Searcharxiv⌕ Search

arXiv subjects

Alexandre Pyvovarov

Publications and source records attributed to Alexandre Pyvovarov.

6 recordsLinked to original sources
arXiv2026

Space of norms on locally algebraic representations

Let $F$ and $E$ be finite extensions of $\mathbb Q_p$, let $\mathbb G$ be a reductive group over $F$, and put $G=\mathbb G(F)$. Let $V$ be a locally algebraic representation of the form $V=\pi_{\mathrm{sm}}\otimes_E\sigma_{\mathrm{alg}}$, where $\pi_{\mathrm{sm}}$ is smooth admissible and $\sigma_{\mathrm{alg}}$ is finite-dimensional algebraic. We study the extended Goldman--Iwahori distance on the set of non-Archimedean norms on $V$. After fixing a reference norm $\alpha_0$, its finite-distance component $\mathscr N_{\alpha_0}(V)$ is the bounded projective limit of the extended Bruhat--Tits buildings attached to $V_K=\pi_{\mathrm{sm}}^K\otimes_E\sigma_{\mathrm{alg}}$. It is complete for the resulting uniform sup metric; this metric is of $\ell^\infty$ type and is generally not CAT(0). We prove directly that a $G$-orbit in $\mathscr N_{\alpha_0}(V)$ is bounded if and only if this component contains a $G$-invariant norm. The invariant norm is the pointwise supremum of the orbit. We formulate an integral group-algebra and type-Hecke condition necessary for an invariant norm. For $G=GL_n(F)$ we specialise to $V=\operatorname{BS}(r)=\pi_{\mathrm{gen}}(r)\otimes_E \pi_{\mathrm{alg}}(r)$.

math.RT↗
arXiv2026

A few remarks on the Baez-Duarte Criterion

We study exponentially damped M\"obius approximants in $\mathscr H=L^2([1,\infty),dt/t^{-2})$. With \[ \gamma_n(t)=\left\lfloor\frac tn\right\rfloor -\frac{\lfloor t\rfloor}{n},\qquad f(u)(t)=\sum_{n\ge1}\mu(n)e^{-nu}\gamma_n(t),\] we compute the relevant scalar products, characterize the M\"obius coefficients as the unique coefficients giving pointwise convergence to the constant function, and prove $\langle1 \mid f(u)\rangle\to1$. Vasyunin's formula expresses $F(e^{-u})=\|f(u)\|_2^2$ as an arithmetic cotangent sum. To analyze $F(x)$ as $x\uparrow 1$, we define the canonical third-order truncation $\mathcal F_{[3]}$ of $F$ by deleting the sole remainder $\rho_3$. We prove exact edge and residue-character cancellations, initial-edge asymptotics, finite-scale formulas, and \[ \mathcal F_{[3]}(x)\ll \frac{\log^2\!\bigl(e/(1-x)\bigr)}{1-x}. \] For the terms containing $\rho_3$, we prove initial-edge asymptotics, and a finite-scale criterion. The unresolved boundedness problem is thereby reduced to explicit global bilinear cancellation.

math.NT↗
arXiv2019

On the Breuil-Schneider conjecture II: Potentially crystalline non-generic case

This paper improves some results of the author's previous work. We investigate the case of non-smooth points on automorphic components and prove the Breuil--Schneider conjecture under a generic Fontaine--Laffaille hypothesis on the residual Galois representation. The residual hypothesis is formulated in terms of an extremal modular Serre weight and the maximal shape of the associated Fontaine--Laffaille module; it is the hypothesis under which weight elimination makes the relevant patched intertwining operator invertible.

math.NT↗
arXiv2018

Generic smooth representations

Let $F$ be a non-archimedean local field. In this paper we explore genericity of irreducible smooth representations of $GL_n(F)$ by restriction to a maximal compact subgroup $K$ of $GL_n(F)$. Let $(J, \lambda)$ be a Bushnell--Kutzko type for a Bernstein component $\Omega$. The work of Schneider--Zink gives an irreducible $K$-representation $\sigma_{min}(\lambda)$, which appears with multiplicity one in $\mathrm{Ind}_J^K \lambda$. Let $\pi$ be an irreducible smooth representation of $GL_n(F)$ in $\Omega$. We will prove that $\pi$ is generic if and only if $\sigma_{min}(\lambda)$ is contained in $\pi$ with multiplicity one.

math.NT↗
arXiv2018

The endomorphism ring of projectives and the Bernstein centre

Let $F$ be a local non-archimedean field and $\mathcal{O}_F$ its ring of integers. Let $\Omega$ be a Bernstein component of the category of smooth representations of $GL_n(F)$, let $(J, \lambda)$ be a Bushnell-Kutzko $\Omega$-type, and let $\mathfrak{Z}_{\Omega}$ be the centre of the Bernstein component $\Omega$. This paper contains two major results. Let $\sigma$ be a direct summand of $\mathrm{Ind}_J^{GL_n(\mathcal{O}_F)} \lambda$. We will begin by computing $\mathrm{c\text{--} Ind}_{GL_n(\mathcal{O}_F)}^{GL_n(F)} \sigma\otimes_{\mathfrak{Z}_{\Omega}}\kappa(\mathfrak{m})$, where $\kappa(\mathfrak{m})$ is the residue field at maximal ideal $\mathfrak{m}$ of $\mathfrak{Z}_{\Omega}$, and the maximal ideal $\mathfrak{m}$ belongs to a Zariski-dense set in $\mathrm{Spec}\: \mathfrak{Z}_{\Omega}$. This result allows us to deduce that the endomorphism ring $\mathrm{End}_{GL_n(F)}(\mathrm{c\text{--} Ind}_{GL_n(\mathcal{O}_F)}^{GL_n(F)} \sigma)$ is isomorphic to $\mathfrak{Z}_{\Omega}$, when $\sigma$ appears with multiplicity one in $\mathrm{Ind}_J^{GL_n(\mathcal{O}_F)} \lambda$.

math.NT↗