp if \(R\) is a discrete valuation ring and by Bosch, Raynaud and the author in the relative case. The latter approach is quite natural as it also works over a general admissible formal scheme in Theorem 3.4.8.

In Sect. 3.4 we explain the major steps of this approach. In particular, it is a first step to provide a semi-stable \(R\)-model of a curve in Theorem 4.4.3 and of a curve fibration; cf. Theorem 7.5.2.

In the last Sect. 3.6 we provide new methods about approximations which are only used in Chap. 7. This part is deeply related to the significance of properness of rigid analytic spaces and to Elkik’s method on approximation of solutions of equations in restricted power series.

In the whole chapter let \(K,R,k,\pi\) be the standard notations as defined in the Glossary of Notations.

Werner Lütkebohmert

Chapter 4. Rigid Analytic Curves

Abstract

The main objective of this chapter is the Stable Reduction Theorem 4.5.3 for smooth projective \(K\)-curves \(X_{K}\). Its proof is split into two problems. In a first step, dealt with in Sect. 3.4, we provide a projective \(R\)-model \(X\) of \(X_{K}\) such that its special fiber \(X\otimes_{R}k\) is reduced. In a second step we will now analyze the singularities of \(X\otimes _{R}k \). This part is related to the resolution of singularities in dimension 2.

For each point \(\tilde{x}\) of the special fiber \(X\otimes_{R}k\) we have the formal fiber \(X_{+}(\tilde{x})\); cf. Definition 3.1.6(d). A cornerstone towards the Stable Reduction Theorem is the presentation of the periphery of \(X_{+}(\tilde{x})\) in Proposition 4.1.11. This is a precise identification how the interior of the formal fiber is connected to the remainder of the curve. Noteworthy, we do not make use of a desingularization result (Lipman in Ann. Math. 107:151–207, 1978) as the usual proofs do in Artin and Winters (in Topology 10:373–383, 1971) or Deligne and Mumford (in Publ. Math. IHES 36:75–109, 1969), see also Raynaud (in Proceedings of the Conference on Fundamental Groups of Curves in Algebraic Geometry Held in Luminy, vol. 187, 2000; Chap. 5).

In Sect. 4.2 the result on the periphery is used to constitute a genus formula in Proposition 4.2.6 which relates the genus of a projective rigid analytic curve to geometric data of the reduction. The formula allows us to define the genus of a formal fiber which serves as a measure for the quality of the singularity in the reduction. From these results we deduce the Stable Reduction Theorem in Sect. 4.4 for smooth projective curves by studying the behavior of meromorphic functions in Sect. 4.3. Blowing-up and blowing-down of components in the reduction can easily be handled by changing formal analytic structures.

Finally, the Stable Reduction Theorem leads in Sect. 4.6 to a construction of a universal covering of a curve. In the case of a split rational reduction the universal covering can be embedded into the projective line and its deck transformation group is a subgroup of \(\operatorname{PGL}(2,K)\), which actually is a Schottky group. Finally we obtain a characterization of Mumford curves by conditions on its stable reduction.

We want to mention that there is also a rigid-analytic proof of the Stable Reduction Theorem by van der Put (in Proc. K. Ned. Akad. Wet., Ser. A, Indag. Math. 46(4):461–478, 1984).

In the sections Sect. 4.1 till Sect. 4.3 we assume that our non-Archimedean field \(K\) is algebraically closed.

Werner Lütkebohmert

Chapter 5. Jacobian Varieties

Abstract

The main objective of this chapter is the uniformization of the Jacobian of a smooth projective curve \(X_{K}\) over a non-Archimedean field \(K\) and its relationship to a semi-stable reduction \(\widetilde{X}\) of \(X_{K}\).

We assume that the reader is familiar with the notion of the Jacobian variety of a smooth projective curve over a field; see for instance the article (Milne in Arithmetic Geometry, Springer, Berlin, 1986) or (Bosch et al. in Néron Models, vol. 21, Springer, Berlin/Heidelberg/New York, 1990; Chap. 9). For our purpose it is necessary to have analyzed the generalized Jacobian of a semi-stable curve \(\widetilde{X}\), especially its representation as a torus extension of the Jacobian of its normalization \(\widetilde{X}'\). In Sects. 5.1 and 5.2 we reassemble the main results we need in the sequel.

