Variations on a theorem of Tate Stefan Patrikis Princeton University,Department of Mathematics,Fine Hall,Washington Road,Princeton, NJ 08544 Current address: Department of Mathematics, The University of Utah, Salt Lake City, UT 84112 E-mail address: [email protected] URL: http://www.math.utah.edu/~patrikis/ NumberTheory Jul 31 2016 19:25:06 EDT Version 4 - Submitted to MEMO 2000 Mathematics Subject Classification. Primary 11R39, 11F80, 14C15 Key words and phrases. Galois representations, algebraic automorphic representations, motives for motivated cycles, monodromy, Kuga-Satake construction, hyperkahler¨ varieties Partially supported by an NDSEG fellowship and NSF grant DMS-1303928. Abstract. Let F be a number field. These notes explore Galois-theoretic, automorphic, and mo- tivic analogues and refinements of Tate’s basic result that continuous projective representations Gal(F=F) ! PGLn(C) lift to GLn(C). We take special interest in the interaction of this result with al- gebraicity (for automorphic representations) and geometricity (in the sense of Fontaine-Mazur). On the motivic side, we study refinements and generalizations of the classical Kuga-Satake construction. Some auxiliary results touch on: possible infinity-types of algebraic automorphic representations; comparison of the automorphic and Galois “Tannakian formalisms”; monodromy (independence- of-`) questions for abstract Galois representations. NumberTheory Jul 31 2016 19:25:06 EDT Version 4 - Submitted to MEMO Contents Chapter 1. Introduction 1 1.1. Introduction 1 1.2. What is assumed of the reader: background references 11 1.2.1. Automorphic representations 11 1.2.2. `-adic Galois representations 12 1.2.3. Motives 12 1.2.4. Connecting the dots 12 1.3. Acknowledgments 14 1.4. Notation 15 Chapter 2. Foundations & examples 17 2.1. Review of lifting results 17 2.2. `-adic Hodge theory preliminaries 23 2.2.1. Basics 23 2.2.2. Labeled Hodge-Tate-Sen weights 25 2.2.3. Induction and `-adic Hodge theory 27 2.3. GL1 27 2.3.1. The automorphic side 27 2.3.2. Galois GL1 33 2.4. Coefficients: generalizing Weil’s CM descent of type A Hecke characters 36 2.4.1. Coefficients in Hodge theory 36 2.4.2. CM descent 37 2.5. W-algebraic representations 39 2.6. Further examples: the Hilbert modular case and GL2 × GL2 −! GL4 43 2.6.1. General results on the GL2 × GL2 −! GL4 functorial transfer 43 2.6.2. The Hilbert modular case 46 2.6.3. A couple of questions 50 2.7. Galois lifting: Hilbert modular case 50 2.7.1. Outline 50 2.7.2. GL2(Q`) ! PGL2(Q`) 51 2.8. Spin examples 54 Chapter 3. Galois and automorphic lifting 61 3.1. Lifting W-algebraic representations 61 3.1.1. Notation and central character calculation 61 3.1.2. Generalities on lifting from G(AF) to Ge(AF) 63 3.1.3. Algebraicity of lifts: the ideal case 67 iii NumberTheory Jul 31 2016 19:25:06 EDT Version 4 - Submitted to MEMO 3.2. Galois lifting: the general case 70 3.3. Applications: comparing the automorphic and Galois formalisms 77 3.3.1. Notions of automorphy 78 3.3.2. Automorphy of projective representations 79 3.4. Monodromy of abstract Galois representations 86 3.4.1. A general decomposition 86 3.4.2. Lie-multiplicity-free representations 88 Chapter 4. Motivic lifting 95 4.1. Motivated cycles: generalities 95 4.1.1. Lifting Hodge structures 95 4.1.2. Motives for homological equivalence and the Tannakian formalism 97 4.1.2.1. Neutral Tannakian categories 98 4.1.2.2. Homological motives 101 4.1.3. Motivated cycles 103 4.1.4. Motives with coefficients 107 4.1.5. Hodge symmetry in MF;E 108 4.1.6. Motivic lifting: the potentially CM case 112 4.2. Motivic lifting: the hyperkahler¨ case 115 4.2.1. Setup 115 4.2.2. The Kuga-Satake construction 117 4.2.3. A simple case 121 4.2.4. Arithmetic descent: preliminary reduction 124 4.2.5. Arithmetic descent: the generic case 126 4.2.6. Non-generic cases: dim(TQ) odd 129 4.2.7. Non-generic cases: dim TQ even 131 4.3. Towards a generalized Kuga-Satake theory 132 4.3.1. A conjecture 132 4.3.2. Motivic lifting: abelian varieties 133 4.3.3. Coda 137 Index of symbols 139 Index of terms and concepts 141 Bibliography 143 iv NumberTheory Jul 31 2016 19:25:06 EDT Version 4 - Submitted to MEMO CHAPTER 1 Introduction 1.1. Introduction Let F be a number field, with ΓF = Gal(F=F) its Galois group relative to a choice of algebraic 2 closure. Tate’s theorem that H (ΓF; Q=Z) vanishes (see [Ser77, Theorem 4] or Theorem 2.1.1 be- × low) encodes one of the basic features of the representation theory of ΓF: since coker(µ1(C) ! C ) 2 × is uniquely divisible, H (ΓF; C ) vanishes as well, and therefore all obstructions to lifting projec- tive representations ΓF ! PGLn(C) vanish. More generally, as explained in [Con11] (Lemma 2.1.4 below), for any surjection He H of linear algebraic