Networks via Hilbert Modular Forms

Ramanujan graphs, introduced by Lubotzky, Phillips and Sarnak, allow the design of efficient communication networks. In joint work with B. Jordan we gave a higher-dimensional generalization. Here we explain how one could use this generalization to construct efficient communication networks which allow for a number of verification protocols and for the distribution of information along several channels. The efficiency of our network hinges on the Ramanujan-Petersson conjecture for certain Hilbert modular forms. We obtain this conjecture in sufficient generality to apply to some particularly appealing constructions, which were not accessible before. The concept of a Ramanujan graph was introduced and studied by Lubotzky, Phillips and Sarnak (LPS) in [16]. These are r-regular graphs for which the nontrivial eigenvalues λ = ±k of the adjacency matrix satisfy the bounds |λ| ≤ 2 √ r − 1. In many aspects these bounds are optimal and natural. For example, the adjacency matrix is the com-binatorial analog of the Laplace operator, and the bounds parallel the (conjectured) Selberg bounds for the Laplacian on Riemann surfaces. The main result of LPS was an explicit construction of p + 1-regular such graphs, p ≡ 1 mod 4 a prime, through the arithmetic of quater-nion algebras over the rational numbers. From the point of view of Communication Network Theory, the arithmetic examples are particularly interesting: all Ramanujan graphs are super-expanders; but in addition the examples have many other useful properties, for example very good expansion constants and large girth. Thus they can be used to design efficient communication networks. The Ramanujan property for the LPS examples hinges on the truth of the Ramanujan-Petersson conjecture for an appropriate space of modular forms of weight 2 over Q. Let f be a weight 2 holomorphic cuspidal Hecke eigenform on a congruence subgroup. The conjecture is that for any prime p not dividing the level the Hecke eigenvalue a p satisfies |a p | ≤ 2 √ p. Eichler and Shimura reduced the conjecture to Weil's