News & Events


Tuesday, July 17, 2018 - 16:30
M-4 (SMS)
Dr. Shalini Bhattacharya
Max Planck Institute for Mathematics, Bonn
On the reduction of Galois representations and local constancy with respect to weight

Abstract: Let $p$ be a prime number. The two-dimensional crystalline representations of the local Galois group $\mathrm{Gal}(\bar\Q_p|\Q_p)$ are parametrized by the pairs $(k,a)$ up to twists, where $k\geq 2$ is an integer and $a\in m_{\bar\Z_p}$, the maximal ideal in the ring of integers of $\bar\Q_p$. We are interested in studying the map$(k,a)\mapsto \bar V_{k,a}$, where $\bar V_{k,a}$ denotes the semisimplified mod $p$ reduction of a typical crystalline representation $V_{k,a}$. These reductions have been computed when $k\leq 2p+1$ or when $a$ has a small $p$-adic valuation. Using the theory of Wach modules, Laurent Berger has also shown that the map displayed above is locally constant with respect to both the variables and found an explicit bound on the radius of local constancy when %$k$ is fixed and$a$ varies (2012). However, if $a$ is fixed and $k$ varies, nothing more than the existence is known about the radius of local constancy. So we ask the following simple question: for any given $p$-adic integer $a$, how close do $k$ and $k'$ need to be in the weight space to ensure that $\bar V_{k,a}\cong\bar V_{k',a}$? We give a partial answer to this question using some explicit computations in the automorphic side of the $p$-adic and mod $p$ Local Langlands Correspondences for $\GL_2(\Q_p)$. 

Contact us

School of Mathematical Sciences

NISERPO- Bhimpur-PadanpurVia- Jatni, District- Khurda, Odisha, India, PIN- 752050

Tel: +91-674-249-4081

Corporate Site - This is a contributing Drupal Theme
Design by WeebPal.