1. Course Background & Purposes

The present course is a subsequent of 2022 Spring Course on Drinfeld modules and modular varieties, where we discussed analytic and algebraic theory of Drinfeld modules and their modular varieities and the construction of maximal abelian extensions. In this course we shall introduce Anderson’s t-modules and abelian modules, which are higher dimensional generalizations of Drinfeld modules and function field analogs of abelian varieties. As their modern counterpart and generalizations, shtukas and their moduli spaces play a role in the Langlands correspondence for function fields, which also motivates the present course.   

2. Course Outline & Descriptions

Our first and main goal is to cover contents of t-motives, following Anderson’s fundamental paper [1]. Our supplementary materials will be Chapter 5 of Goss [2] and the survey paper by Brownawell and Papanikolas [3], for which we shall also introduce dual t-motives and their connection with transcendental theory. We shall briefly introduce shtukas, for this we will go over the Chapter 6 of Goss [2] and also explain its connection with Drinfeld modules and abelian t-modules. We  also hope to discuss a small part of the paper by Hartl and Juschka [4].  

Prerequisites: Drinfeld modules

References:

[1] G. Anderson, t-Motives. Duke Math J. 53 (1986), 457—502.

[2] D. Goss, Basic Structures of Function Field Arithmetic. (1996)

[3] W. Brownawell and M. Papanikolas, A rapid introduction to Drinfeld modules, t-modules and t-motives, arXiv:1806.03919.

[4] U. Hartl, A.-K. Juschka: Pink's Theory of Hodge Structures and the Hodge Conjecture over Function Fields, in t-motives: Hodge structures, transcendence and other motivic aspects, Editors G. Böckle, D. Goss, U. Hartl, M. Papanikolas, EMS Congress Reports, European Mathematical Society 2020, pp. 31-182.  Also available as arXiv:math.NT/1607.01412.

3. Grading

There is no grading and no credits offered in this short course. Students and audience

will be given opportunities of giving reading reports on [2] and [4] and participating discussions.

4. Registration

https://forms.gle/W8advRfJqUviYK1r6

5. Live Stream

TBA