410 W.-C. W. LI, RAMANUJAN GRAPHS AND RAMANUJAN HYPERGRAPHS
Let S be a K 8 -double coset of GL 2 (1F'q) with cardinality greater than that of K.,. It
can be shown that S has coset representatives ( ~ ~) , where ( x, y) runs through
all IF'q-rational points of an ellipse x^2 = o (ay + (y - 1)^2 ) for some a E IF'q. Further,
there are q- 2 such double cosets, parametrized by the elements a in IF'q - {0,4},
which we denote by Sac5.
The Terras graph Xa is the Cayley graph Cay(GL 2 (1F'q)/ K.,, Sac5/ K.,). Different
choices of o result in isomorphic graphs. It is a (q+l)-regular graph whose nontrivial
eigenvalues, arising from representations of GL 2 (1F'q) with a K.,-fixed vector, are
character sums of the following two types. The first type is
x(x),
associated to nontrivial characters x of IF':. The second type is associated to regular
multiplicative characters w of IF' q ( vf6) x:
Aa,w = E(a - 2 - Tr(z)) w(z).
zEIFq ( vfo),N(z)=l
Here Tr and N are the trace and norm maps from IF' q ( vf6) to IF' q, and E( x) is 1, 0, -1
according to x E (IF':)^2 , x = 0, or x E IF' q \(IF' q)^2. Using the Riemann hypothesis
for curves, one can show that l>-a,xl ::; 2 .fa and l>-a,wl ::; 2 .fa; see [22], p. 204.
(This was first proved by H. Stark, R. Evans [13] and N. Katz [17].) Hence Terras
graphs are Ramanujan graphs.
Similar to the norm graphs, Terras graphs are also quotients of Ramanujan
graphs constructed from a quaternion group.
Proposition 1.8. [Li, [21]] For b E IF'q, b -/:-0, 1, the Terras graph X 4 (b-l)/b is a
quotient of the Morgenstern graph XKb with Kb= Kt-b ITw#t,oo,t-b D(Ow), where
Kt-b is the subgroup of D(Ot-b) consisting of elements congruent to the identity
element mod t - b.
Therefore, Aa,x, Aa,w are also eigenvalues of the Hecke operator Tt at t on
automorphic forms on D(Ax). We have results similar to Theorems 1.6 and 1.7.
Theorem 1.9. [Chai-Li [6]] Let K be a function field with the field of constants IF'
a finite field. Given a E Kx, and a nontrivial character x of IF'x, at places w which
is not a pole of a and where a ¢. 0, 1 mod w, let
Then there is an automorphic form fa,x of GL2(Ax) which is an eigenfunction of
the Hecke operator Tw with eigenvalue >-x(IF'w, a) for all good places w as above.
The L-function off a,x at good places is given by
1
w IJ good 1 - A X (IF'w, a) Nw-s + Nwl-2s ·