Communication Networks And Hilbert Modular Forms

Ramanujan graphs, defined and constructed 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.