We discuss the metric theory of simultaneous diophantine approximations in the non-archimedean case. First, we show a Gallagher type 0-1 law. Then by using this theorem, we prove a Duffin-Schaeffer type theorem.
Let K be a complete, algebraically closed nonarchimedean valued field, and let φ(z) ∈ K(z) have degree d ≥ 2. We study how the resultant of φ varies under changes of coordinates. For γ ∈ GL₂(K), we show that the map factors through a function on the Berkovich projective line, which is piecewise affine and convex up. The minimal resultant is achieved either at a single point in , or on a segment, and the minimal resultant locus is contained in the tree in spanned by the fixed points and poles...
Let be a curve over a field with a rational point . We define a canonical cycle . Suppose that is a number field and that has semi-stable reduction over the integers of with fiber components non-singular. We construct a regular model of and show that the height pairing is well defined where and are correspondences. The paper ends with a brief discussion of heights and -functions in the case that is a modular curve.
Let denote a finite index subgroup of the modular group and let denote a finite-dimensional complex representation of Let denote the collection of holomorphic vector-valued modular forms for and let denote the collection of modular forms on . Then is a -graded -module. It has been proven that may not be projective as a -module. We prove that is Cohen-Macaulay as a -module. We also explain how to apply this result to prove that if is a polynomial ring, then is a free...
The coefficients of the moments of the monotone Poisson law are shown to be a type of Stirling number of the first kind; certain combinatorial identities relating to these numbers are proved and a new derivation of the Cauchy transform of this law is given. An investigation is begun into the classical Azéma-type martingale which corresponds to the compensated monotone Poisson process; it is shown to have the chaotic-representation property and its sample paths are described.