1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE l. RAMANUJAN GRAPHS 413

L(f, s) for some cuspidal newform f of weight 2 for the congruence subgroup I'o(N)

with N being the conductor of E. In general, a normalized cuspidal newform f of

weight k for I' 0 (N) has the associated £-function given by

L(f, s) =II 1 _ a p-s

1

+ pk-l-2s II 1 _al p-s'

ptN p PIN p

and the Ramanujan conjecture (proved by Eichler [12], Shimura [35] and Igusa

[14] for k = 2, Deligne [7], [9] for k 2'. 2, and Deligne-Serre [10] for k = 1) implies

that

lapl :::; 2 p(k-1)/2

Hence one can ask a similar question.

forpf N.

Sato-Tate Conjecture for cuspidal newforms If fas above is not of CM type,

then the set { ap/p(k-l)/^2 } is uniformly distributed with respect to the Sato-Tate

measure μsT.

Up-to-date, there are many numerical examples suggesting the truth of the

Sato-Tate conjecture over Q, nonetheless it is not established for a single elliptic

curve or modular form.

The situation is quite different when the base field K is a function field, which

we now discuss. First of all, the corresponding Sato-Tate conjecture for elliptic

curves is known to hold. More precisely,

Theorem 1.13. [Yoshida [40]] The Sato-Tate conjecture holds for any elliptic

curve E defined over a function field K having nonconstant j -invariant.

Remark. The condition of nonconstant j-invariant in the case of a function field
is parallel to the condition of not having complex multiplications in the case of Q.
Applying results by Grothendieck on t he analytic behavior of L(E, s) and its
twists, by Deligne [8] on the behavior of the local constants occurring in these func-
tional equations, and the converse theorem for GL2 a la Weil [38], one concludes
that L(E, s) = L(f, s) for an automorphic cuspidal newform f of GL2 over K.
Drinfeld [11] has established the Ramanujan conjecture for cuspidal automorphic
newforms for GL 2 over a function field. Hence one can state the Sato-Tate conj ec-
ture for such forms as before. In particular, we know that the Sato-Tate conjecture
holds for those automorphic forms corresponding to elliptic curves with nonconstant
j -invariants. In the previous section, we exhibited explicit cuspidal automorphic
newforms with eigenvalues of the Hecke operators given by character sums which
are eigenvalues of (Ramanujan) norm graphs or Terras graphs. The theorem be-
low says t hat they provide examples of automorphic forms satisfying the Sato-Tate
conjecture, other than those arising from elliptic curves. It also implies that the
eigenvalues of the norm graphs and Terras graphs are uniformly distributed with
respect to the Sato-Tate measure μsT, in accordance with the observation made by
Terras numerically in [37].

Theorem 1.14. [Chai-Li [5], [6]] Let K be a function field, and let a be a non-

constant element in K. Then the automorphic forms fa in Theorem 1.6, fa,x in

Theorem 1. 9, and f a,<,w in Theorem 1.11 satisfy the Sato-Tate conjecture.