The Affine Frame in -adic Analysis
In this paper, we will introduce the concept of affine frame in wavelet analysis to the field of -adic number, hence provide new mathematic tools for application of -adic analysis.
The Akiyama-Tanigawa algorithm for Bernoulli numbers.
The Akiyama-Tanigawa algorithm for Carlitz's -Bernoulli numbers.
The Akiyama-Tanigawa transformation.
The algebraic groups leading to the Roth inequalities
We determine the algebraic groups which have a close relation to the Roth inequalities.
The algebraic independence of certain numbers to algebraic powers
The algebraic independence of Weierstrass functions and some related numbers
The analytic continuation and the order estimate of multiple Dirichlet series
Multiple Dirichlet series of several complex variables are considered. Using the Mellin-Barnes integral formula, we prove the analytic continuation and an upper bound estimate.
The analytic order of III for modular elliptic curves
In this note we extend the computations described in [4] by computing the analytic order of the Tate-Shafarevich group III for all the curves in each isogeny class ; in [4] we considered the strong Weil curve only. While no new methods are involved here, the results have some interesting features suggesting ways in which strong Weil curves may be distinguished from other curves in their isogeny class.
The Analytic Rank of a Family of Jacobians of Fermat Curves
We study the family of curves , where p is an odd prime and m is a pth power free integer. We prove some results about the distribution of root numbers of the L-functions of the hyperelliptic curves associated to the curves . As a corollary we conclude that the jacobians of the curves with even analytic rank and those with odd analytic rank are equally distributed.
The annihilation of divisor classes in abelian extensions of the rational function field
The appearence of tens of billion of integers x with ... 24, 13 (x) ... 24, 1 (x) in the vicinity of 10 12.
The application of a new mean value theorem to the fractional parts of polynomials
The Approximation of Solutions to Linear Homogeneous Differential Equations by Rational Functions.
The Arithmetic Genus of the Hubert Modular Variety and the Elliptic Fixed Points of the Hilbert Modular Group.
The arithmetic hyperbolic 3-manifold of smallest volume
The arithmetic of certain del Pezzo surfaces and K3 surfaces
We construct del Pezzo surfaces of degree violating the Hasse principle explained by the Brauer-Manin obstruction. Using these del Pezzo surfaces, we show that there are algebraic families of surfaces violating the Hasse principle explained by the Brauer-Manin obstruction. Various examples are given.
The arithmetic of curves defined by iteration
We show how the size of the Galois groups of iterates of a quadratic polynomial f can be parametrized by certain rational points on the curves Cₙ: y² = fⁿ(x) and their quadratic twists (here fⁿ denotes the nth iterate of f). To that end, we study the arithmetic of such curves over global and finite fields, translating key problems in the arithmetic of polynomial iteration into a geometric framework. This point of view has several dynamical applications. For instance, we establish a maximality theorem...
The Arithmetic of Elliptic Curves.
The arithmetic of Fermat curves.