Speaker: V. Ravitheja
Affiliation: IISc, Bengaluru
Title: Reductions of Crystalline representations for slopes in (2,p)
Date and Time: June 11, 2024, 14:30:00 Hours
Venue: AG-77
Asbtract: Let $p$ be an odd prime and let $G_p$ denote the local Galois group Gal($\bar{\mathbb Q}_p|\mathbb Q_p$). Colmez-Fontaine proved that a two-dimensional representation $W$ of $G_p$ is crystalline if and only if $D_{\mathrm{cris}}(W)$ is an admissible filtered $\varphi$-module of dimension $2$. Let $k \geq 2$ be an integer and $a_p \in \bar{\mathbb Q}_p$ have positive $p$-adic valuation $v(a_p)$, where $v$ is normalized so that $v(p) = 1$. Let $V_{k,a_p}$ be the two-dimensional $p$-adic crystalline representation of $G_p$ defined over $\bar{\mathbb Q}_p$ with Hodge-Tate weights $(0,k-1)$ and the trace of Frobenius acting on $D_{\mathrm{cris}}(V_{k,a_p})$ equals $a_p$. Let $\bar{V}_{k,a_p}$ denote the semi-simplified mod $p$ reduction of any $G_p$-stable lattice in $V_{k,a_p}$. We are interested in the explicit description of $\bar{V}_{k,a_p}$ as a function of $k$ and $a_p$. Now, the structure of $\bar{V}_{k,a_p}$ is known by the works of Buzzard-Gee and Ghate and his co-authors for slopes in the interval $(0,2)$. In this talk, we will report on some results describing the shape of $\bar{V}_{k,a_p}$ when $v(a_p) \in (2,p)$ and $v(a_p) \not \in\mathbb Z$. This is based on an ongoing joint work with Bhattacharya and Ghate.
Affiliation: IISc, Bengaluru
Title: Reductions of Crystalline representations for slopes in (2,p)
Date and Time: June 11, 2024, 14:30:00 Hours
Venue: AG-77
Asbtract: Let $p$ be an odd prime and let $G_p$ denote the local Galois group Gal($\bar{\mathbb Q}_p|\mathbb Q_p$). Colmez-Fontaine proved that a two-dimensional representation $W$ of $G_p$ is crystalline if and only if $D_{\mathrm{cris}}(W)$ is an admissible filtered $\varphi$-module of dimension $2$. Let $k \geq 2$ be an integer and $a_p \in \bar{\mathbb Q}_p$ have positive $p$-adic valuation $v(a_p)$, where $v$ is normalized so that $v(p) = 1$. Let $V_{k,a_p}$ be the two-dimensional $p$-adic crystalline representation of $G_p$ defined over $\bar{\mathbb Q}_p$ with Hodge-Tate weights $(0,k-1)$ and the trace of Frobenius acting on $D_{\mathrm{cris}}(V_{k,a_p})$ equals $a_p$. Let $\bar{V}_{k,a_p}$ denote the semi-simplified mod $p$ reduction of any $G_p$-stable lattice in $V_{k,a_p}$. We are interested in the explicit description of $\bar{V}_{k,a_p}$ as a function of $k$ and $a_p$. Now, the structure of $\bar{V}_{k,a_p}$ is known by the works of Buzzard-Gee and Ghate and his co-authors for slopes in the interval $(0,2)$. In this talk, we will report on some results describing the shape of $\bar{V}_{k,a_p}$ when $v(a_p) \in (2,p)$ and $v(a_p) \not \in\mathbb Z$. This is based on an ongoing joint work with Bhattacharya and Ghate.