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CORRECTIONS TO RIGID LOCAL SYSTEMS MANUSCRIPT

NICHOLAS M. KATZ

Corrections to Introduction, local page numbers

page 6, line -3: should read

λ(1− λ)(d/dλ)2f + (c− (a + b + 1)λ)(d/dλ)f − abf = 0

Corrections to Chapter 2, local page numbers

page 32, line 1 of Remarks 2.10.4: should begin

1) Here is a slightly variant...

page 48, lines 2-3 of Corollary 2.13.3 should end/begin

Then K = Lχ(x−1)[1] satisfies K?mid×D(inv?K) = δ1.

Corrections to Chapter 3, local page numbers

page 10, last line (proof of Lemma 3.3.1) should read

proof This was proven in 2.10.2 and 2.10.8 above. QED

Corrections to Chapter 4, local page numbers

page 7, lines 2-3 of Lemma 4.3.8 should end/begin

For F lisse on X −D and tame along D, j : X −D → X and i : D → X the inclusions, we have

Corrections to Chapter 8, local page numbers

page 4, lines 10-11 of the proof of 2) of Lemma 8.2.2 should end/begin

By proper base change

page 14, line 11 of 8.5.1 should begin

of ⊗iLχ2,i(X2−Ti). So essentially... page 14, line 11 of 8.5.1: This is now correct, but still a bit confusing:

the characters χ2,i occurring in “⊗iLχ2,i(X2−Ti)” were defined on line -2 of the previous page as

χa,i = χ e(a,i).

1

2 NICHOLAS M. KATZ

Princeton University, Mathematics, Fine Hall, NJ 08544-1000, USA E-mail address: [email protected]

Rigid Local Systems NNNNiiiicccchhhhoooollllaaaassss MMMM.... KKKKaaaattttzzzz

Contents-1

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Introduction

Chapter 1-First results on rigid local systems 1.0 Generalities concerning rigid local systems over ^ 1.1 The case of genus zero 1.2 The case of higher genus 1.3 The case of genus one 1.4 The case of genus one: detailed analysis

Chapter 2-The theory of middle convolution

2.0 Transition from irreducible local systems on open sets of @1 to

irreducible middle extension sheaves on !1.

2.1 Transition from irreducible middle extension sheaves on !1 to

irreducible perverse sheaves on !1

2.2 Review of Dbc(X, ä$…)

2.3 Review of perverse sheaves 2.4 Review of Fourier Transform 2.5 Review of convolution 2.6 Convolution operators on the category of perverse sheaves: middle convolution 2.7 Interlude: middle direct images (relative dimension one) 2.8 Middle additive convolution via middle direct image 2.9 Middle additive convolution with Kummer sheaves 2.10 Interpretation of middle additive convolution via Fourier Transform

2.11 Invertible objects on !1 in characteristic zero 2.12 Musings on *mid -invertible objects in ∏ in the ´m case

2.13 Interlude: surprising relations between *mid on ! 1 and on ´m

2.14 Interpretive remark: Fourier-Bessel Transform 2.15 Questions about the situation in several variables 2.16 Questions about the situation on elliptic curves 2.17 Appendix 1: the basic lemma on end-exact functors 2.18 Appendix 2: twisting representations by characters

Contents-2

Chapter 3-Fourier Transform and rigidity 3.0 Fourier Transform and index of rigidity 3.1 Lemmas on representations of inertia groups 3.2 Interlude: the operation ‚mid 3.3 Applications to middle additive convolution 3.4 Some open questions about local Fourier Transform

Chapter 4-Middle convolution: dependence on parameters 4.0 Good schemes 4.1 The basic setting 4.2 Basic results in the basic setting 4.3 Middle convolution in the basic setting

Chapter 5-Structure of rigid local systems 5.0 Cohomological rigidity 5.1 The category Ê…, and the functors MCç and MTÒ 5.2 The main theorem on the structure of rigid local systems 5.3 Applications and Interpretations of the main theorem 5.4 Some open questions 5.5 Existence of universal families of rigids with given local monodromy 5.6 Remark on braid groups 5.7 Universal families without quasiunipotence 5.8 The complex analytic situation 5.9 Return to the original question

Chapter 6-Existence algorithms for rigids 6.0 Numerical invariants 6.1 Numerical incarnation: the group NumData 6.2 A compatibility theorem 6.3 Realizable and plausible elements 6.4 Existence algorithm for rigids 6.5 An example 6.6 Open questions 6.7 Action of automorphisms 6.8 A remark and a question

