1549380232-Automorphic_Forms_and_Applications__Sarnak_

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

the sum of the invariants of H over the places in S is also zero. Denote by D' the

multiplicative group of H' modulo its center. We proceed to compare the infinite-

dimensional irreducible automorphic representations of D(AK) to those of D' (AK)

using Selberg trace formula, which is known and simple since both D(K)\D(AK)

and D'(K)\D'(AK) are compact. The two groups D(AK) and D'(AK) differ lo-

cally only at two places w = oo, oo' such that D(Kw) is the multiplicative group

of a division algebra mod center while D'(Kw) is isomorphic to PGLn(Kw)· At

such a place w, the local Jacquet-Langlands correspondence, proved by Badulescu

in [1] for local fields of positive characteristic, asserts the existence of a unique map

JL from the set of equivalence classes of admissible irreducible representations of

D(Kw) to the equivalence classes of essentially square integrable admissible irre-

ducible representations of D' (Kw) such that if 7r~ = J L(7rw), then the character of

7r w is equal to ( -1 r-^1 times the character of 7r~ at the elements which have the

same separable characteristic polynomials. Here Kw denotes the completion of K

at the place w.

By carefully comparing the geometric and the spectral sides of the two trace

formulas, one concludes that

Theorem 2.11. [Li [23]] Given an infinite-dimensional admissible irreducible au-

tomorphic representation 7r = ©~1rw of D(AK ), there exists an infinite-dimensional

admissible irreducible automorphic representation 7r^1 = ©~ 7r~ of D' (AK) such that

at places w = oo, oo', 7r~ = JL(7rw) and at places w-=/=-oo,oo', the representations

1l"w and 7r~ are isomorphic.

Now let the representation 7r of Theorem 2.11 correspond to an automorphic

form in A(D, JC). Then the component 7r 00 is the trivial representation of D 00 (K 00 ).

Its image under the map JL is a Steinberg representation of D'oo(K 00 ). By Theorem

(14.12) of the paper [18] by Laumon-Rapoport-Stuhler, the Ramanujan conjecture

holds at the places where 7r^1 is unramified. As both 7r v and 7r~ are unramified and

isomorphic, we conclude that XK. is Ramanujan.

This completes the proof of the following theorem, which is the goal of this

lecture.

Theorem 2.12. [Li [23]] For n 2 3 and a prime power q, there exists an infinite

family of finite (q + 1)-regular Ramanujan n-hypergraphs.

2.4. Open questions

We conclude our lectures with a few open questions.

(1) The known explicit constructions of infinite family of k-reqular Ramanujan
graphs, as discussed in the previous lecture, are for k = q + 1 with q a prime power.
For remaining values of k, it is unknown if such an infinite family exists, and, if it
does, how to construct it explicitly.

(2) Describe the universal cover for k-regular n-hypergraphs for k -=/=- q + l.
Similar questions for Ramanujan hypergraphs for k -=/=-q + l.