arXiv subjects

Anand Chitrao

Publications and source records attributed to Anand Chitrao.

7 recordsLinked to original sources

arXiv2026

On the $I(1)$-invariants: Non-abelian Hecke algebra case

Let $F$ be a finite extension of $\mathbb{Q}_p$. The so-called supersingular representations are the basic building blocks in the theory of mod $p$ representations of ${\rm GL}_2(F)$. The space of pro-$p$-Iwahori invariants of a universal module played a crucial role in the construction of the supersingular representations of ${\rm GL}_2(\mathbb{Q}_p)$. In this paper, we give an explicit description of the pro-$p$-Iwahori invariants of the universal module $\pi_r$ for $r = 0, q - 1$ using the Iwahori-Hecke model. We also determine the action of the pro-$p$-Iwahori-Hecke algebra on these newly found invariants. As an application, we recover $\pi_r$ functorially from its space of $I(1)$-invariants and extend a theorem of Ollivier for any totally ramified extension of $\mathbb{Q}_p$ other than itself.

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arXiv2026

Reduction mod $p$ of semi-stable representations of some super-Breuil weights

We determine the mod $p$ reductions of the semi-stable representations $V_{k, \mathcal{L}}$ of weight $k \in [p + 5, 2p]\cup[2p + 6, 3p + 1]$ and $v_p(\mathcal{L}) < 1-k/2$ for primes $p \geq 5$. In particular, this shows that the techniques introduced in [CG24] involving the $p$-adic and mod $p$ local Langlands correspondences can be used to compute the reduction of $V_{k, \mathcal{L}}$ outside the range $k \in [3, p + 1]$. Moreover, this shows that the bound on $v_p(\mathcal{L})$ given by Bergdall-Levin-Liu [BLL23] can be improved, at least for weights $k \in [2p + 6, 3p + 1]$.

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arXiv2026

Explicit isomorphisms for a Herr-type complex over a metabelian extension

Let $S$ be a Banach algebra over $\mathbb{Q}_p$ whose residue fields are finite extensions of $\mathbb{Q}_p$. Given an arithmetic family $V$ of Galois representations, i.e., a finite free $S$-module $V$ with a continuous action of the absolute Galois group of a $p$-adic number field, we construct a complex associated to $V$ over false-Tate extensions and construct explicit isomorphisms between its cohomology and the Galois cohomology. This recovers earlier results by Tavares Ribeiro when $S$ is a finite extension of $\mathbb{Q}_p$.

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arXiv2025

Iwahori-Hecke model for the universal supersingular representation

Let $F$ be a non-archimedean local field with residue field $\mathbb{F}_q$. When $F$ is a finite extension of $\mathbb{Q}_{p}$, Anandavardhanan-Borisagar and Anandavardhanan-Jana introduced an Iwahori-Hecke model for the universal supersingular representation $\pi_r$ in the regular case $ 0 < r < q-1$. When $F=\mathbb{Q}_{p}$, the first author introduced an Iwahori-Hecke model for $\pi_r$ when $r = 0, p - 1$. We extend this result to an arbitrary local field $F$ for $r = 0, q - 1$. We also write down an explicit non-split self-extension of $\pi_r$ which has a four-dimensional space of $I(1)$-invariants when $F = \mathbb{Q}_p$ and $r = 0, p-1$.

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arXiv2025

Semi-stable representations as limits of crystalline representations

We construct an explicit sequence $V_{k_n,a_n}$ of crystalline representations of exceptional weights converging to a given irreducible two-dimensional semi-stable representation $V_{k,{\mathcal{L}}}$ of $\mathrm{Gal}({\overline{\mathbb{Q}}}_p/{\mathbb{Q}}_p)$. The convergence takes place in the blow-up space of two-dimensional trianguline representations studied by Colmez and Chenevier. The process of blow-up is described in detail in the rigid analytic setting and may be of independent interest. Also, we recover a formula of Stevens expressing the ${\mathcal{L}}$-invariant as a logarithmic derivative. Our result can be used to compute the reduction of $V_{k,{\mathcal{L}}}$ in terms of the reductions of the $V_{k_n,a_n}$. For instance, using the zig-zag conjecture we recover (resp. extend) the work of Breuil-Mézard and Guerberoff-Park computing the reductions of the $V_{k,{\mathcal{L}}}$ for weights at most $p-1$ (resp. $p+1$), at least on the inertia subgroup. In the cases where zig-zag is known, we are further able to obtain some new information about the reductions for small odd weights. Finally, we explain some apparent violations to local constancy in the weight of the reductions of crystalline representations of small weight.

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arXiv2024

Reductions of semi-stable representations using the Iwahori mod $p$ Local Langlands Correspondence

We determine the mod $p$ reductions of all two-dimensional semi-stable representations $V_{k,\mathcal{L}}$ of the Galois group of $\mathbb{Q}_p$ of weights $3 \leq k \leq p+1$ and $\mathcal{L}$-invariants $\mathcal{L}$ for primes $p \geq 5$. In particular, we describe the constants appearing in the unramified characters completely. The proof involves computing the reduction of Breuil's $\mathrm{GL}_2(\mathbb{Q}_p)$-Banach space $\tilde{B}(k,\mathcal{L})$, by studying certain logarithmic functions using background material developed by Colmez, and then applying an Iwahori theoretic version of the mod $p$ Local Langlands Correspondence.

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arXiv2023

An Iwahori theoretic mod $p$ Local Langlands Correspondence

We extend a comparison theorem of Anandavardhanan-Borisagar between the quotient of the induction of a mod $p$ character by the image of an Iwahori-Hecke operator and compact induction of a weight to the case of the trivial character. This involves studying the corresponding non-commutative Iwahori-Hecke algebra. We use this to give an Iwahori theoretic reformulation of the (semi-simple) mod $p$ Local Langlands Correspondence discovered by Breuil and reformulated functorially by Colmez. This version of the correspondence is expected to have applications to computing the mod $p$ reductions of semi-stable Galois representations.

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