In Sect. 5.3 it is shown that the generalized Jacobian \(\widetilde{J}:=\operatorname{Jac}{\widetilde{X}}\) has a lifting \(\overline{J}_{K}\) as an open rigid analytic subgroup of \(J_{K}:=\operatorname{Jac}{X_{K}}\) and that \(\overline{J}_{K}\) has a smooth formal \(R\)-model \(\overline{J}\) with semi-abelian reduction. \(\overline{J}\) is a formal torus extension of a formal abelian \(R\)-scheme \(B\) with reduction \(\widetilde{B}=\operatorname{Jac}{\widetilde{X}'}\).

The generic fiber \(\overline{J}_{K}\) of \(\overline{J}\) is the largest connected open subgroup of \(J_{K}\) which admits a smooth formal \(R\)-model; this is discussed in Sect. 5.4 in a more general context. The relationship between the maximal formal torus \(\overline {T}\) of \(\overline{J}\) and the group \(H^{1}(X_{K},\mathbb{Z})\) shows that the inclusion map \(\overline{T}_{K}\hookrightarrow\overline{J}_{K}\) from the generic fiber \(\overline{T}_{K}\) of the formal torus \(\overline{T}\) to \(\overline{J}_{K}\) extends to a rigid analytic group homomorphism \(T_{K}\to J_{K}\), where \(T_{K}\) is the affine torus which contains \(\overline {T}_{K}\) as the torus of units.

The push-out \(\widehat{J}_{K}:=T_{K}\amalg_{\overline{T}}\overline {J}_{K}\) is a rigid analytic group which contains \(\overline{J}_{K}\) as an open rigid analytic subgroup and the inclusion \(\overline {J}_{K}\hookrightarrow J_{K}\) extends to a surjective homomorphism \(\widehat{J}_{K}\to J_{K}\) of rigid analytic groups. The kernel of the latter map is a lattice \(M\) in \(\widehat{J}_{K}\) and makes \(J_{K}=\widehat {J}_{K}/M\) into a quotient of the “universal covering” \(\widehat{J}_{K}\). The representation \(J_{K}=\widehat{J}_{K}/M\) is called the Raynaud representation of \(J_{K}\).

Since every abelian variety is isogenous to a subvariety of a product of Jacobians, one can transfer the results to abelian varieties. This implies Grothendieck’s semi-abelian reduction theorem for abelian varieties; cf. (Grothendieck et al. in Groupes de Monodromie en Géométrie Algébrique, vols. 288, 340, Springer, Berlin/Heidelberg/New York, 1972).

We want to mention that there are also contributions by Fresnel, Reversat and van der Put (in Rigid Analytic Geometry and Its Applications, vol. 218, Birkhäuser Boston, Inc., Boston 2004) and (in Bull. Soc. Math. Fr. 117(4):415–444, 1989).

Werner Lütkebohmert

Chapter 6. Raynaud Extensions

Abstract

In the last chapter we presented the uniformization \(J_{K}=\widehat {J}_{K}/M\) of the Jacobian variety \(J_{K}\) of a connected smooth projective curve. The universal covering \(\widehat{J}_{K}\) is a Raynaud extension; i.e. an affine torus extension of the generic fiber of a formal abelian \(R\)-scheme. The new topic in this chapter is the algebraization result for \(\widehat{J}_{K}\); i.e., that \(\widehat{J}_{K}\) is an algebraic torus extension of an abelian variety with good reduction.

We study this in the more general setting of uniformized abeloid varieties; i.e., of rigid analytic groups in Raynaud representation \(E_{K}/M\), where \(E_{K}\) is a Raynaud extension and where \(M\subset E_{K}\) is a lattice of rank equal to the torus part of \(E_{K}\). This requires a systematic study of Raynaud extensions and their line bundles with \(M\)-action. Thus, one is led to the construction of the dual of a uniformized abeloid variety. The algebraization of a uniformized abeloid variety is related to the existence of a polarization.

Of special interest are the polarizations of Jacobians \(\operatorname{Jac}(X)\). There are two, the usual theta polarization and the canonical polarization which is related to a pairing on the homology group \(H_{1}(X,\mathbb{Z})\) of the curve \(X\). In Sect. 6.5 we discuss these polarizations. This is related to Riemann’s vanishing theorem Corollary 2.9.16 for Mumford curves.

