Vivek Rai | Searcharxiv

arXiv2022

Towards a mod-$p$ Lubin-Tate theory for $\GL_2$ over totally real fields

We show that the conjectural mod $p$ local Langlands correspondence can be realised in the mod $p$ cohomology of the Lubin-Tate towers. The proof utilizes a well known conjecture of Buzzard-Diamond-Jarvis \cite[Conj. 4.9]{BDJ10}, a study of completed cohomology of the ordinary and supersingular locus of the Shimura curves for a totally real field $F$ and of mod $l(\neq p)$ local Langlands correspondence as given by Emerton-Helm \cite{EmertonHelm14}. %And then we connect the completed cohomlgy with the cohomology of Lubin-Tate towers. In the case of modular curves a similar theorem was obtained by Chojecki \cite{Cho15}.

math.NT↗

arXiv2020

Reductions of Galois representations of Slope $\frac{3}{2}$

We prove a zig-zag conjecture describing the reductions of irreducible crystalline two-dimensional representations of $G_{\mathbb{Q}_p}$ of slope $\frac{3}{2}$ and exceptional weights. This along with previous works completes the description of the reduction for all slopes less than $2$. The proof involves computing the reductions of the Banach spaces attached by the $p$-adic LLC to these representations, followed by an application of the mod $p$ LLC to recover the reductions of these representations.

math.NT↗