402 W.-C. W. LI, RAMANUJAN GRAPHS AND RAMANUJAN HYPERGRAPHS
also eigenvalues of certain Ramanujan graphs. We end this lecture by showing that
these newly discovered automorphic forms satisfy the Sato-Tate conjecture, provid-
ing new classes of evidence to this celebrated conjecture since these automorphic
forms do not arise from elliptic curves. The rich interplay between combinatorics
and number theory is highly interesting.
Extending what we did for graphs to hypergraphs is the focus of the second
lecture. The results contained in this lecture are new. We begin by introducing
combinatorial assumptions and operators. Briefly speaking, a finite k-regular graph
has its universal cover an infinite k-regular tree, while a finite (q + 1)-regular n-
hypergraph has its universal cover the Bruhat-Tits building on GLn(F), where F
is a nonarchimedean local field with q elements in its residue field. Naturally q is
a prime power here. There are n - 1 adjacency operators on an n-hypergraph. We
study the behavior of the eigenvalues of each adjacency operator acting on functions
on (vertices of) a family of regular hypergraphs whose radii tend to infinity along
with the size, and obtain a hypergraph analogue of the Alon-Boppana theorem for
regular graphs. This justifies our definition of Ramanujan hypergraphs. Then we
give explicit constructions of Ramanujan hypergraphs based on the multipicative
group of division algebras. The Ramanujan conjecture for GLn over function fields
could be used for primes n for which the J acquet-Langlands map is known to exist.
For arbitrary n, we adopt a suggestion of Clozel to use t he result by Laumon-
Rapoport-Stuhler and compare trace formulas for different division algebras.