arXiv subjects

Amit Ophir

Publications and source records attributed to Amit Ophir.

6 recordsLinked to original sources

arXiv2024

Projective smooth representations in natural characteristic

We investigate under which circumstances there exists nonzero {\it{projective}} smooth $\field[G]$-modules, where $\field$ is a field of characteristic $p$ and $G$ is a locally pro-$p$ group. We prove the non-existence of (non-trivial) projective objects for so-called {\it{fair}} groups -- a family including $\bf{G}(\frak{F})$ for a connected reductive group $\bf{G}$ defined over a non-archimedean local field $\frak{F}$. This was proved in \cite{SS24} for finite extensions $\frak{F}/\Bbb{Q}_p$. The argument we present in this note has the benefit of being completely elementary and, perhaps more importantly, adaptable to $\frak{F}=\Bbb{F}_q(\!(t)\!)$. Finally, we elucidate the fairness condition via a criterion in the Chabauty space of $G$.

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arXiv2024

On the relation between pseudocharacters and Chenevier's determinants

Consider a commutative unital ring $A$ and a unital $A$-algebra $R$. Let $d$ be a positive integer. Chenevier proved that when $(2d)!$ is invertible in $A$, the map associating to a determinant its trace is a bijection between $A$-valued $d$-dimensional determinants of $R$ and $A$-valued $d$-dimensional pseudocharacters of $R$. In this paper, we show that assuming $d!$ is invertible in $A$ is sufficient. This assumption is already made in the definition of a $d$-dimensional pseudocharacter. Our proof involves establishing a product formula for pseudocharacters, which might be of independent interest.

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arXiv2024

Maximal stable lattices in representations over discretely valued fields

Let $\rho\colon G\to \mathrm{GL}_n(K)$ be an continuous irreducible representation of a compact group over a complete discretely valued field $K$. Let $W_i,W_j$ be two irreducible subrepresentations of $\overline{\rho}^{ss}$, the semisimplification of the residual representation. We study the structure of the $G$-stable lattices $\Lambda\subseteq K^n$ with a view to understanding the question of when $\rho$ realises a non-split extension of $W_i$ by $W_j$. In particular, we introduce the notion of a maximal $G$-stable lattice and prove that any non-split extension of $W_i$ by $W_j$ that can be realised by $\rho$ can also be realised by a maximal lattice. As applications, we give a new proof and a strengthening of Bella\"iche's generalisation of Ribet's Lemma, which assures the abundancy of non-split extensions that can be realised by $\rho$. On the other hand, we also show that, if the representations $W_i, W_j$ occur with multiplicity one in $\overline{\rho}^{ss}$, then $\rho$ can realise at most one non-split extension of $W_i$ by $W_j$.

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arXiv2021

On Ribet's Lemma for $\mathrm{GL}_2$ modulo prime powers

Let $\rho\colon G\to \mathrm{GL}_2(K)$ be a continuous representation of a compact group $G$ over a complete discretely valued field $K$, with ring of integers $\mathcal O$ and uniformiser $\pi$. We prove that $\operatorname{tr}\rho$ is reducible modulo $\pi^n$ if and only if $\rho$ is reducible modulo $\pi^n$. More precisely, there exist characters $\chi_1,\chi_2 \colon G\to(\mathcal O/\pi^n\mathcal O)^{\times}$ such that $\det(t - \rho(g))\equiv (t-\chi_1(g))(t-\chi_2(g))\pmod{\pi^n}$ for all $g\in G$, if and only if there exists a $G$-stable lattice $\Lambda\subset K^2$ such that $\Lambda/\pi^n\Lambda$ contains a $G$-invariant, free, rank one $\mathcal O/\pi^n\mathcal O$-submodule. Our result applies in the case that $\rho$ is not residually multiplicity free, in which case it answers a question of Bella\"iche--Chenevier. As an application, we prove an optimal version of Ribet's Lemma, which gives a condition for the existence of a $G$-stable lattice $\Lambda$ that realises a non-split extension of $\chi_2$ by $\chi_1$

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arXiv2021

On the rigidity of invariant norms on the $p$-adic Schrödinger representation

Motivated by questions about $\mathbb{C}_p$-valued Fourier transform on the locally compact group $(\mathbb{Q}_p^d,+)$, we study invariant norms on the $p$-adic Schrödinger representation of the Heisenberg group. Our main result is a minimality and rigidity property for norms in a family of invariant norms parameterized by a Grassmannian. This family is the orbit of the sup norm under the action of the symplectic group, acting via intertwining operators. We also prove general fundamental properties of quotients of the universal unitary completion of cyclic algebraic representations. Combined with the rigidity property, we are able to show that the completion of the Schrödinger representation in any of the norms in that family satisfies a strong notion of irreducibility and a strong version of Schur's lemma. Norms that can be formed as the maximum of a finite number of norms from that family are also studied. We conclude this paper with a list of open questions.

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arXiv2016

Q-binomials and non-continuity of the p-adic Fourier transform

Let F be a finite extension of Q_p. We show that every Schwartz function on F, with values in an algebraic closure of Q_p, is the uniform limit of a sequence of Schwartz functions, whose Fourier transforms tend uniformly to 0. The proof uses the notion of q-binomial coefficients and some classical identities between them.

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