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Local-Global Principles for Quadratic Forms and Strong Linkage

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Local-Global Principles for Quadratic Forms and Strong Linkage Local-global principles for quadratic forms and strong linkage Lokaal-globaal-principes voor kwadratische vormen en sterke koppeling Thesis for the degree of Doctor of Science: Mathematics at the University of Antwerp Thesis for the degree of Doctor of Natural Sciences at the University of Konstanz presented by Parul Gupta Promotors: Prof. Dr. Karim Johannes Becher Prof. Dr. Arno Fehm Antwerp, 2018 To my parents Rekha and Umesh Chand Gupta Table of Contents Acknowledgements i Introduction iii Nederlandse samenvatting xiii 1Discretevaluations 1 1.1 Valuations . 1 1.2 Gaussextensions ............................ 8 1.3 Rationalfunctionfields. 12 1.4 Algebraic function fields . 16 2Symbolsandramification 23 2.1 Milnor K-groups ............................ 23 2.2 Ramification .............................. 27 2.3 Symbolalgebras ............................ 31 2.4 The unramified Brauer group . 35 2.5 Quasi-finitefields ............................ 38 3Divisorsonsurfaces 47 3.1 Regularrings .............................. 47 3.2 Regular surfaces . 50 3.3 Divisorial valuations . 56 3.4 Ramification on surfaces . 59 3.5 Surfaces over henselian local domains . 60 4Stronglinkageforfunctionfields 65 4.1 Functionfieldswithfinitesupportproperty . 66 4.2 Function fields of surfaces . 69 5 Quadratic forms and the u-invariant 77 5.1 Basic concepts . 77 5.2 Quadraticformsandvaluations . 82 5.3 TheWittinvariant ........................... 85 5.4 The u-invariant............................. 89 5.5 Criteria for bounds on the u-invariant . 91 6Local-globalprinciplesforisotropy 99 6.1 Knownlocal-globalprinciples . 99 6.2 Rational function fields over global fields . 105 6.3 Hidden local anisotropy on an arithmetic surface . 110 7Formsoverfunctionfieldsofsurfaces 115 7.1 Forms over two-dimensional regular domains. 116 7.2 Fields of u-invarianteight ....................... 133 7.3 A refinement for four-dimensional forms . 137 8Local-globalprinciplesoverrationalfunctionfields 145 8.1 Square-reflexivepolynomials. 145 8.2 Rational function fields over 2-special fields . 153 Bibliography 161 Index 167 Acknowledgements First and foremost, I wish to express my gratitude to my supervisors, Karim Johannes Becher and Arno Fehm, for their patience, encouragement and criticism. I thank Wendy Lowen, Freddy Van Oystaeyen, David Leep, Jean-Pierre Tignol and Suresh Venapally for having accepted to be members of the jury of my thesis and for all their comments and suggestions. I am grateful to Amit Kulshrestha for having introduced me to quadratic form theory. Over the years, my research has benefited enormously from the kind and pa- tient input and encouragement of David Leep, Raman Parimala, Suresh Venapally and David Grimm. I have further obtained valuable suggestions and comments from Jean-Louis Colliot-thélène, Annette Bachmayr, Yong Hu, Adam Chapman and Asher Auel. To Raman Parimala and Suresh Venapally I further am grateful for the invi- tation to Emory University and for their hospitality. Equally I wish to thank Eva Bayer-Fluckiger for her hospitality at EPFL. Over the years, a couple of institutions and dedicated staffmembers have shown crucial support in helping to deal with the requests of other institutions. I thank the Zukunftskolleg for a first invitation to Konstanz in 2012. The Welcome Center of the University of Konstanz and in particular Johannes Dingler have helped me enormously in getting started administratively at Kon- stanz. Similarly, the International StaffOffice of University of Antwerp, and in particular Erika Leunens, have helped me crucially in problems related to visa in Belgium and Germany during the last phase of my PhD. The last year of my PhD I spent at TU Dresden, where most of the writing of this thesis was done. I thank TU Dresden and in particular Christian Zschalig for administrative support. I am grateful to my colleagues from University of Konstanz, University of i Antwerp and TU Dresden. I especially thank Sten Veraa, Kader Bingöl, Gonzalo Manzano Flores and Nikolaas Verhulst for reading parts of the first draft of my thesis. I thank Julia Ramos González for many tips concerning the final phase of the thesis. I wish to thank my parents and my siblings for all their love, support and faith in me. Finally, I am grateful to my friends from India and friends whom I got to know in Konstanz, Antwerp and Dresden for being such good friends. Introduction A key problem in the study of quadratic forms is to determine whether a given quadratic form is isotropic i.e. whether it has a non-trivial zero. Initially, the problem was considered over the ring of integers Z and over the field of rational numbers Q. In 1890, Minkowski showed that a quadratic form over Q is isotropic if and only if it is indefinite and has, for every prime number p, a non-trivial