t-motives: Hodge structures, transcendence and other motivic aspects

Drinfeld in 1974, in his seminal paper [10], revolutionized the contribution to arithmetic of the area of global function fields. He introduced a function field analog of elliptic curves over number fields. These analogs are now called Drinfeld modules. For him and for many subsequent developments in the theory of automorphic forms over function fields, their main use was in the exploration of the global Langlands conjecture over function fields. One of its predictions is a correspondence between automorphic forms and Galois representations. The deep insight of Drinfeld was that the moduli spaces of Drinfeld modules can be assembled in a certain tower such that the corresponding direct limit of the associated `-adic cohomologies would be an automorphic representation which at the same time carries a Galois action. This would allow him to realize the correspondence conjectured by Langlands in geometry. Building on this, in [11] Drinfeld himself proved the global Langlands’ correspondence for function fields for GL2 and later in [18] Lafforgue obtained the result for all GLn. In a second direction, the analogy of Drinfeld modules with elliptic curves over number fields made them interesting objects to be studied on their own right. One could study torsion points and Galois representations, one could define cohomology theories such as de Rham or Betti cohomology and thus investigate their periods as well as transcendence questions. A main advance in this direction is the introduction of t-motives by Anderson in [1]. Passing from Drinfeld modules to t-motives may be compared to the passage from elliptic curves to abelian varieties. But more is true. The category of t-motives is also a simple function field analog of Grothendieck’s conjectured category of motives over number fields. It is this second direction which constitutes the main theme of the present workshop on t-motives. Intended subtopics were Galois representations, L-functions, transcendence results, Hodge structures and period domains.