In Sect. 6.6, following the article (Bosch and Lütkebohmert in Topology 30:653–698, 1991) we discuss the results of this chapter on the degeneration data of abelian varieties and compare them with the ones established in Faltings and Chai (Degeneration of Abelian Varieties, vol. 22, Springer, Berlin/Heidelberg/New York, 1990). Prerequisites on torus extensions and cubical structures are surveyed in the Appendix.

Werner Lütkebohmert

Chapter 7. Abeloid Varieties

Abstract

Every connected compact complex Lie group of dimension \(g\) can be presented as a quotient \(\mathbb{C}^{g}/\varLambda\) of the affine vector group \(\mathbb{C}^{g}\) by a lattice \(\varLambda\) of rank \(2g\). From the multiplicative point of view, it can be presented as a quotient \(\mathbb{G}_{m,\mathbb{C}}^{g} /M\) of the affine torus \(\mathbb{G}_{m,\mathbb{C}}^{g}\) by a multiplicative lattice \(M\) of rank \(g\). In the rigid analytic case the situation is more complicated because of the phenomena of good and multiplicative reduction, which in general occur in a twisted form. For example look at the rigid analytic uniformization of abelian varieties of Theorem 5.6.5.

The fundamental example of a proper rigid analytic group \(A_{K}\) is the analytic quotient \(A_{K} = E_{K} / M_{K}\) in Raynaud representation; cf. Definition 6.1.5, where \(E_{K}\) is an extension of a proper rigid analytic group \(B_{K}\) with good reduction by an affine torus \(T_{K}\), where \(M_{K}\) is a lattice in \(E_{K}\) of rank equal to \(\dim T_{K}\); cf. Proposition 6.1.4. The main result of this chapter is that every smooth rigid analytic group, which is proper and connected, is of the form \(E_{K} / M_{K}\) after a suitable extension of the base field. This is a generalization of Grothendieck’s Stable Reduction Theorem (Grothendieck in Groupes de Monodromie en Géométrie Algébrique, vols. 288, 340, Springer, Berlin/Heidelberg/New York, 1972; I, Exp. IX, 3.5) as well as of the rigid analytic uniformization of abelian varieties.

The proof requires advanced techniques; it mainly relies on the stable reduction theorem for smooth curve fibrations which are not necessarily proper. In Sect. 7.5 we compactify such a curve fibration by using the Relative Reduced Fiber Theorem 3.4.8 and approximation techniques provided in Sect. 3.6. Then we can apply the moduli space of marked stable curves. Therefore, one can cover the given group \(A_{K}\) by a finite family of smooth curve fibrations with semi-stable reduction.

In a second step one deduces from such a covering the largest open subgroup \(\overline{A}_{K}\) which admits a smooth formal \(R\)-model \(\overline{A}\) by well-known techniques on group generation dating back to A. Weil; cf. Sect. 7.2. The formal group \(\overline{A}\) is a formal torus extension of a formal abelian \(R\)-scheme \(B\). The prolongation of the embedding \(\overline{T}\hookrightarrow\overline{A}\) of the formal torus to a group homomorphism \(T_{K}\to A_{K}\) of the associated affine torus \(T_{K}\) follows by the approximation theorem and a discussion on the convergence of group homomorphisms; cf. Sect. 7.3.

Thus, the group homomorphism \(\overline{A}_{K}\to A_{K}\) extends to a group homomorphism from the push-out \(\widehat{A}_{K}:=T_{K}\amalg_{\overline {T}}\overline {A}\) to \(A_{K}\). The surjectivity of the map \(\widehat{A}_{K}\to A_{K}\) is shown by an analysis of the map from the curve fibration to \(A_{K}\). In fact, the whole torus part is induced by the double points in the reduction of the stable curve fibration; cf. Sect. 7.4.

So far we are concerned only with the case, where the base field is algebraically closed. But it is not difficult to see that the whole approach can be done after a suitable finite separable field extension if one starts with a non-Archimedean field which is not algebraically closed.

If the non-Archimedean field in question has a discrete valuation, there is a notion of a formal Néron model. Then our result implies a semi-abelian reduction theorem for such Néron models. As a further application one can deduce that every abeloid variety has a dual; i.e., the Picard functor of translation invariant line bundles on \(A_{K}\) is representable by an abeloid variety.

Werner Lütkebohmert

Backmatter

Titel: Rigid Geometry of Curves and Their Jacobians
verfasst von: Werner Lütkebohmert
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-27371-6
Print ISBN: 978-3-319-27369-3
DOI: https://doi.org/10.1007/978-3-319-27371-6