Colloquium on the Geometry of Groups and Numbers

Taking place at Université Pierre et Marie Curie, 8th of June, 2017.

Günter Harder (Hausdorff Center for Mathematics, Bonn)

Divisibility of L-values and the existence of non trivial cohomology classes

We consider a very special class of congruence subgroups of Sl_2(Z[i]) ; for each of these congruence subgroups, the Eisenstein cohomology is essentially described in terms of special values of a specific Hecke character (which is determined by the subgroup).
We analyze these special values and show that the divisibility of these special values by certain primes ℓ implies that the cohomology of the congruence subgroup contains some non trivial classes. These classes are either cuspidal classes or they are ℓ-torsion classes.

In an experiment always the second case occurs (computation by D. Yasaki).

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