High-codimension vanishing conjecture for Chevalley–Demazure groups over Euclidean number rings

Establish that for any Chevalley–Demazure group scheme G and any Euclidean number ring R, the rational cohomology H^{vcd(G(R))−i}(G(R); Q) vanishes for all integers i with 0 ≤ i < rk(G).

Background

Motivated by integral-generation results for Steinberg modules and top-degree vanishing in specific types (e.g., SL_n and Sp_{2n}), the authors propose a unified vanishing phenomenon for arithmetic groups G(R) associated to Chevalley–Demazure group schemes over Euclidean rings.

The conjecture generalizes the Church–Farb–Putman SL_n(Z) conjecture and is consistent with known cases (i = 0 for many types; i = 1 for several families; i = 2 for SL_n(Z)), as well as with low-rank computations across different series.