Lecture Course on "Anderson Motives"

Time Schedule

Monday, March 16 and Tuesday, March 17

11:00 a.m. to 12:00 p.m.

Lunch break

2:00 p.m. to 3:00 p.m.

 I want to discuss some construction of objects which could be called ”mixed Tate” motives. This can be

justified because these objects have some cohomological realizations and these realizations produce numbers.

These numbers are the values of elements in certain extension groups of mixed Hodge-structures or mixed

Galois-modules.

     We have some other constructions of objects of the same nature using K-theory or what is called motivic

cohomology. These objects also produce some numbers and among these numbers we find the values ζ '(−n)

where n runs over the set of even positive integers. So the question arises whether the above construction

provides the same numbers.

     In my lectures I will not speak about K-theory. I will show that we can find mixed (Tate-) motives

inside the cohomology of Shimura-vatieties, some of these motives are not mixed Tate but rather ”Heegner-

motives”.

     If time permits I will explain the computation of the extension classes of these motives in the category

of mixed Hodge structures and I will show that they are in fact given by the values ζ '(−n).

     I will explain that some speculative ideas about these mixed motives - namely the non existence of

”exotic” mixed Tate motives- predicts interesting congruences between Siegel and elliptic modular forms,

these congruences have been tested experimentally.