Abstract

That there is an intimate connection between algebra (representations of Galois groups) and analysis (representations of matrix groups) is the content of the Langlands program. We recall the mod p Local Langlands Correspondence using the language of Iwahori induction.
We use the Iwhaori mod p LLC to compute the reductions of all 2-dimensional semi-stable representations of the Galois group of Q_p of weights up to p+1. We show that the reduction varies through an alternating sequence of irreducible and reducible representations.
In principle, our method works for all weights. In particular, it lets us go beyond the earlier glass ceiling of weight p-1 which occurs in the work of Breuil-Mezard and Guerberoff-Park, allowing us to complete our proof of our zig-zag conjecture.

This is joint work with Anand Chitrao (https://aus01.safelinks.protection.outlook.com/?url=https%3A%2F%2Farxiv.org%2Fpdf%2F2311.03740.pdf&data=05%7C02%7Cjames.borger%40anu.edu.au%7C3f06e548d4894f61d6e008dc22581e90%7Ce37d725cab5c46249ae5f0533e486437%7C0%7C0%7C638423009285829644%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C0%7C%7C%7C&sdata=knWNFBIiUHFCaxKXg3gtZjPcXHYaYb0QwXqDAI2f5dw%3D&reserved=0).