NOTES ON LOW DEGREE L-DATA (Analytic Number Theory and Related Areas)

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 Lsim-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 din mathrm{N} then the L-datum is that of a mathrm{G}mathrm{L}_{n}(mathrm{A}_{F})-automorphic representation for nin mathrm{N} and a number field F (if F=mathbb{Q}, then n=d This statement was shown to be true for 0displaystyle leq d<frac{5}{3} by Booker in his pioneering paper [3], and in these notes we consider an extension of his methods to 0leq 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 [11].