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.

- 1. Lecture 1: General Introduction. - 2. Lecture 2: Kronecker’s Rationality Criteria and Grothendieck’s p-Curvature Conjecture. - 3. Lecture 3: Malcev-Grothendieck’s Theorem, Its Variants in Characteristic p > 0, Gieseker’s Conjecture, de Jong’s Conjecture, and the One to Come. - 4. Lecture 4: Interlude on Some Similarity Between the Fundamental Groups in Characteristic 0 and p > 0. - 5. Lecture 5: Interlude on Some Difference Between the Fundamental Groups in Characteristic 0 and p > 0. - 6. Lecture 6: Density of Special Loci. - 7. Lecture 7: Companions, Integrality of Cohomologically Rigid Local Systems and of the Betti Moduli Space. - 8. Lecture 8: Rigid Local Systems and F-Isocrystals. - 9. Lecture 9: Rigid Local Systems, Fontaine-Laffaille Modules and Crystalline Local Systems. - 10. Lecture 10: Comments and Questions.

Hélène Esnault is an emeritus Einstein Professor at the Freie Universität Berlin. She is currently a part time Professor at the University of Copenhagen and an associate part time Professor at Harvard University. She is working in the field of algebraic and arithmetic geometry. She established bridges between the analytic and arithmetic theories comprising for example the study of vanishing theorems, of rational points over finite fields, of complex and $\ell$-adic local systems, of analytic and arithmetic crystals, of algebraic cycles. She was awarded numerous prizes, including the Cantor medal (2019), and several honorary degrees, including the one of the University of Chicago (2023) for her "vision of algebraic geometry that touches upon most of the active branches of that vast field but is nevertheless recognizable her own."