Assuming the grand Riemann hypothesis, we investigate the distribution of the lowlying zeros of the $L$-functions $L(s,\psi)$, where $\psi$ is a character of the ideal class group of the imaginary quadratic field $\mathbb {Q}(\sqrt{-D}) (D\text {squarefree},D>3,D\equiv 3(\mod 4))$. We prove that, in the vicinity of the central point $s = 1/2$, the average distribution of these zeros (for $D\longrightarrow \infty$) is governed by the symplectic distribution. By averaging over $D$, we go beyond the natural bound of the support of the Fourier transform of the test function. This problem is naturally linked with the question of counting primes $p$ of the form $4p = m\sp 2+Dn\sp 2$, and sieve techniques are applied.

The purpose of the paper is to extend and refine earlier results of the author on nonvanishing of the $L$-functions associated to modular forms in the anticyclotomic tower of conductor $p\sp \infty$ over an imaginary quadratic field. While the author's previous work proved that such $L$-functions are generically nonzero at the center of the critical strip, provided that the sign in the functional equation is $+1$, the present work includes the case where the sign is $-1$. In that case, it is shown that the derivatives of the $L$-functions are generically nonzero at the center. It is also shown that when the sign is $+1$, the algebraic part of the central critical value is nonzero modulo $\ell$ for certain $\ell$. Applications are given to the mu-invariant of the $p$-adic $L$-functions of M. Bertolini and H. Darmon. The main ingredients in the proof are a theorem of M. Ratner, as in the author's previous work, and a new "Jochnowitz congruence," in the spirit of Bertolini and Darmon.

Lagrangian submanifolds of symplectic manifolds are very central objects in classical mechanics and microlocal analysis. These manifolds are frequently singular (integrable systems, bifurcations, reduction). There have been many works on singular Lagrangian manifolds initiated by V. Arnold, A. Givental, and others. The goal of our paper is to extend the classical and semiclassical normal forms of completely integrable systems near nondegenerate (Morse-Bott) singularities to more singular systems. It turns out that there is a nicely working way to do that, leading to normal forms and universal unfoldings. We obtain in this way natural ansatzes extending the Wentzel-Kramers-Brillouin(WKB)-Maslov ansatz. We give more details on the simplest non-Morse example, the cusp, which corresponds to a saddle-node bifurcation.

We establish a connection between a specialization of the nonsymmetric Macdonald polynomials and the Demazure characters of the corresponding affine Kac-Moody algebra. This allows us to obtain a representation-theoretical interpretation of the coefficients of the expansion of the specialized symmetric Macdonald polynomials in the basis formed by the irreducible characters of the associated finite Lie algebra.

The notions of orbifold elliptic genus and elliptic genus of singular varieties are introduced, and the relation between them is studied. The elliptic genus of singular varieties is given in terms of a resolution of singularities and extends the elliptic genus of Calabi-Yau hypersurfaces in Fano Gorenstein toric varieties introduced earlier. The orbifold elliptic genus is given in terms of the fixed-point sets of the action. We show that the generating function for the orbifold elliptic $\sum {\rm Ell\sp {orb}}(X\sp n,\Sigma\sb n)p\sp n$ for symmetric groups $\Sigma\sb n$ acting on $n$-fold products coincides with the one proposed by R. Dijkgraaf, G. Moore, E. Verlinde, and H. Verlinde. The two notions of elliptic genera are conjectured to coincide.

In this paper we consider the ergodic properties of linear actions on the plane of (2×2)-matrices with entries in the complex numbers, quaternions, or Clifford numbers.