Number Theory Seminar
Monday, 23 October 2023, 14:30
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Venue: Ramanujan hall
Host: U. K. Anandavardhanan
Speaker: Anand Chitrao
Affiliation: TIFR Mumbai
Title: Reductions mod $p$ of semi-stable representations.
Abstract: We compute the reductions mod $p$ of irreducible two-dimensional semi-stable representations of the absolute Galois group $\GQp$ of $\Qp$. We use the compatibility with respect to reduction mod $p$ between the $p$-adic Local Langlands Correspondence and an Iwahori theoretic version of the mod $p$ Local Langlands Correspondence. By estimating certain logarithmic functions on $\Qp$ by polynomials on open subsets of $\Zp$, we compute the reductions mod $p$ completely for weights at most $p + 1$. We also state how this method can be used, in theory, to compute the reductions mod $p$ of semi-stable representations of arbitrarily large weights. In passing, we extend some results on Iwahori induction to the case of non-commutative Hecke algebras.