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arXiv · 2504.17683

On the locally analytic $\text{Ext}^1$-conjecture in the $\text{GL}_2(L)$ case

Benchao Su
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Abstract

Let $L$ be a finite extension of $\mathbb{Q}_p$. We calculate the dimension of $\text{Ext}^1$-groups of certain locally analytic representations of $\text{GL}_2(L)$ defined using coherent cohomology of Drinfeld curves. Furthermore, let $\rho_p$ be a $2$-dimensional continuous representation of $\text{Gal}(\bar L/L)$, which is de Rham with parallel Hodge-Tate weights $0,1$ and whose underlying Weil-Deligne representation is irreducible. We prove Breuil's locally analytic $\text{Ext}^1$ conjecture for such $\rho_p$. As an application, we show that the isomorphism class of the multiplicity space $\Pi^{\text{an}}_{\text{geo}}(\rho_p)$ of $\rho_p$ in the pro-\'etale cohomology of Drinfeld curves uniquely determines the isomorphism class of $\rho_p$.

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Benchao Su. 2025-04-24. On the locally analytic $\text{Ext}^1$-conjecture in the $\text{GL}_2(L)$ case. https://arxiv.org/abs/2504.17683

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