Given a smooth subvariety of dimension greater than $(2/3)(r-1)$ in $\mathbb {P}\sp r$, we show that the double locus (upstairs) of its generic projection to $\mathbb {P}\sp {r-1}$ is irreducible. This implies a version of Zak's linear normality theorem.

We study the asymptotic limit of solutions of the Ginzburg-Landau equations in two dimensions with or without magnetic field. We first study the Ginzburg-Landau system with magnetic field describing a superconductor in an applied magnetic field, in the "London limit" of a Ginzburg-Landau parameter $\kappa$ tending to $\infty$. We examine the asymptotic behavior of the "vorticity measures" associated to the vortices of the solution, and we prove that passing to the limit in the equations (via the "stress-energy tensor") yields a criticality condition on the limiting measures. This condition allows us to describe the possible locations and densities of the vortices. We establish analogous results for the Ginzburg-Landau equation without magnetic field.

This paper focuses on the connection between the Brauer group and the $0$-cycles of an algebraic variety. We give an alternative construction of the second $l$-adic Abel-Jacobi map for such cycles, linked to the algebraic geometry of Severi-Brauer varieties on $X$. This allows us then to relate this Abel-Jacobi map to the standard pairing between $0$-cycles and Brauer groups (see [M], [L]), completing results from [M] in this direction. Second, for surfaces, it allows us to present this map according to the more geometrical approach devised by M. Green in the framework of (arithmetic) mixed Hodge structures (see [G]).

Needless to say, this paper owes much to the work of U. Jannsen and, especially, to his recently published older letter [J4] to B. Gross.

We prove that Fredholm determinants of the form $\det(1-K\sb s)$, where $K\sb s$ is the restriction of either the discrete Bessel kernel or the discrete $\sb 2F\sb 1$-kernel to $\{s, s + 1,\ldots\}$, can be expressed, respectively, through solutions of discrete Painlevé II (dPII) and Painlevé V (dPV) equations.

These Fredholm determinants can also be viewed as distribution functions of the first part of the random partitions distributed according to a Poissonized Plancherel measure and a $z$-measure, or as normalized Toeplitz determinants with symbols $e\sp {\eta(\zeta+\zeta\sp {-1})}$ and $(1 +\sqrt {\xi}\zeta)\sp z(1 +\sqrt {\xi}/\zeta)\sp {z\sp \prime}$.

The proofs are based on a general formalism involving discrete integrable operators and discrete Riemann-Hilbert problems. A continuous version of the formalism has been worked out in [BD].

Let $K$ be a function field of odd characteristic, and let $\pi$ (resp., $\eta$) be a cuspidal automorphic representation of ${\rm GL}\sb 2(\mathbb {A}\sb K)$ (resp., ${\rm GL}\sb 1(\mathbb {A}\sb K)$). Then we show that a weighted sum of the twists of $L(s,\pi)$ by quadratic characters $\chi\sb D,\sum \sb DL(s,\pi\otimes \sp \chi\sb D)a\sb 0(s,\pi,D)\eta(D)|D|\sp {-w}$, is a rational function and has a finite, nonabelian group of functional equations. A similar construction in the noncuspidal cases gives a rational function of three variables. We specify the possible denominators and the degrees of the numerators of these rational functions. By rewriting this object as a multiple Dirichlet series, we also give a new description of the weight functions $a\sb 0(s,\pi,D)$ originally considered by D. Bump, S. Friedberg and J. Hoffstein.

We show that in the semiclassical limit the eigenfunctions of quantized ergodic symplectic toral automorphisms cannot concentrate in measure on closed orbits of the dynamics. More generally, we show that the mass of the pure point component of the limit measure must be smaller than two thirds of the total mass. The proofs use only the algebraic (i.e., not the number-theoretic) properties of the toral automorphisms together with the exponential instability of the dynamics and therefore work in all dimensions.