Let E be a CM-elliptic curve over ℚ with good ordinary reduction at a prime p≥5. The purpose of this paper is to construct the p-adic elliptic polylogarithm of E, following the method of A. Beĭlinson and A. Levin. Our main result is that the specializations of this object at torsion points give the special values of the one-variable p-adic L-function of the Grössencharakter associated to E.

Let G be a split adjoint group defined over F_q, let F_q(t), and let A be the adèles of F. We describe the local constituents at two points of automorphic representations of G in the discrete part of L²(G(F)\G(A)) which have vectors invariant under Iwahori subgroups at these two points and are unramified at all other points.

We prove that the existence of an accretive system in the sense of M. Christ is equivalent to the boundedness of a Calderón-Zygmund operator on L²(μ)$. We do not assume any kind of doubling condition on the measure $\mu$, so we are in the nonhomogeneous situation. Another interesting difference from the theorem of Christ is that we allow the operator to send the functions of our accretive system into the space bounded mean oscillation (BMO) rather than L\sp ^∞. Thus we answer positively a question of Christ as to whether the L\sp ^∞-assumption can be replaced by a BMO assumption.

We believe that nonhomogeneous analysis is useful in many questions at the junction of analysis and geometry. In fact, it allows one to get rid of all superfluous regularity conditions for rectifiable sets. The nonhomogeneous accretive system theorem represents a flexible tool for dealing with Calderón-Zygmund operators with respect to very bad measures.

We examine the effects of certain hypothetical configurations of zeros of Dirichlet L-functions lying off the critical line on the distribution of primes in arithmetic progressions.

Let $\mathcal{O}$ be a nilpotent orbit for a semisimple Lie group which appears as the leading orbit in the wave-front set of an $A_{\mathfrak{q}}(\lambda )$ -module. We establish a limit formula for the computation of the canonical measure on $\mathcal{O}$ through differentiation of the canonical measures on elliptic orbits.

To what extent does the eigenvalue spectrum of the Laplace-Beltrami operator on a compact Riemannian manifold determine the geometry of the manifold? We present a method for constructing isospectral manifolds with different local geometry, generalizing an earlier technique. Examples include continuous families of isospectral negatively curved manifolds with boundary as well as various pairs of isospectral manifolds. The latter illustrate that the spectrum does not determine whether a manifold with boundary has negative curvature, whether it has constant Ricci curvature, and whether it has parallel curvature tensor, and the spectrum does not determine whether a closed manifold has constant scalar curvature.

In this note we prove local L^p-regularity for the highest-order derivatives of an elliptic system of arbitrary order in nondivergence form where the coefficients of the principal part are taken in the space of Sarason vanishing mean oscillation (VMO). Lower-order coefficients and the known term belong to suitable Lebesgue spaces. As a consequence, we obtain Hölder regularity results.