Comptage de repr\'esentations cuspidales congruentes
Vincent S\'echerre (LM-Versailles)

TL;DR
This paper establishes a numerical criterion for when integral $ell$-adic cuspidal representations of inner forms of $GL_n(F)$ have supercuspidal reductions mod $ell$, and analyzes their structure and invariants, aiding the study of congruences under the Jacquet-Langlands correspondence.
Contribution
It generalizes previous results by providing a counting criterion for supercuspidal reductions and introduces an invariant $w( ilde ho)$ to analyze their properties and behavior.
Findings
Provides a criterion for supercuspidal irreducible reduction mod $ell$.
Develops a counting method for inertial classes congruent to a given class.
Defines an invariant $w( ilde ho)$ with expected compatibility under Jacquet-Langlands.
Abstract
Let be a non-Archimedean locally compact field of residue characteristic , be an inner form of , , and be a prime number different from . We give a numerical criterion for an integral -adic irreducible cuspidal representation of to have a super\-cuspidal irreducible reduction mod , by counting inertial classes of cuspidal representations that are congruent to the inertial class of , generalizing results by Vign{\'e}ras and Dat. In the case the reduction mod of is not super\-cuspidal irreducible, we show that this counting argument allows us to compute its length and the size of the supercuspidal support of its irreducible components. We define an invariant | the product of this length by this size | which is expected to behave nicely through the local Jacquet-Langlands…
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Taxonomy
TopicsAdvanced Algebra and Geometry · Algebraic Geometry and Number Theory
