Variations on a Theorem of Tate [1 ed.] 9781470450670, 9781470435400

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Number 1238

Variations on a Theorem of Tate Stefan Patrikis

March 2019 • Volume 258 • Number 1238 (second of 7 numbers)

Number 1238

Variations on a Theorem of Tate Stefan Patrikis

March 2019 • Volume 258 • Number 1238 (second of 7 numbers)

Library of Congress Cataloging-in-Publication Data Cataloging-in-Publication Data has been applied for by the AMS. See http://www.loc.gov/ publish/cip/. DOI: https://doi.org/10.1090/memo/1238

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Memoirs of the American Mathematical Society (ISSN 0065-9266 (print); 1947-6221 (online)) is published bimonthly (each volume consisting usually of more than one number) by the American Mathematical Society at 201 Charles Street, Providence, RI 02904-2213 USA. Periodicals postage paid at Providence, RI. Postmaster: Send address changes to Memoirs, American Mathematical Society, 201 Charles Street, Providence, RI 02904-2213 USA. c Stefan Patrikis. Permission to copy for private use granted.  This publication is indexed in Mathematical Reviews , Zentralblatt MATH, Science Citation Index , Science Citation IndexTM-Expanded, ISI Alerting ServicesSM, SciSearch , Research Alert , CompuMath Citation Index , Current Contents /Physical, Chemical & Earth Sciences. This publication is archived in Portico and CLOCKSS. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines 

established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

24 23 22 21 20 19

To my parents

Contents Chapter 1. Introduction 1.1. Introduction 1.2. What is assumed of the reader: Background references 1.3. Acknowledgments 1.4. Notation

1 1 12 15 16

Chapter 2. Foundations & examples 2.1. Review of lifting results 2.2. -adic Hodge theory preliminaries 2.3. GL1 2.4. Coefficients: Generalizing Weil’s CM descent of type A Hecke characters 2.5. W-algebraic representations  2.6. Further examples: The Hilbert modular case and GL2 × GL2 − → GL4 2.7. Galois lifting: Hilbert modular case 2.8. Spin examples

19 19 25 30 39 42 46 54 58

Chapter 3. Galois and automorphic lifting 3.1. Lifting W -algebraic representations 3.2. Galois lifting: The general case 3.3. Applications: Comparing the automorphic and Galois formalisms 3.4. Monodromy of abstract Galois representations

65 65 75 83 92

Chapter 4. Motivic lifting 4.1. Motivated cycles: Generalities 4.2. Motivic lifting: The hyperk¨ahler case 4.3. Towards a generalized Kuga-Satake theory

101 101 122 140

Bibliography

147

Index of symbols

153

Index of terms and concepts

155

v

Abstract Let F be a number field. These notes explore Galois-theoretic, automorphic, and motivic 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 algebraicity (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.

Received by the editor April 22, 2013 and, in revised form, December 6, 2013, December 2, 2015, and July 31, 2016. Article electronically published on February 7, 2019. DOI: https://doi.org/10.1090/memo/1238 2010 Mathematics Subject Classification. Primary 11R39, 11F80, 14C15. Key words and phrases. Galois representations, algebraic automorphic representations, motives for motivated cycles, monodromy, Kuga-Satake construction, hyperk¨ ahler varieties. Partially supported by an NDSEG fellowship and NSF grant DMS-1303928. c 2019 Stefan Patrikis

vii

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 closure. Tate’s theorem that H 2 (ΓF , Q/Z) vanishes (see [Ser77, Theorem 4] or Theorem 2.1.1 below) encodes one of the basic features of the representation theory of ΓF : since coker(μ∞ (C) → C× ) is uniquely divisible, H 2 (ΓF , C× ) vanishes as well, and therefore all obstructions to lifting projective representations ΓF → PGLn (C) vanish. More generally, as explained in [Con11]   H of linear algebraic groups over Q (Lemma 2.1.4 below), for any surjection H with central torus kernel, any homomorphism ΓF → H(Q ) lifts (continuously) to   ). The simplicity of Tate’s theorem is striking: replacing Q/Z by some finite H(Q group Z/m of coefficients, the vanishing result breaks down. Two other groups, more or less fanciful, extend ΓF and conjecturally encode the key structural 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 Andr´e’s theory of motivated cycles (see §4.1.3). 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 Langlands group into a concrete and well-defined question about auto be an inclusion of connected reductive F morphic representations. Let G ⊂ G 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 general ∨ denote the respective Langlands dual groups, which are ity. We let G∨ and G  ∨ → G∨ whose kernel is a central torus related by a surjective homomorphism G ∨  (in the SLn ⊂ GLn case, this dual homomorphism is the usual surjection in G 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 Langlands 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 1

