Local Systems in Algebraic-Arithmetic Geom...

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Beschreibung

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.

Über den Autor

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 l-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."

Zusammenfassung

(Over)view on the relation between local systems in complex algebraic geometry and in arithmetic geometry

Suggests deep conjectures which at present are out or reach

Suggests sub-conjectures some of which might be reachable

Inhaltsverzeichnis

- 1. Lecture 1: General Introduction. - 2. Lecture 2: Kronecker's Rationality Criteria and Grothendieck's p-Curvature Conjecture. - 3. Lecture 3: Mal¿ev-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.

Details

Erscheinungsjahr:	2023
Fachbereich:	Arithmetik & Algebra
Genre:	Mathematik
Rubrik:	Naturwissenschaften & Technik
Medium:	Taschenbuch
Seiten:	104
Reihe:	Lecture Notes in Mathematics
Inhalt:	vii 94 S. 15 s/w Illustr. 94 p. 15 illus.
ISBN-13:	9783031408397
ISBN-10:	303140839X
Sprache:	Englisch
Ausstattung / Beilage:	Paperback
Einband:	Kartoniert / Broschiert
Autor:	Esnault, Hélène
Auflage:	1st ed. 2023
Hersteller:	Springer Nature Switzerland Springer International Publishing Lecture Notes in Mathematics
Maße:	235 x 155 x 7 mm
Von/Mit:	Hélène Esnault
Erscheinungsdatum:	20.09.2023
Gewicht:	0,172 kg

preigu-id: 127243535