THE IHARA–SELBERG ZETA FUNCTION FOR PGL₃ AND HECKE OPERATORS

Abstract

A weak version of the Ihara formula is proved for zeta functions attached to quotients of the Bruhat-Tits building of PGL 3 . This formula expresses the zeta function in terms of Hecke-Operators. It is the first step towards an arithmetical interpretation of the combinatorially defined zeta function.

The Zeta Function of a Hypergraph

“…We recommend the series of articles by Harold Stark and Audrey Terras to the reader interested in current theory on Iharatype zeta functions on graphs [21,22,23]. Recently, there have also been a number of generalizations of the zeta functions to digraphs as well as buildings [17,18,7]. To define our zeta function, we need the appropriate concept of a "prime cycle."…”

A prime geodesic theorem for SL4

“…We recall that for λ ∈ a * we write λ for the norm given by the form b in (12). It then follows from Proposition 2.5 that there exist m 1 ∈ N and C > 0 such that for every π ∈Ĝ and every λ ∈ a * we have |m n,λ | ≤ C(1 + λ ) m1 .…”

“…In [9,12,11,13], the prime geodesic theorem is applied to derive class number asymptotics. The last open case for these is the case of complex quartic fields for which the PGT required is the one of SL 4 (R).…”

Zeta functions of group based graphs and complexes

“…On the other hand, despite several attempts [6,7,8], the complex zeta function with similar properties was not defined until the work of Kang-Li [23]. Like the graph zeta function, the complex zeta function counts a certain type of geodesic cycles of given lengths in a complex.…”