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arXiv · 1804.00136

Shimura varieties at level $\Gamma_1(p^\infty)$ and Galois representations

Ana Caraiani·Daniel R. Gulotta·Chi-Yun Hsu·Christian Johansson·Lucia Mocz·Emanuel Reinecke·Sheng-Chi Shih
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Abstract

We show that the compactly supported cohomology of certain $\mathrm{U}(n,n)$ or $\mathrm{Sp}(2n)$-Shimura varieties with $\Gamma_1(p^\infty)$-level vanishes above the middle degree. The only assumption is that we work over a CM field $F$ in which the prime $p$ splits completely. We also give an application to Galois representations for torsion in the cohomology of the locally symmetric spaces for $\mathrm{GL}_n/F$. More precisely, we use the vanishing result for Shimura varieties to eliminate the nilpotent ideal in the construction of these Galois representations. This strengthens recent results of Scholze and Newton-Thorne.

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Ana Caraiani, Daniel R. Gulotta, Chi-Yun Hsu, Christian Johansson, Lucia Mocz, Emanuel Reinecke, Sheng-Chi Shih. 2018-03-31. Shimura varieties at level $\Gamma_1(p^\infty)$ and Galois representations. https://doi.org/10.1112/s0010437x20007149

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