## Automorphic forms and Ramanujan hypergraphs

#### Wen-Ching LiPennsylvania State UniversityMathematics

Ramanujan graphs are known to have
and computer science. These are regular graphs
whose nontrivial eigenvalues are small in absolute
value. The theory of automorphic forms played an
essential role in the explicit construction of
Ramanujan graphs.

In this talk we extend the above to regular $n$-
hypergraphs. The first key result is an analogue
of the Alon-Boppana theorem which describes the
distribution of eigenvalues of the $n-1$ adjacency
operators acting on a family of $n$-hypergraphs.
This leads to the definition of Ramanujan hypergraphs.
The second key result is the explicit construction
of Ramanujan $n$-hypergraphs using the theory of
automorphic forms. The method is somewhat different
from the case of graphs: instead of using the
Jacquet-Langlands correspondence, which is not
established for arbitrary $n$, we appeal to the
result by Laumon-Rapoport-Stuhler.

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