The present paper is devoted to studying a class of smoothly (C^∞) finitely determined vector fields on ℝ³. Given any such generic local system of the form $\dot{x}$=Ax+⋯, where A is a 3×3 matrix, we find the minimal possible number i(A) such that the vector field is i(A)-jet determined, and we find the number μ(A) of moduli in the C^∞ classification. We also give a list of the simplest normal forms, that is, polynomials of degree i(A) containing exactly μ(A) parameters.

A recent conjecture of Kottwitz predicts that certain central elements in Iwahori-Hecke algebras play an important role in the bad reduction of Shimura varieties with Iwahori level structure. Namely, the function trace of Frobenius on nearby cycles is conjecturally expressible in terms of the so-called Bernstein function z_μ corresponding to the cocharacter μ coming from the Shimura data. In this paper we prove an explicit formula for z_μ in terms of the standard basis for the Iwahori-Hecke algebra, for any minuscule cocharacter μ of any p-adic group G. We then use this formula to prove Kottwitz's conjecture for a particular Shimura variety, attached to G=GU(1,d−1) and μ=(1,0^d−1), known as the "Drinfeld case."

Consider a volume preserving vector field defined in some compact domain of 3-space and tangent to its boundary. A long piece of orbit can be made into a knot by connecting its endpoints by some arc whose length is less than the diameter of the domain. In this paper, we study the behaviour of the signatures of these knots as the lengths of the pieces of orbits go to infinity. We relate this "asymptotic signature" to the "asymptotic Hopf invariant" that has been studied by Arnold.

In this paper, we prove an isogeny criterion for abelian varieties that involves conditions on the formal groups of the varieties (see Theorem 1.1). In the particular case of abelian varieties over ℚ with real multiplication, we easily deduce from our criterion a new proof of the Tate conjecture which is independent of G. Faltings's work [11], as well as a bound for the minimal degree of an isogeny between two isogenous abelian varieties, as in the paper of D. Masser and G. Wüstholz [17]. To this end, we use C. Deninger and E. Nart's result giving the link between the L-functions and the formal groups of such varieties (see [9]). Our method generalizes D. and G. Chudnovsky's transcendental proof of the isogeny theorem for elliptic curves over ℚ [6, Prop. 2.3] to the case of abelian varieties, with a systematic use of the Arakelov formalism of J.-B. Bost (see [1]).

The cohomology of Drinfeld's p-adic symmetric domain was computed by P. Schneider and U. Stuhler in 1991. Here we propose a more explicit and combinatorial approach based on a notion of residue of a closed form along simplices in the Bruhat-Tits building. We identify the cohomology with a certain space of harmonic cochains on the building. We also answer a few questions left open in the original approach.

In this note we investigate the relationship between the conformal boundary at infinity of a hyperbolic 3-manifold and the boundary of its convex core. In particular, we prove that the length of a curve in the conformal boundary gives an upper bound on the length of the corresponding curve in the boundary of the convex core.