\(t\)-motives: Hodge structures, transcendence and other motivic aspects. (English) Zbl 1441.14003

Böckle, Gebhard (ed.); Goss, David (ed.); Hartl, Urs (ed.); Papanikolas, Matthew (ed.)

EMS Series of Congress Reports. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-198-9/hbk; 978-3-03719-698-4/ebook). xi, 461 p. (2020).

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Publisher’s description: This volume contains research and survey articles on Drinfeld modules, Anderson \(t\)-modules and t-motives. Much material that had not been easily accessible in the literature is presented here, for example the cohomology theories and Pink’s theory of Hodge structures attached to Drinfeld modules and t-motives. Also included are survey articles on the function field analogue of Fontaine’s theory of \(p\)-adic crystalline Galois representations and on transcendence methods over function fields, encompassing the theories of Frobenius difference equations, automata theory, and Mahler’s method. In addition, this volume contains a small number of research articles on function field Iwasawa theory, 1-\(t\)-motifs, and multizeta values.
This book is a useful source for learning important techniques and an effective reference for all researchers working in or interested in the area of function field arithmetic, from graduate students to established experts.
The articles of this volume will be reviewed individually.
Indexed articles:
Brownawell, W. Dale; Papanikolas, Matthew A., A rapid introduction to Drinfeld modules, \(t\)-modules, and \(t\)-motives, 3-30 [Zbl 1440.14020]
Hartl, Urs; Juschka, Ann-Kristin, Pink’s theory of Hodge structures and the Hodge conjecture over function fields, 31-182 [Zbl 07242506]
Hartl, Urs; Kim, Wansu, Local shtukas, Hodge-Pink structures and Galois representations, 183-259 [Zbl 1440.14113]
Chang, Chieh-Yu, Frobenius difference equations and difference Galois groups, 261-295 [Zbl 1440.11098]
Pellarin, Federico, An introduction to Mahler’s method for transcendence and algebraic independence, 297-349 [Zbl 1440.11137]
Thakur, Dinesh S., Automata methods in transcendence, 351-372 [Zbl 1441.11052]
Bandini, Andrea; Bars, Francesc; Longhi, Ignazio, Aspects of Iwasawa theory over function fields, 375-416 [Zbl 07242511]
Taelman, Lenny, 1-\(t\)-motifs, 417-439 [Zbl 1441.11151]
Thakur, Dinesh S., Multizeta in function field arithmetic, 441-452 [Zbl 1441.11222]

MSC:

14-06	Proceedings, conferences, collections, etc. pertaining to algebraic geometry
11-06	Proceedings, conferences, collections, etc. pertaining to number theory
14C15	(Equivariant) Chow groups and rings; motives
14C30	Transcendental methods, Hodge theory (algebro-geometric aspects)
11G09	Drinfel’d modules; higher-dimensional motives, etc.
11J93	Transcendence theory of Drinfel’d and \(t\)-modules
11R58	Arithmetic theory of algebraic function fields
13A35	Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
00B15	Collections of articles of miscellaneous specific interest