LECTURE 2. RAMANUJAN HYPERGRAPHS 4212.3. An explicit construction of Ramanujan n-hypergraphs
The construction here is parallel to the construction of Ramanujan graphs based on
quaternion groups. Fix n 2: 3 and a prime power q. Let K be a function field with
q elements in its field of constants. Fix a place v of K of degree l. Fix another place
oo of K different from v. Let H be a division algebra defined over K, with center
K and of dimension n^2 over K, which is unramified at v and totally ramified at
oo (that is, H(K 00 ) is a genuine division algebra). Denote by D the multiplicative
group of H divided by its center. Let
XK = D(K)\D(AK )/ D(K 00 ) D(Ov) K,
where K is a compact open subgroup of ITw,,ooo,v D(Ow)· As before, by strong
approximation theory, we may choose double coset representatives locally at v so
that we get the second expression of XK:where
rK = D(K) nK
is a discrete subgroup of D(Kv)· By construction, H is unramified at v so that
D(Kv) = PGLn(Kv) and D(Kv)/D(Ov) is the building Bn,Kv· Hence we obtain
the hypergraph structure on XK:
XK = fK\Bn,Kv·If ordv(detfK) ~ nZ, we may replace fK by a congruence subgroup f of finite
index so that ordv(detr) ~ nZ. Hence we may assumer= r K to begin with so
that XK is a finite (q + 1)-regular n-hypergraph.
As before, the space A(D, K) of functions on vertices of XK are certain auto-
morphic forms of D(AK ). Since the 1-skeleton of XK is an n-partite graph, A(D, K)
contains constant functions as well as their twists by powers of (n. They are eigen-
functions of Ai, 1 ~ i ~ n - 1, with eigenvalues (;;" qn,i· The automorphic forms
in the orthogonal complement of these functions correspond to infinite-dimensional
automorphic irreducible representations u of D(AK ). The Jacquet-Langlands corre-
spondence predicts that these representations "should" correspond to automorphic
irreducible representations of GLn(AK). Because the representations u of D(AK)
are trivial at oo (where H ramifies) by construction, the corresponding representa-
tions of GLn(AK), if exist, are cuspidal. Then, by the work of Lafforgue [19], the
Ramanujan conjecture holds for these cuspidal representations of GLn(AK), and
hence we can conclude that the graph XK is Ramanujan.
To-date, the Jacquet-Langlands correspondence over a function field is proved
for n = 3 by Jacquet, Piatetskii-Shapiro, and Shalika [16] using converse theorem,
and for prime n by Bernstein and Kazhdan. It is not settled for composite n. To
circumvent this problem, we adopt a suggestion of Clozel by considering two division
algebras and using the result by Laumon, Rapoport, and Stuhler [18] instead.
We may assume that the division algebra H above is totally ramified at an-
other place oo' with invariant opposite to that at oo, and H is also ramified at a
nonempty set S of even number of places other than v, oo, oo'. Let H' be another
division algebra over K of degree n^2 and center K which ramifies exactly at the
places in S such that at each place other than oo and oo', D' and D are locally
isomorphic. Such H' exists since H has opposite invariants at oo and oo' and hence