Notes on Low Degree L-Data

Thomas Oliver

Research output: Chapter in Book/Report/Conference proceedingConference Contribution (Conference Proceeding)

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Abstract

These notes are an extended version of a talk given by the author at the conference “Analytic Number Theory and Related Areas”, held at Research Institute for Mathematical Sciences, Kyoto University in November 2015. We are interested in “L-data”, an axiomatic framework for L-functions introduced by Andrew Booker in 2013 [Boo15]. Associated to each L-datum, one has a real number invariant known as the degree. Conjecturally the degree d is an integer, and if dN then the L-datum is associated to a GLn(AF )- automorphic representation for nN and a number field F (if F = Q and there is no scaling, then n = d.). This statement was shown to be true for 0 ≤ d < 5/3 by Booker in his pioneering paper [Boo15], and in these notes we consider an extension of his methods to 0 ≤ d < 2. This is simultaneously a generalisation of Booker’s result and the results and techniques of Kaczorowski–Perelli in the Selberg class [KP11]. Furthermore, we consider applications to zeros of automorphic L-functions. In these notes we review Booker’s results and announce new ones to appear elsewhere shortly [Oli16]. 
Original languageEnglish
Title of host publicationAnalytic Number Theory and Related Areas
PublisherRIMS Kôkyûroku
Pages48-59
Number of pages11
Volume2014
Publication statusPublished - Jan 2017

Publication series

NameRIMS Kôkyûroku Bessatsu
PublisherRIMS
ISSN (Print)1880-2818
ISSN (Electronic)1881-6193

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    Oliver, T. (2017). Notes on Low Degree L-Data. In Analytic Number Theory and Related Areas (Vol. 2014, pp. 48-59). [5] (RIMS Kôkyûroku Bessatsu). RIMS Kôkyûroku. http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/2014.html