TY - GEN

T1 - Notes on Low Degree L-Data

AU - Oliver, Thomas

PY - 2017/1

Y1 - 2017/1

N2 - 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 d ∈ N then the L-datum is associated to a GLn(AF )-
automorphic representation for n ∈ N 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].

AB - 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 d ∈ N then the L-datum is associated to a GLn(AF )-
automorphic representation for n ∈ N 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].

M3 - Conference Contribution (Conference Proceeding)

VL - 2014

T3 - RIMS Kôkyûroku Bessatsu

SP - 48

EP - 59

BT - Analytic Number Theory and Related Areas

PB - RIMS Kôkyûroku

ER -