1549380232-Automorphic_Forms_and_Applications__Sarnak_

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 ·