2

1. INTRODUCTION

 F ), i.e. the fibers of the functorial transfer associated to G  ∨ → G∨ are of G(A non-empty. We show (Proposition 3.1.4) that cuspidal representations of G(AF )  F ). This result is elementary, but our analogies insist that do indeed lift to G(A we investigate it further. For instance, the analogous lifting result is false for GF , and 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 representations 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 automorphic 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. To continue, then, we must discuss notions of algebraicity for automorphic representations; then we can see whether the automorphic lifting analogous to Tate’s theorem can be made to respect this ‘algebraicity.’ Weil laid the foundation for this discussion in his paper [Wei56], by showing that for Hecke characters (automorphic representations of GL1 (AF )), an integrality condition on the archimedean component suffices to imply algebraicity of the coefficients of the corresponding L-series (i.e. algebraicity of its Satake parameters). Waldschmidt ([Wal82]) later proved necessity of this condition. Weil, Serre, and others also showed that a subset, the ‘type A0 ’ Hecke characters (Definition 2.3.2), moreover give rise to compatible sys× tems of Galois characters ΓF → Q (and, via the theory of CM abelian varieties, motives underlying these compatible systems). The general feeling since has been that the most obvious analogue of Weil’s type A0 condition should govern the existence of associated -adic representations. To be precise, let π be an automorphic representation of our F -group G, and let T be a maximal torus of G. We will use the terminology ‘L-algebraic’ of [BG11] as the general analogue of type A0 char∼ → C, we can write (in acters. That is, fixing at each v|∞ an isomorphism ιv : F v −

1.1. INTRODUCTION

3

Langlands’ normalization of [Lan89]) the restriction to WF v of its L-parameter as recv (πv ) : z → ιv (z)μιv ¯ιv (z)νιv ∈ T ∨ (C). with μιv , νιv ∈ X • (T )C and μιv − νιv ∈ X • (T ) (here and throughout, ¯ιv denotes the complex conjugate of ιv ). Unless there is risk of confusion, we will omit reference to the embedding ιv , writing μv = μιv , etc. Definition 1.1.2. The automorphic representation π is L-algebraic if for all v|∞, μv and νv lie in X • (T ). The na¨ıve reason for focusing on this condition is that, for G = GLn , it lets us see the Hodge numbers of the (hoped-for) corresponding motive. Clozel ([Clo90]), noticing that cohomological representations need not satisfy this condition, but have parameters μv , νv ∈ ρ + X • (T ), where ρ is the half-sum of the positive roots,1 studied this alternative integrality condition for G = GLn . Buzzard and Gee ([BG11]) have recently elaborated on this latter condition (‘C-algebraic’) and its relation with L-algebraicity. For GL1 , the two notions C and L-algebraic coincide, and in general they are both useful and distinct generalizations of Weil’s type A0 condition. But Weil’s ‘integrality condition,’ which he calls ‘type A’ (Definition 2.3.2), is more general than the type A0 condition, and its analogues in higher rank seem largely to have been neglected.2 It is of independent interest to resuscitate this condition, but we moreover find it essential for understanding the interaction of descent problems and algebraicity, including our original lifting question. By this we mean the following problem: given connected reductive (say quasi-split) F groups H and G with a morphism of L-groups L H → L G, and given a cuspidal L-algebraic representation π of G(AF ) which is known to be in the image of the associated functorial transfer, is π the transfer of an L-algebraic representation? The answer is certainly ‘no,’ even for GL1 , but it fails to be ‘yes’ in a tightly constrained way. I believe the most useful general notion is the following: Definition 1.1.3. We say that π is W -algebraic if for all v|∞, μv and νv lie in 12 X • (T ). This is more restrictive than Weil’s type A condition (which allows twists | · |r for r ∈ Q as well), and it is easy to concoct examples of automorphic representations that have algebraic Satake parameters but are not even twists of W -algebraic representations. It is one of our guiding principles, suggested by the (archimedean) Ramanujan conjecture, that all such examples are degenerate, and up to unwinding these degeneracies, all representations with algebraic Satake parameters should be built up from W -algebraic pieces. We can now describe the algebraic refinement for automorphic representations of the aforementioned (Proposition 3.1.4) lifting result. The answer depends on whether the field F has real embeddings, so we state it in two parts: Proposition 1.1.4 (See Proposition 3.1.12). Let F be a CM field, and for sim be a split F -group containing plicity let G be a split semi-simple F -group, and let G 1 For some choice of Borel containing T . Such a choice was implicit in defining a dual group, with its Borel B ∨ containing a maximal torus T ∨ , and the above archimedean L-parameters. 2 One substantial use of type A but not A Hecke characters since the early work of Weil 0 and Shimura occurs in the paper [BR93], in which Blasius-Rogawski associate motives to certain tensor products of Hilbert modular representations.

