1549380232-Automorphic_Forms_and_Applications__Sarnak_

412 W.-C. W. LI, RAMANUJAN GRAPHS AND RAMANUJAN HYPERGRAPHS

two, there is an automorphic form 9b,<,w on D(K)\D(AK )/ D(K 00 ) D( Ot) Kb whose

associated £-function is

1

L(s,gb,<,w) = 1-w(-l)Noo-s

1

l-E(-l)N(t-1)-s

1

w#t-l,t-b,oo II 1-(-E(-l))degw,A E,W (lF Wl 2t-4+2b)Nw-s+Nwl-2s· t-b

In particular, 9b,e,w is an eigenfunction of the H ecke operator Tt at place t with

eigenvalue equal to A4(b-l)

-b-,w

The proofs of Theorems 1.6 - 1.12 are geometrical, using the theory of R-
adie cohomology. We investigate the actions of Gal(Ksep / K) on certain sheaves,
compute vanishing cycles and local factors at bad places. Then use the converse
theorem to conclude the existence of automorphic forms on GLz.

1.3. Ramanujan graphs and Sato-Tate conjecture

We end this lecture by discussing the distribut ion of the Eigenvalues of t he norm
graphs and Ter ras graphs. It is related to the Sato-Tate conjecture which we now
explain. Let E be an elliptic curve defined over a global field K. We may assume
that E is defined by a polynomial equation f ( x, y , z) = 0 over K. Then at almost
all places v of K, the equation f (x, y, z) = 0 mod v defines an elliptic curve Ev
over the residue field at v, in which case we say that E has a good reduction at v,
and call v a good place. At a good place v, t he zeta function of Ev has the form

1 - av Nv-s + Nvl-Zs

Z(Ev, s) = (1 - Nv-s)(l - Nvl-s)'

where av is an integer given by 1 + N v minus the number of points of Ev lying in

the residue field at v. The Hasse-Weil £-function attached to E is defined as

1 1

L(E, s) = II 1 - av Nv-s + N v (^1) - 2 8 II 1 - av Nv-s ·
v good v bad
Here the value of av at a bad place is eit her 0, 1 or -1, depending on the reduction
type. Hasse showed that at a good place v , t he absolute value of av is majorized
by 2'1NV. Therefore av/VfiV lies in the interval [-2, 2]. We are interested in the
distr ibution of these quantities as the good places v vary and E is fixed. Since the
distribution is not affected if finitely many av 's are cha nged, we shall be ambiguous
about the set of places v as long as the set contains almost all places of K. In
particular, we can ignore t he av from bad places.
When K = Q, the expected distribution is described as follows.
Sato-Tate Conjecture for elliptic curves Let Ebe an elliptic curve defined over
Q with av as above. Suppose that E does not have complex multiplications. Then
the set {av/ VNV} is uniformly distributed with respect to the Sato-Tate measure
μ sT defined by equation (1.3).
Because of the Taniya ma-Shimura conjecture established through the collective
efforts by Wiles [39], Taylor-Wiles [36], and Breuil-Conrad-Diamond-Taylor [3], we
now know that the Hasse-Weil £ -function of E in the conjecture satisfies L(E, s) =