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Colin J. Bushnell

Publications and source records attributed to Colin J. Bushnell.

9 recordsLinked to original sources
arXiv2019

Tame multiplicity and conductor for local Galois representations

Let $F$ be a non-Archimedean locally compact field of residual characteristic $p$. Let $σ$ be an irreducible smooth representation of the absolute Weil group $\Cal W_F$ of $F$ and $\sw(σ)$ the Swan exponent of $σ$. Assume $\sw(σ) \ge1$. Let $\Cal I_F$ be the inertia subgroup of $\Cal W_F$ and $\Cal P_F$ the wild inertia subgroup. There is an essentially unique, finite, cyclic group $\varSigma$, of order prime to $p$, so that $σ(\Cal I_F) = σ(\Cal P_F)\varSigma$. In response to a query of Mark Reeder, we show that the multiplicity in $σ$ of any character of $\varSigma$ is bounded by $\sw(σ)$.

math.NT↗
arXiv2019

Local Langlands correspondence and ramification for Carayol representations

Let $F$ be a non-Archimedean locally compact field of residual characteristic $p$ with Weil group $\Cal W_F$. Let $σ$ be an irreducible smooth complex representation of $\Cal W_F$, realized as the Langlands parameter of an irreducible cuspidal representation $π$ of a general linear group over $F$. In an earlier paper, we showed that the ramification structure of $σ$ is determined by the fine structure of the endo-class $\varTheta$ of the simple character contained in $π$, in the sense of Bushnell-Kutzko. The connection is made via the {\it Herbrand function} $Ψ_\varTheta$ of $\varTheta$. In this paper, we concentrate on the fundamental Carayol case in which $σ$ is totally wildly ramified with Swan exponent not divisible by $p$. We show that, for such $σ$, the associated Herbrand function satisfies a certain symmetry condition or functional equation, a property that essentially characterizes this class of representations. We calculate $Ψ_\varTheta$ explicitly, in terms of a classical Herbrand function coming from the Bushnell-Kutzko classification of simple characters. We describe exactly the class of functions arising as Herbrand functions $Ψ_\varXi$, as $\varXi$ varies over totally wild endo-classes of Carayol type. In a separate argument, we get a complete description of $σ$ restricted to any ramification subgroup. This provides a different, more Galois-centred, view on $Ψ_\varTheta$.

math.RT↗
arXiv2017

Explicit local Jacquet-Langlands correspondence: the non-dyadic wild case

Let $F$ be a non-Archimedean locally compact field of residual characteristic $p$ with $p\neq 2$. Let $n$ be a power of $p$ and let $G$ be an inner form of the general linear group $\text{\rm GL}_n(F)$. We give a transparent parametrization of the irreducible, totally ramified, cuspidal representations of $G$ of parametric degree $n$. We show that the parametrization is respected by the Jacquet-Langlands correspondence, relative to any other inner form. This expresses the Jacquet-Langlands correspondence for such representations within a single, compact formula.

math.RT↗
arXiv2016

Higher ramification and the local Langlands correspondence

Let $F$ be a non-Archimedean locally compact field. We show that the local Langlands correspondence over $F$ has a strong property generalizing the higher ramification theorem of local class field theory. If $π$ is an irreducible cuspidal representation of a general linear group $GL_n(F)$ and $σ$ the corresponding irreducible representation of the Weil group $W_F$ of $F$, the restriction of $σ$ to a ramification subgroup of $W_F$ is determined by a truncation of the simple character $θ_π$ contained in $π$, and conversely. Numerical aspects of the relation are governed by a Herbrand-like function $Ψ_Θ$ depending on the endo-class $Θ$ of $θ_π$. We give a method for determining $Ψ_Θ$. Consequently, the ramification-theoretic structure of $σ$ can be predicted from the simple character $θ_π$ alone.

math.NT↗
arXiv2013

To an effective local Langlands Corrspondence

Let $F$ be a non-Archimedean local field. Let $\Cal W_F$ be the Weil group of $F$ and $\Cal P_F$ the wild inertia subgroup of $\scr W_F$. Let $\hat{\Cal W}_F$ be the set of equivalence classes of irreducible smooth representations of $\Cal W_F$. Let $\Cal A^0_n(F)$ denote the set of equivalence classes of irreducible cuspidal representations of $\roman{GL}_n(F)$ and set $\hat{\roman{GL}}_F = \bigcup_{n\ge1} \Cal A^0_n(F)$. If $σ\in \hat{\Cal W}_F$, let $\upr Lσ\in \hat{\roman{GL}}_F$ be the cuspidal representation matched with $σ$ by the Langlands Correspondence. If $σ$ is totally wildly ramified, in that its restriction to $\Cal P_F$ is irreducible, we treat $\upr Lσ$ as known. From that starting point, we construct an explicit bijection $\Bbb N:\hat{\Cal W}_F \to \hat{\roman{GL}}_F$, sending $σ$ to $\upr Nσ$. We compare this "naïve correspondence" with the Langlands correspondence and so achieve an effective description of the latter, modulo the totally wildly ramified case. A key tool is a novel operation of "internal twisting" of a suitable representation $π$ (of $\Cal W_F$ or $\roman{GL}_n(F)$) by tame characters of a tamely ramified field extension of $F$, canonically associated to $π$. We show this operation is preserved by the Langlands correspondence.

math.RT↗
arXiv2011

A congruence property of the local Langlands correspondence

Let $F$ be a non-Archimedean local field of residual characteristic $p$, and $\ell$ a prime number, $\ell \neq p$. We consider the Langlands correspondence, between irreducible, $n$-dimensional, smooth representations of the Weil group of $F$ and irreducible cuspidal representations of $\text{\rm GL}_n(F)$. We use an explicit description of the correspondence from an earlier paper, and otherwise entirely elementary methods, to show that it respects the relationship of congruence modulo $\ell$. The $\ell$-modular correspondence thereby becomes as effective as the complex one.

math.NT↗
arXiv2011

Intertwining of simple characters in GL(n)

Let $F$ be a non-Archimedean local field and let $G$ be the general linear group $G = \text{\rm GL}_n(F)$. Let $θ_1$, $θ_2$ be simple characters in $G$. We show that $θ_1$ intertwines with $θ_2$ if and only if $θ_1$ is endo-equivalent to $θ_2$. We also show that any simple character in $G$ is a $G$-type.

math.RT↗
arXiv2004

Functorial products for $GL_2\times GL_3$ and the symmetric cube for $GL_2$

In this paper we prove two new cases of Langlands functoriality. The first is a functorial product for cusp forms on $GL_2\times GL_3$ as automorphic forms on $GL_6$, from which we obtain our second case, the long awaited functorial symmetric cube map for cusp forms on $GL_2$. We prove these by applying a recent version of converse theorems of Cogdell and Piatetski-Shapiro to analytic properties of certain $L$-functions obtained from the method of Eisenstein series (Langlands-Shahidi method). As a consequence, we prove the bound 5/34 for Hecke eigenvalues of Maass forms over any number field and at every place, finite or infinite, breaking the crucial bound 1/6 (see below and Section 7 and 8) towards Ramanujan-Petersson and Selberg conjectures for $GL_2$. Many other applications are obtained.

math.NT↗