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Luis Alberto Lomelí

Publications and source records attributed to Luis Alberto Lomelí.

4 recordsLinked to original sources
arXiv2018

Langlands Program and Ramanujan Conjecture: a survey

We present topics in the Langlands Program to graduate students and a wider mathematically mature audience. We study both global and local aspects in characteristic zero as well as characteristic $p$. We look at modern approaches to the generalized Ramanujan conjecture, which is an open problem over number fields, and present known cases over function fields. In particular, we study automorphic $L$-functions via the Langlands-Shahidi method. Hopefully, our approach to the Langlands Program can help guide the interested reader into an exciting field of mathematical research.

math.NT↗
arXiv2013

On automorphic L-functions in positive characteristic

We give a proof of the existence of Asai, exterior square, and symmetric square local $L$-functions, $γ$-factors and root numbers in characteristic $p$, including the case of $p = 2$. Our study is made possible by developing the Langlands-Shahidi method over a global function field in the case of a Siegel Levi subgroup of a split classical group or a quasi-split unitary group. The resulting automorphic $L$-functions are shown to satisfy a rationality property and a functional equation. A uniqueness result of the author and G. Henniart allows us to show that the definitions provided in this article are in accordance with the local Langlands conjecture for ${\rm GL}_n$. Furthermore, in order to be self contained, we include a treatise of $L$-functions arising from maximal Levi subgroups of general linear groups.

math.NT↗
arXiv2012

The LS method for the classical groups in positive characteristic and the Riemann Hypothesis

We provide a definition for an extended system of $γ$-factors for products of generic representations $τ$ and $π$ of split classical groups or general linear groups over a non-archimedean local field of characteristic $p$. We prove that our $γ$-factors satisfy a list of axioms (under the assumption $p \neq 2$ when both groups are classical groups) and show their uniqueness (in general). This allows us to define extended local $L$-functions and root numbers. We then obtain automorphic $L$-functions $L(s,τ\times π)$, where $τ$ and $π$ are globally generic cuspidal automorphic representations of split classical groups or general linear groups over a global function field. In addition to rationality and the functional equation, we prove that our automorphic $L$-functions satisfy the Riemann Hypothesis.

math.NT↗