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The Topology of Conjugate Berkovich Spaces

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The Topology of Conjugate Berkovich Spaces Freie Universitat¨ Berlin Fachbereich Mathematik und Informatik The topology of conjugate Berkovich spaces Pedro A. Castillejo Blasco Dissertation zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) 2020 Betreuerin: Prof. Dr. Dr. h. c. mult. H´el`eneESNAULT Erstgutachterin: Prof. Dr. Dr. h. c. mult. H´el`eneESNAULT Zweitgutachter: Prof. Dr. Johannes NICAISE Tag der Disputation: 10.06.2020 Abstract We want to understand how the topology of Berkovich spaces varies when we conjugate the non-archimedean base field. After a short introduction with a discussion of the original prob- lem solved by Serre [Ser64] in the complex setting, we explain some background material about non-archimedean geometry and non-archimedean analytifications. We are able to construct ex- amples of non-homeomorphic conjugate Berkovich spaces by controlling the homotopy type of the Berkovich analytification via its skeleton and its tropicalization. In the appendix we include some useful programs written in SAGE that compute the examples of the last section. Zussamenfassung Wir wollen verstehen, wie sich die Topologie von Berkovich-R¨aumenver¨andert,wenn wir der nichtarchimedische Basisk¨orper konjugieren. Nach einer kurzen Einf¨uhrungmit einer Diskussion des urspr¨unglichen Problems, das von Serre [Ser64] in der komplexen Umgebung gel¨ostwurde, erl¨auternwir etwas Hintergrundmaterial ¨uber nichtarchimedische Geometrie und nichtarchime- dische Analysis. Wir konstruieren Beispiele f¨urnicht-hom¨oomorphkonjugierte Berkovich-R¨aume, indem wir den Homotopietyp der Berkovich-Analytifizierung ¨uber ihr Berkovich-Skelett und ihre Tropikalisierung kontrollieren. Im Anhang gibt es in SAGE geschriebene Programme, die die Beispiele des letzten Abschnitts berechnen. Between 2016 and 2020, (at least) 12,685 people died in the Mediter- ranean trying to reach Europe according to the International Organiza- tion for Migrants of the United Nations. This is a low estimate, since many deaths are not even reported nor heard by anyone who was not on the sinking boat. Since World War II there have never been so many unburied bodies in Europe. This thesis, done in this time period, is dedicated to the memory of all migrants who died on their way to Europe, so much at origin as during voyage or after arrival. Contents 1 Introduction9 2 Non-archimedean geometry and Berkovich spaces 11 2.1 Non-archimedean fields............................ 11 2.2 The topology of non-archimedean fields.................. 16 2.3 Analysis over the rational numbers..................... 18 2.4 Analysis and geometry over a non-archimedean field........... 21 2.4.1 Rigid geometry............................ 21 2.4.2 Berkovich spaces........................... 25 2.4.3 Berkovich analytification of algebraic varieties........... 31 2.5 Skeleton of Berkovich spaces......................... 34 3 Non-homeomorphic conjugate Berkovich spaces 37 3.1 Example of the elliptic curve........................ 37 3.2 More examples................................ 39 3.3 Existence of a contractible conjugate.................... 41 4 Tropicalizing towards explicit examples of curves of higher genus 42 4.1 Tropicalizations................................ 43 4.2 Faithful and homotopic tropicalizations.................. 47 4.3 Faithful tropicalization of an elliptic curve................. 48 4.4 Faithful tropicalizations via modifications................. 50 5 Smooth planar curves of higher genus 52 5.1 Candidates of arbitrary degree....................... 53 5.2 Smooth non-homeomorphic planar curves of arbitrary degree...... 55 6 Acknowledgments 59 A Appendix: Programs to make computations in SAGE 61 A.1 Program that computes the discriminant.................. 61 A.2 Program that defines the curve that we expect to be Mumford..... 62 A.3 Program that defines a Mumford curve of arbitrary degree........ 64 References 67 The topology of conjugate Berkovich spaces 1 Introduction Given a projective algebraic variety X over a number field K, and choosing an embed- ¢ 0 ¢ ding φ K C, we can consider its base change Xφ X K,φ C so that it becomes a complex algebraic variety. Now we can follow GAGA [Ser56] and consider its complex an analytification Xφ , whose underlying topological space is just the set of complex points ¦ Xφ C together with the euclidean topology. Note that considering just the underlying an topological space is equivalent to forgetting the analytic structure of Xφ : for example, two elliptic curves with different j-invariants are different as analytic spaces, but they both have the same underlying topological space (homeomorphic to a doughnut). ¦ Some of the topological invariants of Xφ C are independent of the chosen embed- ding φ: for example, the comparison between ´etalecohomology and singular cohomology ¦ due to Artin [SGA4, Exp. XI] implies that the Betti numbers of Xφ C are independent of φ. ¦ If X is a smooth projective curve of genus g, we know that Xφ C is a Riemann surface with g holes. Hence, the variation of φ doesn't change the topology of the analytification, since choosing a different φ doesn't change the genus of the curve. In arbitrary dimension, we can also say something about the topological fundamental ¦ " ¦ group of Xφ C . We first fix a base point x X C , and denote also by x its pullback to ¦ Xφ C . Since the work of Grothendieck and others (see for example [SGA1], or [Sza09, Theorem 5.7.4] for an easier introduction), we know that the profinite completion of top ¦ ¦ ´et ¦ π1 Xφ C ; x is isomorphic to the ´etalefundamental group π1 Xφ; x , and this group is independent of the chosen φ (see for example [Esn17, Proposition 6.1]). Indeed, all base changes of X to an algebraic closure of the base field K are isomorphic as K- ´et schemes; in particular, the geometric ´etalefundamental group π1 Xφ; x¦ is independent top ¦ ¦ of the chosen φ up to isomorphism. Hence, the profinite completion of π1 Xφ C ; x is independent of the chosen embedding φ. With this kind of examples in mind, it was a plausible question back in the 60's ¦ whether the topology of Xφ C is independent of the chosen φ. In other words, the ¦ scheme Xφ is equipped with a structure morphism Xφ Spec C given by the Carte- sian square1 ¦ Xφ Spec C φ X Spec K¦; ¦ and the question is whether the topology of Xφ C is intrinsic to the abstract scheme ¢ 0 Xφ, or if it depends on the structure morphism induced by the embedding φ K C. Serre answered this question [Ser64]: he constructs a variety X over a number field K and embeddings φ, ¢ K 0 C such that the topological fundamental group top ¦ ¦ top ¦ ¦ π1 Xφ C ; x is different from π1 X C ; x . Let's say a word on Serre's example: Construction of K. The numberÓ field K is the Hilbert class field of a certain ¦ imaginary quadratic field Q p , whereÓ p is a prime number congruent to 1 modulo 4 and the class number of Q p¦ satisfies an extra condition that we 1As usual, we abuse notation and write φ instead of Spec φ¦. 9 don'tÓ explain here; for example, we canÓ take K to be the Hilbert class field of 3 Q ¡ 23§, which happens to be K Q ¡ 23§ x© x x 1¦. Construction of the embeddings φ, ¢ K 0 C. Due to the theory of complex multiplication of elliptic curves, there exists an elliptic curve E defined over K whose ring of endomorphisms is precisely the ring of integers of the imaginary quadratic field. In this situation, there exist embeddings φ, ¢ K 0 C such that top Ó top ¦ ¦ ¦ ¦ ¦ π1 Eφ C ; e is a free OQ p -module of rank 1, but π1 E C ; e is not free Ó ¦ as an OQ p -module. p1 Construction of X defined over K. Let A be the abelian variety given by A E 2 , p1 <p p and let Y be the hypersurface inside PK given by the equation i 1 xi 0; in our above example, we would have A E11 and Y the 21-dimensional hypersurface <23 23 given by i 1 xi 0. Let G be the cyclic group of order p, which acts both on Y by permutation of the coordinates and on A by a different action that we don't describe here. Then, Serre defines X as the quotient of Y A by G, where G acts on Y A via g y; a¦ g 1y; ga¦. Now we base change X via φ and and we get ¦ ¦ Xφ and X such that Xφ C and X C have different topological fundamental groups. Serre calls the varieties Xφ and X conjugate, as we can go from Xφ to X via " ¦ a conjugation of the complex numbers σ AutK C (indeed, σ is an extension of a particular element in the absolute Galois group Gal K;K¦). In other words, if φ σ` , then we have a commutative diagram Xφ X σ Spec C¦ Spec C¦ φ Spec K¦ which is Cartesian, i.e. Xφ X C,σ C. Note that Xφ and X are isomorphic as K-schemes, but not as C-schemes. We have seen our first example of non-homeomorphic conjugate varieties, where the topological fundamental group of the analytification of one variety is different than the one of its conjugate's. In particular, since the profinite completion of the topological fundamental group is independent of the embedding φ, we see that the difference be- top ¦ ¦ tween π1 Xφ C ; x and its profinite completion can be far from trivial. Actually, we ´et don't know whether there exists a complex projective variety V such that π1 V; x¦ is top ¦ ¦ j trivial but π1 V C ; x 1 (c.f. [Sza09, Remark 5.7.5]). Another example in this direction is given by Toledo [Tol93], who constructs a smooth projective variety V over top ¦ ¦ ´et ¦ C such that the profinite completion map π1 V C ; x π1 V; x contains a free kernel of infinite rank.
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