4

1. INTRODUCTION

G as its derived group. Let π be a cuspidal representation of G(AF ). Assume π∞ is tempered. (1) If π is L-algebraic, then there exists an L-algebraic lift to a cuspidal au F ). tomorphic representation π ˜ of G(A (2) If π is W -algebraic, then there exists a W -algebraic lift π ˜. In §2.4 we develop a conjectural framework that allows this result to be extended to all totally imaginary F . See page 10 of this introduction. In contrast, over totally real fields, we have the following: Proposition 1.1.5 (See Proposition 3.1.14). Now suppose F is totally real,  for the center of with π as before. Continue to assume π∞ is tempered. Write Z 0 3   G, and Z for its connected component. (1) If π is L-algebraic, then it admits a W -algebraic lift π ˜. (2) For the ‘only if ’ direction of the following statement assume Hypothesis 3.1.8. Then the images of μv and νv under X • (T ) → X • (ZG ∩ Z0 ) lie in X • (ZG ∩ Z0 )[2], and π admits an L-algebraic lift if and only if these images are independent of v|∞. The basic example in which an L-algebraic representation does not lift to an L-algebraic representation is given by a mixed-parity Hilbert modular form π on GL2 (AF ) (F a totally real field not equal to Q). That is, π is the cuspidal automorphic representation associated to a classical Hilbert modular cusp form whose weights (in the classical sense) at two different infinite places of F have different parities: such a π restricts to an L-algebraic representation of SL2 (AF ), but no twist of π itself is L-algebraic. In contrast Proposition 1.1.5 tells us that, when F is totally real, there is no obstruction to finding L-algebraic lifts for G a simple split group of type A2n , E6 , E8 , F4 or G2 . For other groups the obstructions are in fact realizable: Corollary 3.1.15 applies limit multiplicity formulas (as in [Clo86]) to produce many discrete-series examples generalizing the basic case of mixed-parity Hilbert modular forms. These examples imply that Arthur’s conjectural construction ([Art02]) of a morphism LF → GF requires modification (see Remark 3.1.16). Propositions 1.1.4 and 1.1.5 give the basic answer to our refined lifting question for automorphic representations. We next turn to the corresponding question for geometric Galois representations; Conjecture 1.1.1 leads us to expect conclusions analogous to those of these two automorphic results. First, we recall the definition (due to Fontaine-Mazur) of a geometric Galois representation: Definition 1.1.6. A semi-simple representation ρ : ΓF → GLn (Q ) is geometric if it is unramified at all but a finite set of primes and is de Rham at all places v|. (See §2.2.1 for further discussion of the de Rham condition.) More generally, if H is a linear algebraic group over Q , then a representation ρ : ΓF → H(Q ) is geometric if it is almost everywhere unramified, de Rham at all places above , and if the Zariski closure of ρ(ΓF ) in H is reductive. The Galois question parallel to our refined automorphic lifting results has been raised by Brian Conrad in [Con11]. In that paper, Conrad, building on previous 3 Note

that we use somewhat different notation in §3.1; see the introduction to that section.

1.1. INTRODUCTION

5

work of Wintenberger ([Win95]) addresses lifting problems of the form

ρ˜

ΓF

y

y

y ρ

y

 ) , H(Q y