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arXiv · 2409.17812

Singularities of Steinberg deformation rings

Daniel Funck·Jack Shotton
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Abstract

Let $l$ and $p$ be distinct primes, let $F$ be a local field with residue field of characteristic $p$, and let $\mathfrak{X}$ be the irreducible component of the moduli space of Langlands parameters for $GL_3$ over $\mathbb{Z}_l$ corresponding to parameters of Steinberg type. We show that $\mathfrak{X}$ is Cohen-Macaulay and compute explicit equations for it. We also compute the Weil divisor class group of the special fibre of $\mathfrak{X}$, motivated by work of Manning for $GL_2$. Our methods involve the calculation of the cohomology of certain vector bundles on the flag variety, and build on work of Snowden, Vilonen-Xue, and Ngo.

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Daniel Funck, Jack Shotton. 2024-09-26. Singularities of Steinberg deformation rings. https://arxiv.org/abs/2409.17812

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