  Motivated by problems of mathematical physics (quantum chaos) questions of
equidistribution of eigenfunctions of the Laplace operator on a Riemannian
manifold have been studied by several authors. We consider here, in analogy
with arithmetic hyperbolic surfaces, orthonormal bases of eigenfunctions of the
Laplace operator on the two dimensional unit sphere which are also
eigenfunctions of an algebra of Hecke operators which act on these spherical
harmonics. We formulate an analogue of the equidistribution of mass conjecture
for these eigenfunctions as well as of the conjecture that their moments tend
to moments of the Gaussian as the eigenvalue increases. For such orthonormal
bases we show that these conjectures are related to the analytic properties of
degree eight arithmetic L-functions associated to triples of eigenfunctions.
Moreover we establish the conjecture for the third moments and give a
conditional (on standard analytic conjectures about these arithmetic
L-functions) proof of the equdistribution of mass conjecture.