Contents-3

Chapter 7-Diophantine aspects of rigidity 7.0 Diophantine criterion for irreducibility 7.1 Diophantine criterion for rigidity 7.2 Appendix: a counterexample

Chapter 8-Motivic description of rigids 8.0 The basic setting 8.1 Interlude: Kummer sheaves

8.2 Naive convolution on (!1- {T1, ... , Tn})SN,n,… .

8.3 Middle convolution on (!1- {T1, ... , Tn})SN,n,… .

8.4 "Geometric" description of all tame rigids with quasi-unipotent local monodromy 8.5 A remark and a question

Chapter 9-Grothendieck's p-curvature conjecture for rigids 9.0 Introduction 9.1 Review of Grothendieck's p-curvature conjecture 9.2 Interlude: Picard-Fuchs equations and some variants 9.3 The main result of [Ka-ASDE] and a generalization 9.4 Application to rigid local systems 9.5 Comments and questions

References

Introduction-1

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It is now nearly 140 years since Riemann introduced [Rie-EDL]

the concept of a "local system" on @1 - {a finite set of points}. His idea was that one could (and should) study the solutions of an n'th order linear differential equation by studying the rank n local system (of its local holomorphic solutions) to which it gave rise.

Riemann knew that a rank n local system on @1 - {m points} was "nothing more" than a collection of m invertible matrices Ai in

GL(n, ^) which satisfy the matrix equation A1A2...Am = (idn), such

collections taken up to simultaneous conjugation by a single element of GL(n, ^). He also knew each individual Ai was, up to GL(n, ^)

conjugacy, just the effect of analytic continuation along a small loop encircling the i'th missing point.

His first application of these then revolutionary ideas was to study the classical Gauss hypergeometric function [Rie-SG], which he

did by studying rank two local systems on @1 - {three points}. His investigation was a stunning success, in large part because any such (irreducible) local system is rrrriiiiggggiiiidddd in the sense that it is determined up to isomorphism as soon as one knows separately the individual conjugacy classes of all its local monodromies. By exploiting this rigidity, Riemann was able to recover Kummer's transformation theory of hypergeometric functions "almost without calculation" [Rie-APM].

It soon became clear that Riemann had been "lucky", in the sense that the most local systems are not rigid. For instance, rank

two irreducible local systems on @1 - {m points}, all of whose local monodromies are non-scalar, are rigid precisely for m=3. And rank

n irreducible local systems on @1 - {three points}, each of whose local monodromies has n distinct eigenvalues, are rigid precisely for n=1 and n=2.

On the other hand, some of the best known classical functions are solutions of differential equations whose local systems are rigid, including both of the standard generalizations of the hypergeometric

function, namely nFn-1, which gives a rank n local system on @ 1 -

{0,1,‘}, and the Pochhammer hypergeometric functions, which give

rank n local systems on @1 - {n+1 points}. In the classical literature, rigidity or its lack is expressed in

Introduction-2

terms of the vanishing or nonvanishing of the "number of accessory parameters". But the object whose rigidity is classically in question

is not a rank n local system on @1 - {m points}, but rather an n'th order linear differential equation with rational function coefficients which has regular singularities at the m missing points, and no other singularities. In practice, one assumes also that each of the local monodromies has n distinct eigenvalues, expressed classically by saying that no two exponents differ by integers. One then looks for the most general n'th order linear differential equation with rational function coefficients which has regular singularities at the m missing points, no other singularities, and whose indicial polynomial at each missing point is the same as for the equation we started with. [This game with indicial polynomials makes sense for any equation with regular singularities, but it is only a meaningful game in the case where each of its local monodromies has n distinct eigenvalues, for only then can we be sure that any equation with the same indicial polynomials automatically has isomorphic local monodromies.] The "number of parameters" upon which such an equation depends is called the "number of accessory parameters", or the "number of constants in excess" [Ince, 20.4]. For example, in the classical literature one finds that for second order equations with m regular singularities, the number of accessory parameters is m-3 [Ince, top of page 506].

From a modern point of view, what corresponds precisely to a

rank n local system on @1 - {m points} is an n≠n first order system of differential equations with regular singularities at the named points (i.e., an algebraic vector bundle with integrable connection on

@1 - {m points} with regular singularities at the m missing points) [De-ED]. So what corresponds to rigidity for a local system is the absence of deformations of the corresponding n≠n system which preserve local monodromy. But in the classical literature, the q