groups over Q` with central torus kernel, any homomorphism ΓF ! H(Q`) lifts (continuously) to He(Q`). The simplicity of Tate’s theorem is striking: replacing Q=Z by some finite group Z=m of coefficients, the vanishing result breaks down. Two other groups, more or less fanciful, extend ΓF and conjecturally encode the key struc- tural features of, respectively, automorphic representations and (pure) motives. The more remote automorphic Langlands group, LF, at present exists primarily as heuristic, whereas the motivic Galois group GF would take precise form granted Grothendieck’s Standard Conjectures, and in the meantime can be approximated by certain unconditional substitutes, for instance via Deligne’s theory of absolute Hodge cycles, or Andre’s´ theory of motivated cycles (see x4.1.3). In either case, however, we can ask for an analogue of Tate’s lifting theorem and attempt to prove unconditional results. The basic project of this book is to pursue these analogies wherever they may lead; the fundamental importance of Tate’s theorem is affirmed by the fruitfulness of these investigations, which lead us to reconsider and reinvigorate classic works of Weil ([Wei56]) and Kuga and Satake ([KS67]). We begin by translating the heuristic lifting problem for the (conjectural) automorphic Lang- lands group into a concrete and well-defined question about automorphic representations. Let G ⊂ Ge be an inclusion of connected reductive F-groups having the same derived group; basic cases to keep in mind are SLn ⊂ GLn or Sp2n ⊂ GSp2n. For simplicity, we will throughout assume these groups are split, although many of the results described continue to hold in greater generality. We let G_ and Ge_ denote the respective Langlands dual groups, which are related by a surjective _ _ _ homomorphism Ge ! G whose kernel is a central torus in Ge (in the SLn ⊂ GLn case, this dual homomorphism is the usual surjection GLn(C) ! PGLn(C)); thus on the dual side we have a setup analogous to the one in Tate’s lifting theorem, and indeed the whole point of the Lang- lands group LF is that, loosely, automorphic representations of G(AF) should correspond to maps _ LF ! G . The automorphic analogue of Tate’s theorem is then the assertion that automorphic representations of G(AF) extend to automorphic representations of Ge(AF), i.e. the fibers of the functorial transfer associated to Ge_ ! G_ are non-empty. We show (Proposition 3.1.4) that cusp- idal representations of G(AF) do indeed lift to Ge(AF). This result is elementary, but our analogies insist that we investigate it further. For instance, the analogous lifting result is false for GF, and 1 NumberTheory Jul 31 2016 19:25:06 EDT Version 4 - Submitted to MEMO a priori there is no reason to believe that ‘lifting problems’ for the three groups ΓF; LF, and GF should qualitatively admit the same answer: a general automorphic representation has seemingly no connection with either `-adic representations or motives; and a general `-adic representation has no apparent connection with classical automorphic representations or motives. Two quite distinct kinds of transcendence–one complex, one `-adic–prevent these overlapping theories from being reduced to one another. A key problem, therefore, is to identify the overlap and ask how the lifting problem behaves when restricted to this (at least conjectural) common ground; it is only in posing this refined form of the lifting problem that the relevant structures emerge. What automorphic representations, then, do we expect to correspond to `-adic Galois represen- tations or motives? Putting off for the time being precise definitions, we now informally state the fundamental conjectures, due to Fontaine-Mazur, Langlands, Tate, Serre, and Grothendieck, that lay out the basic expectations. Conjecture 1.1.1 (Preliminary form: see Conjecture 1.2.1 for the precise version). There is a bijection, characterized by local compatibilities, between • “Algebraic” cuspidal automorphic representations of GLn(AF). • Irreducible “geometric” Galois representations ΓF ! GLn(Q`). • Irreducible pure motives over F (with Q-coefficients). (The only direction of this correspondence whose construction is known is from motives to Galois representations: simply take `-adic cohomology.) This book will focus on studying the analogues of Tate’s lifting theorem at this conjectural intersection of the three subjects of automor- phic forms, motives, and Galois representations. The automorphic and Galois-theoretic aspects will be more or less fully treated, and we will find that the motivic aspect is a deep and largely unconsidered problem; but our Galois-theoretic and automorphic investigations will allow us to frame a precise conjectural response to the motivic lifting question.