Let A be the adeles of the function eld Fq (T). If E(g; s) is the Eisenstein series associated with the spherical vector in the principal series representation of the double cover of GL2(A), then the Fourier coeecients (Whittaker functions) of E(g; s) have an Euler product and satisfy a `Riemann Hypothesis' condition. Namely, the zeros occur only on the line <(s) = 0. These Whittaker functions ...

We investigate the rigidity and asymptotic properties of quantum SU(2) representations of mapping class groups. In particular we prove that the hyperelliptic mapping class groups do not have Kazhdan’s property (T). On the other hand, the quantum SU(2) representations of the mapping class group of the torus do not have almost invariant vectors, in fact they converge to the metaplectic representa...

In the present paper we find explicit formulas for the degrees of Heegner divisors on arithmetic quotients of the orthogonal group O(2, p) and for the integrals of certain automorphic Green’s functions associated with Heegner divisors. The latter quantities are important in the study of the arithmetic degrees of Heegner divisors in the context of Arakelov geometry. In particular, we obtain a di...

We’ll now turn to a topic which is a precise analog of the previous discussion of the Clifford algebra and spinor representations. By replacing the symmetric two-form (the inner product) in the earlier discussion by an antisymmetric two-form, we get a new algebra, the Heisenberg algebra. The group of automorphism of this algebra is now a symplectic group, and we again get a projective represent...