zero modulo sufficiently large powers of p. The latter criterion can be formulated in a more elegant way using the fields of p-adic numbers Qp, which were introduced by Hensel in 1897. In 1924, Hasse generalized Minkowski’s result to arbitrary number fields: Theorem A (Hasse-Minkowski). A quadratic form over a number field F is isotropic if and only if it is isotropic over all completions of F . Here, completions are taken with respect to places of F , which are equivalence classes of absolute values. In fact, the proof of this theorem also works for algebraic function fields over finite fields. By an algebraic function fields we mean a finitely generated field extension of transcendence degree one. Number fields and algebraic function fields over finite fields are called global fields and their completions are called local fields. It is an easy task to determine the isotropy of a quadratic form over local fields, which makes the Hasse-Minkowski Theorem very useful to determine isotropy of quadratic fields over global fields. Theorems of this type are referred to as local-global principles or Hasse principles. In order to prove the theorem, Hasse introduced an invariant of quadratic forms over number fields, which can best be explained in terms of quaternion algebras. Quaternion algebras over an arbitrary field of characteristic different from two were introduced by Dickson in 1912. Let F in the sequel be a field with char F 2.Fora, b F ,thequater- p q‰ P ˆ nion algebra a, b is the four-dimensional F -algebra F Fi Fj Fij with p qF ‘ ‘ ‘ iii iv INTRODUCTION multiplication determined by the rules i2 a, j2 b and ij ji. “ “ “´ If a quaternion algebra over F is not a division algebra, then it is isomorphic to the matrix algebra M2 F . Dickson showed that a, b F M2 F if and only if p q p q – p q the quadratic form X2 aX2 bX2 is isotropic over F [Dic12]. 1 ´ 2 ´ 3 The Hasse invariant of a quadratic form over F takes values in the 2-torsion part of the Brauer group of F , and it is given as a certain product of quaternion algebras depending on the coefficients of the quadratic form. In particular, the Hasse invariant of a three-dimensional form is given by a quaternion algebra. For example the Hasse-invariant of the three-dimensional form X2 aX2 bX2 is 1 ´ 2 ´ 3 the algebra a, b . Dickson’s result then says that a three-dimensional quadratic p qF form over F is isotropic if and only if its Hasse invariant is trivial. Let ⌦ be a set of discrete valuations on F .Forv ⌦, we denote by Fv P the completion of F with respect to v. We say that quaternion algebras over F satisfy the Hasse principle with respect to ⌦ if any quaternion algebra over F which becomes isomorphic to M2 Fv by extending scalars to Fv for every v ⌦, p q P is indeed isomorphic to M2 F over F . Similarly, for a class of quadratic forms p q C over F ,wesaythatforms in satisfy the local-global principle for isotropy with C respect to ⌦ if any quadratic form in which is isotropic over Fv for every v ⌦ C P is isotropic over F . As a consequence of Dickson’s result, we obtain that, if quaternion algebras over F satisfy the Hasse principle with respect to ⌦, then three-dimensional quadratic forms over F also satisfy the local-global principle for isotropy with respect to ⌦. Moreover, if quaternion algebras defined over quadratic field ex- tensions of F also satisfy the Hasse principle with respect to the set of discrete valuations which are extensions of valuations in ⌦, then four-dimensional forms over F also satisfy the local-global principle for isotropy with respect to ⌦ (see Proposition 6.1.3). We now give some important examples of fields where quadratic forms of di- mension three and four satisfy the local-global principle for isotropy with respect to the set of all places. These examples play a crucial role in this thesis. Examples I. In each of the following cases, quaternion algebras over F satisfy the Hasse principle with respect to the set ⌦ of all Z-valuations on F .Inallthose v cases this follows from a more general Hasse-principle for central simple algebras, which we mention in parentheses. a F is a nonreal global field (Albert-Brauer-Hasse-Noether, see [Rei03, Theorem p q 32.11]). b F is an algebraic function field over a pseudo-algebraically closed (PAC) field p q (Efrat, see [Efr01, Corollary 3.2]). See Section 2.1 for the definition of a PAC- field. c F is the fraction field of a two-dimensional complete regular local domain with p q separably closed residue field of characteristic different from two, for example F C X, Y , the fraction field of the ring of power series in two variables “ pp qq over C (Colliot-Thélène-Ojanguren-Parimala, see [CTOP02, Corollary 1.10]). d F is an algebraic function field over a complete discretely valued field with p q separably closed residue field of characteristic different from two, for example F C t X , the rational function field in one variable over the Laurent “ pp qq p q series field in one variable over C.
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