Local Systems in Algebraic-Arithmetic Geometry

The topological fundamental group of a smooth complex algebraic variety is poorly understood. One way to approach it is to consider its complex linear representations modulo conjugation, that is, its complex local systems. A fundamental problem is then to single out the complex points of such moduli spaces which correspond to geometric systems, and more generally to identify geometric subloci of the moduli space of local systems with special arithmetic properties. Deep conjectures have been made in relation to these problems. This book studies some consequences of these conjectures, notably density, integrality and crystallinity properties of some special loci. This monograph provides a unique compelling and concise overview of an active area of research and is useful to students looking to get into this area. It is of interest to a wide range of researchers and is a useful reference for newcomers and experts alike. Contents 1 Lecture 1: General Introduction Acknowledgements 2 Lecture 2: Kronecker's Rationality Criteria and Grothendieck's p-Curvature Conjecture 2.1 Kronecker's Criteria 2.1.1 Kronecker's Analytic Criterion, Kro57 2.1.2 Kronecker's Algebraic Criterion, Bau04 2.1.3 Translation of Kronecker's Algebraic Criterion in Terms of Differential Equations 2.2 Grothendieck's p-Curvature Conjecture 2.3 Back to Kronecker's Analytic Criterion: Gauß-ManinConnections 3 Lecture 3: Malčev-Grothendieck's Theorem, Its Variants in Characteristic p>0, Gieseker's Conjecture, de Jong's Conjecture, and the One to Come 3.1 Over C 3.2 Over an Algebraically Closed Field of Characteristic p>0 3.3 Over a p-Adic Field 4 Lecture 4: Interlude on Some Similarity Between the Fundamental Groups in Characteristic 0 and p>0 4.1 Lubotzky's Theorem 4.2 Tame Fundamental Group 4.3 Grothendieck's Specialization Homomorphism, SGA1, Exposé X, Exposé XIII 4.4 Finite Generation 4.5 Proof of Theorem 4.5, the ≠p Part 4.6 Proof of Theorem 4.5, the p Part 5 Lecture 5: Interlude on Some Difference Between the Fundamental Groups in Characteristic 0 and p>0 5.1 Serre's Construction 5.2 Various Obstructions 5.3 An Abstract Obstruction to Lift to Characteristic 0, Based on the Structure of the Fundamental Group 5.4 Main Definition 5.5 Independence of and Schur Rationality 5.6 The Roquette Curve, Combined with Serre's Construction 6 Lecture 6: Density of Special Loci 6.1 Definitions 6.2 Why Quasi-unipotent Monodromies at Infinity 6.3 Density Theorem for Quasi-unipotent Local Systems 6.4 Remarks 6.5 Some Other Dense or Not Dense Loci 6.6 Arithmetic Local Systems Are Not Dense: The Work of Biswas-Gupta-Mj-Whang and Landesman-Litt 6.6.1 Statement 6.6.2 Rigidity 6.6.3 Proof of Theorem 6.7 6.7 Weakly Arithmetic Complex Local Systems 6.7.1 Definitions 6.7.2 Density 7 Lecture 7: Companions, Integrality of Cohomologically Rigid Local Systems and of the Betti Moduli Space 7.1 Motivation on the Complex Side 7.2 Analogy Over a Finite Field 7.3 Geometricity 7.4 On Drinfeld's Proof 7.4.1 Reductions 7.4.2 Boundedness 7.4.3 Moduli 7.4.4 Gluing 7.5 Cohomologically Rigid Local Systems Are Integral, See EG18, Theorem 1.1 7.6 Integrality of the Whole Betti Moduli Space, See dJE22, Theorem 1.1 7.7 Obstruction 8 Lecture 8: Rigid Local Systems and F-Isocrystals 8.1 Crystalline Site, Crystals and Isocrystals 8.2 Nilpotent Crystalline Site, Crystals and Isocrystals 8.3 The Frobenius Action on the Set of Crystals on the Nilpotent Crystalline Site 8.4 The Frobenius Induces an Isomorphism on Cohomology in Characteristic 0 8.5 Proof of EG20, Theorem 1.6 9 Lecture 9: Rigid Local Systems, Fontaine-Laffaille Modules and Crystalline Local Systems 9.1 The Main Theorems 9.1.1 Good Model in the Projective Case 9.1.2 Theorem in the Projective Case 9.1.3 Good Model in the Quasi-projective Case 9.1.4 Theorem in the Quasi-projective Case 9.2 Simpson's Versus Ogus-Vologodsky's Correspondences in the Projective Case 9.3 Periodic de Rham-Higgs Flow on Xs and the GLr(p)-local Systems on XK in the Projective Case 9.4 Periodic de Rham-Higgs Flow on W and the GLr(W(p))-local Systems on XK in the Projective Case 9.5 From Crystalline p-Adic Local Systems on XK to p-Adic Local Systems on X in the Projective Case 9.6 Remarks 10 Lecture 10: Comments and Questions 10.1 With Respect to the p-Curvature Conjecture (Chap.SPIlinkcolor1002) 10.2 With Respect to the Malčev-Grothendieck's Theorem and Its Shadows in Characteristic p>0 (Chap.SPIlinkcolor1003) 10.3 With Respect to Lubotzky's Theorem SPIlinkcolor1004.2 10.4 With Respect to Theorem SPIlinkcolor1004.7 10.5 With Respect to Theorems SPIlinkcolor1005.3 and SPIlinkcolor1005.10 10.6 With Respect to Theorem SPIlinkcolor1006.2 10.7 With Respect to Theorem SPIlinkcolor1007.6 and a More Elaborate Version of Theorem SPIlinkcolor1007.8, See Theorem [Theorem 1.4]dJE22 10.8 With Respect to Theorem SPIlinkcolor1007.8 10.9 With Respect to Chaps.SPIlinkcolor1008 and SPIlinkcolor1009 Reference