Notes on low degree L-data : WestminsterResearch

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Title	Notes on low degree L-data
Authors	Oliver, T.
Type	Conference paper
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\sim‐functions introduced by Andrew Booker in 2013 [3]. Associated to each L‐datum, one has a real number invariant known as the degree. Conjecturally the degree d is an integer, and if d\in \mathrm{N} then the L‐datum is that of a mathrm{G}\mathrm{L}_{n}(\mathrm{A}_{F}) ‐automorphic representation for n\in \mathrm{N} and a number field F (if F=\mathbb{Q} , then n=d This statement was shown to be true for 0\displaystyle \leq d<\frac{5}{3} by Booker in his pioneering paper [3], and in these notes we consider an extension of his methods to 0\leq d<2. This is simultaneously a generalisation of Booker’s result and the results and techniques of Kaczorowski‐Pereli in the Selberg class [10]. 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.
Year	2017
Conference	Analytic Number Theory and Related Areas
Published	2017
Journal	RIMS Kokyuroku
Journal citation	2014, pp. 48-58
Web address (URL) of conference proceedings	https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/2014.html

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Notes on low degree L-data

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