arXiv subjects

Benchao Su

Publications and source records attributed to Benchao Su.

4 recordsLinked to original sources

arXiv2026

Locally analytic vectors in the completed cohomology of quaternionic Shimura curves

We use the methods introduced by Lue Pan to study the locally analytic vectors of the completed cohomology of Shimura curves associated to an indefinite quaternion algebra $D$ which is ramified at a prime number $p$. Let $D_p^{\times}$ be the group of units of $D$ at $p$. Using $p$-adic uniformization of the quaternionic Shimura curves, we compute the Hecke eigenspace of the completed cohomology with the Hecke eigenvalues associated to a classical automorphic form on another quaternion algebra $\bar D$ (switching invariants of $D$ at $p,\infty$). We present this locally analytic $D_p^\times$-representation using the de Rham complex of the Lubin-Tate tower of dimension $1$. This is analogous to the Breuil-Strauch conjecture for the group $\mathrm{GL}_2(\mathbb{Q}_p)$. We show that the locally analytic $D_p^{\times}$-representation does not detect the Hodge filtration of the local de Rham Galois representation at $p$ in the crystalline case, and also give applications for the locally analytic Jacquet--Langlands correspondence for $\mathrm{GL}_2(\mathbb{Q}_p)$ and $D_p^\times$.

math.NT↗

arXiv2025

Locally analytic vectors in the completed cohomology of unitary Shimura curves

We use the methods introduced by Lue Pan to study the locally analytic vectors in the completed cohomology of unitary Shimura curves. As an application, we prove a classicality result on two-dimensional regular $\sigma$-de Rham representations of $\text{Gal}(\bar L/L)$ appearing in the locally $\sigma$-analytic vectors of the completed cohomology, where $L$ is a finite extension of $\mathbb{Q}_p$ and $\sigma:L\hookrightarrow E$ is an embedding of $L$ into a sufficiently large finite extension $E$ of $\mathbb{Q}_p$. We also prove that if a two-dimensional representation of $\text{Gal}(\bar L/L)$ appears in the locally $\sigma$-algebraic vectors of the completed cohomology then it is $\sigma$-de Rham. Finally, we give a geometric realization of some locally $\sigma$-analytic representations of $\mathrm{GL}_2(L)$. This realization has some applications to the $p$-adic local Langlands program, including a locality theorem for Galois representations arising from classical automorphic forms, an admissibility result for coherent cohomology of Drinfeld curves, and some special cases of the Breuil's locally analytic Ext$^1$-conjecture for $\mathrm{GL}_2(L)$.

math.NT↗

arXiv2025

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

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 $\rho_p$.

math.NT↗

arXiv2025

De Rham cohomology of Lubin-Tate spaces and Drinfeld spaces

Let $d\ge 1$ be an integer. We use the methods introduced by Lue Pan to prove that the compactly supported cohomology of Lubin-Tate towers and Drinfeld towers are isomorphic, as $\text{GL}_{d+1}(L)\times D_{L,\frac{1}{d+1}}^\times$-modules.

math.NT↗