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A basic problem of diophantine analysis is to investigate the asymptotics as T of (1.2) N(T, V)= {m V(Z): Ilmll T} where we denote by V(A), for any ring A, the set of A-points of V. Hence I1" is some Euclidean norm on R". The only general method available for such problems is the Hardy-Littlewood circle method, which however has certain limitations,(More)
Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, today the evidence is quite convincing. Firstly, there are the " function field " analogues, that is zeta functions of curves over finite fields and their generalizations. For(More)
1. Background. The aim of this note is to give an explicit construction of a rich family of k-regular (except for k ° =k) of the adjacency matrix satisfy Ikjl < 2 k~-l. graphs for which all the eigenvalues kj This bound is optimal (see Proposition 2.1). We call such graphs Ramanujan graphs. These graphs have many applications in the construction of explicit(More)
We introduce the notion of the automorphic dual of a matrix algebraic group defined over Q. This is the part of the unitary dual that corresponds to arithmetic spectrum. Basic functorial properties of this set are derived and used both to deduce arithmetic vanishing theorems of " Ramanujan " type as well as to give a new construction of automorphic forms.(More)