arXiv subjects

Luis Lomelí

Publications and source records attributed to Luis Lomelí.

8 recordsLinked to original sources

arXiv2024

On Kim's Assumption A over function fields

We prove Kim's Assumtion A for the split classical groups in positive characteristic. Actually, we work in the slightly more general setting of groups of classical kind, which includes quasi-split classical groups and generalized spinor groups. We establish our results whenever a local Ramanujan bound holds; a bound that is known for the split classical groups in characteristic $p$, and we prove it for groups of classical kind under a local-global restriction.

math.NT↗

arXiv2024

On generic representations of quasi-split reductive groups over local fields of positive characteristic

Let $F$ be a locally compact non-Archimedean field, and $\bf G$ a connected quasi-split reductive group over $F$. We are interested in complex irreducible smooth generic representations $π$ of ${\bf G}(F)$. When $F$ has positive characteristic, we prove important properties which previously were only available for $F$ of characteristic 0. The first one is the tempered $L$-function conjecture of Shahidi, stating that when $π$ as above is tempered, then the $L$-functions attached to $π$ by the Langlands-Shahidi method have no pole for ${\rm Re}(s)>0$. We also establish the standard module conjecture of Casselman and Shahidi, saying that if $π$ is written as the Langlands quotient of a standard module, then it is in fact the full standard module. Finally, for a split classical group $\bf G$ we prove a useful result on the unramified unitary spectrum of ${\bf G}(F)$.

math.RT↗

arXiv2021

${\rm SL}_$ over local and adèle rings: $$-euclideanity and Bruhat generators

Let $(R,*)$ be a ring with involution and let $A = {\rm M}(n,R)$ be the matrix ring endowed with the $*$-transpose involution. We study ${\rm SL}_*(2,A)$ and the question of Bruhat generation over commutative and non-commutative local and adèlic rings $R$. An important tool is the property of a ring being $*$-Euclidean. In this regard, we introduce the notion of a $*$-local ring $R$, prove that $A$ is $*$-Euclidean and explore reduction modulo the Jacobson radical for such rings. Globally, we provide an affirmative answer to the question wether a commutative adèlic ring $R$ leads towards the ring $A$ being $*$-Euclidean; while the non-commutative adèlic quaternions are such that $A$ is $*$-Euclidean and ${\rm SL}_*$ is generated by its Bruhat elements if and only if the characteristic is $2$.

math.GR↗

arXiv2016

On the rationality and holomorphy of Langlands-Shahidi L-functions over function fields

We prove three main results: all Langlands-Shahidi automorphic $L$-functions over function fields are rational; after twists by highly ramified characters our automorphic $L$-functions become polynomials; and, if $π$ is a globally generic cuspidal automorphic representation of a split classical group or a unitary group ${\bf G}_n$ and $τ$ a cuspidal (unitary) automorphic representation of a general linear group, then $L(s,π\times τ)$ is holomorphic for $\Re(s) > 1$ and has at most a simple pole at $s=1$. We also prove the holomorphy and non-vanishing of automorphic exterior square, symmetric square and Asai L-functions for $\Re(s) > 1$. Finally, we complete previous results on functoriality for the classical groups over function fields.

math.NT↗

arXiv2016

Globalization of supercuspidal representations over function fields and applications

Let H be a connected reductive group defined over a non-archimedean local field F of characteristic p>0. Using Poincaré series, we globalize supercuspidal representations of H(F) in such a way that we have control over ramification at all other places, and such that the notion of distinction with respect to a unipotent subgroup (indeed more general subgroups) is preserved. In combination with the work of Vincent Lafforgue on the global Langlands correspondence, we present some applications, such as the stability of Langlands-Shahidi γ-factors and the local Langlands correspondence for classical groups.

math.NT↗

arXiv2014

On twisted exterior and symmetric square $γ$-factors

We establish the existence and uniqueness of twisted exterior and symmetric square $γ$-factors in positive characteristic by studying the Siegel Levi case of generalized spinor groups. The corresponding theory in characteristic zero is due to Shahidi. In addition, in characteristic $p$ we prove that these twisted local factors are compatible with the local Langlands correspondence. As a consequence, still in characteristic $p$, we obtain a proof of the stability property of $γ$-factors under twists by highly ramified characters. Next we use the results on the compatibility of the Langlands-Shahidi local coefficients with the Deligne-Kazhdan theory over close local fields to show that the twisted symmetric and exterior square $γ$-factors, $L$-functions and $\varepsilon$-factors are preserved over close local fields. Furthermore, we obtain a formula for Plancherel measures in terms of local factors and we also show that they also preserved over close local fields.

math.NT↗

arXiv2013

Uniqueness of Rankin-Selberg products

In the present paper, we show the equality of the $γ$-factors defined by Jacquet, Piatetski-Shapiro and Shalika with those obtained via the Langlands-Shahidi method. Our results are new in the case of positive characteristic, where we establish a refined version of the local-global principle for ${\rm GL}_n$ which has independent interest. In characteristic zero, the results are due to Shahidi. The comparison of $γ$-factors is made via a uniqueness result for Rankin-Selberg $γ$-factors.

math.NT↗

arXiv2012

Characterization of γ-factors: the Asai case

Let $E$ be a separable quadratic extension of a locally compact field $F$ of positive characteristic. Asai γ-factors are defined for smooth irreducible representations πof ${\rm GL}_n(E)$. If σis the Weil-Deligne representation of $\mathcal{W}_E$ corresponding to πunder the local Langlands correspondence, we show that the Asai γ-factor is the same as the Deligne-Langlands γ-factor of the Weil-Deligne representation of $\mathcal{W}_F$ obtained from σunder tensor induction. This is achieved by proving that Asai γ-factors are characterized by their local properties together with their role in global functional equations for $L$-functions. As an immediate application, we establish the stability property of γ-factors under twists by highly ramified characters.

math.NT↗