The Categorical Local Langlands Correspondence and Anabelomorphy

Abstract

Let $G/\mathbb{Q}_p$ be a connected, split, reductive group over $\mathbb{Q}_p$. In this paper I show that if $K$ and $L$ are anabelomorphic $p$-adic fields i.e. $K$ and $L$ have topologically isomorphic absolute Galois groups, then the stacks of Langlands parameters (for the fields $K$ and $L$) considered in [Fargues and Scholze, 2024], are also isomorphic (Theorem 2.2.1). This leads to Conjecture 3.3.1 which provides a precise relationship between the main conjecture of [Fargues and Scholze, 2024] and anabelomorphy of $p$-adic fields considered in [Joshi, 2020a]. I establish my conjecture for a split torus in Theorem 4.1.