1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 1. RAMANUJAN GRAPHS 411

Note that >-x(lFw, a) is nothing but the eigenvalue A4a,xoNFw/F of a Terras graph

with base field lFw. Hence the eigenvalues of fa,x are eigenvalues of Terras graphs

of first type.

Parallel to Theorem 1.7, we have

Theorem 1.10. [Chai-Li [6]] Let K be JF(t) with odd characteristic, b E JF\{O, 1},

and x be a nontrivial character of JFX. There is an automorphic form gb,x on

D(K)\D(AK )/ D(K:)()) D(Ot) Kt whose associated L-function is

L(s ) - 1 1

,gb,x -l-x(-l)N(t-1)-s 1-Noo-s

1

II 1-x(-l)degw_). (JF i=l)Nw-s+Nwl-2s"

wi't-1,t-b,oo X W> t-b

For eigenvalues of Terras graphs of second type, we have the following parallel

result.

Theorem 1.11. [Chai-Li, [6]] Let K be a function field with the field of constants lF

of odd characteristic. Let a be a nonzero element in K, E be a nontrivial character of

JFx, and w be a regular character of the multiplicative group of a quadratic extension

JF', such that either i:^2 is nontrivial or w^2 is a regular character of JF' x. At each place

w of K, which is not a pole of a and where a "¢. ±2 mod w, define the character

sum

->-e,w(lFw, a) =

L E o NIFw/IF(TrIFw©IF'/IFw(u) +a mod w) W o N!Fw©IF'/IF'(u).

uEIFw©IF' ,NFw®F' /Fw (u)=l

Then there is an automorphic form fa,e,w of GL2(AK ), with central charcter T/a,e,w 1

whose associated L-function at good places is given by

1

w II good 1 - Ae ' w(lFw, a) Nw-s + T/a , ' € w(1rw) Nw-^2 s.

Moreover, if E has order two, then T/a,e,w is the unramified idele class character I· 1-^1.

Let
g a,t:,w (x) = (-i:(-l))deg det(x) f a,£,w (x) '
be the twist of fa,e,w by -E(-1). Then ga,e,w is an automorphic form of GL2(AK)
whose eigenvalues at good places of odd degree are eigenvalues of Terras graphs
of the second type, while those at places of even degree are eigenvalues of Terras
graphs of the first type.
Back to the quaternion group D from Morgenstern graphs, we have

Theorem 1.12. [Chai-Li, [6]] Let K = JF(t) be a rational function field with odd
characteristic. Given a nonzero b E lF not equal to 1, a quadratic character E of
JFX, and a regular character w of a quadratic extension of lF of order greater than