Shalini Bhattacharya
University of Hyderabad
June 14, 2024
Reduction modulo $p$ of certain Galois representations: We will describe the problem of mod $p$ reduction of $p$-adic local Galois representations.For two dimensional crystalline representations of the local Galois group $\mathrm{Gal}(\bar{\mathbb Q_p}|\mathbb Q_p)$, the reduction can be computed using the compatibility of $p$-adic and mod $p$ Local Langlands Correspondences; this method was first introduced by Christophe Breuil in 2003.We will give a sketch of the background and history of the problem and discuss few interesting patterns in the behavior of the reduction map. Most of the above results are available only for odd primes or sufficiently large primes. If time permits, we will look into the curious case of $p=2$, which is an ongoing work with Arathy Venugopal. We will compare our results with some existing results for odd primes, for example with Buzzard-Gee's results for slope $1/2$ and in general with the Zigzag conjecture of Ghate.
University of Hyderabad
June 14, 2024
Reduction modulo $p$ of certain Galois representations: We will describe the problem of mod $p$ reduction of $p$-adic local Galois representations.For two dimensional crystalline representations of the local Galois group $\mathrm{Gal}(\bar{\mathbb Q_p}|\mathbb Q_p)$, the reduction can be computed using the compatibility of $p$-adic and mod $p$ Local Langlands Correspondences; this method was first introduced by Christophe Breuil in 2003.We will give a sketch of the background and history of the problem and discuss few interesting patterns in the behavior of the reduction map. Most of the above results are available only for odd primes or sufficiently large primes. If time permits, we will look into the curious case of $p=2$, which is an ongoing work with Arathy Venugopal. We will compare our results with some existing results for odd primes, for example with Buzzard-Gee's results for slope $1/2$ and in general with the Zigzag conjecture of Ghate.