Groupes d'unités elliptiques
Grundeinheiten für einige unendliche Klassen reiner biquadratischer Zahlkörper mit einer Anwendung auf die diophanitsche Gleichung x4 - ay4 = ... c (c = 1, 2, 4 oder 8).
Hasse’s problem for monogenic fields
In this article we shall give a survey of Hasse’s problem for integral power bases of algebraic number fields during the last half of century. Specifically, we developed this problem for the abelian number fields and we shall show several substantial examples for our main theorem [7] [9], which will indicate the actual method to generalize for the forthcoming theme on Hasse’s problem [15].
Height Functions for Groups of S-units of Number Fields and Reductions Modulo Prime Ideals
A result on the orders of the reductions of an element of the group of S-units of a number field is obtained by investigating three height functions for groups of S-units of number fields which are analogous to local, global and canonical height functions for elliptic curves.
Heights, regulators and Schinzel's determinant inequality
We prove inequalities that compare the size of an S-regulator with a product of heights of multiplicatively independent S-units. Our upper bound for the S-regulator follows from a general upper bound for the determinant of a real matrix proved by Schinzel. The lower bound for the S-regulator follows from Minkowski's theorem on successive minima and a volume formula proved by Meyer and Pajor. We establish similar upper bounds for the relative regulator of an extension l/k of number fields.
Holomorphe Abbildungen zwischen quasiprojektiven Räumen.
Ideal class groups of cyclotomic number fields I
Ideal Criteria for both Ideal Criteria for both X2-dy2 = M1 And X2-dy2 = M2 to have Primitive Solutions for any Integers M1, M2 Prime to D > 0
This article provides necessary and sufficient conditions for both of the Diophantine equations X^2 − DY^2 = m1 and x^2 − Dy^2 = m2 to have primitive solutions when m1 , m2 ∈ Z, and D ∈ N is not a perfect square. This is given in terms of the ideal theory of the underlying real quadratic order Z[√D].
Independent systems of units in certain algebraic number fields.
Index for subgroups of the group of units in number fields
We define a sequence of rational integers for each finite index subgroup E of the group of units in some finite Galois number fields K in which prime p ramifies. For two subgroups E’ ⊂ E of finite index in the group of units of K we prove the formula . This is a generalization of results of P. Dénes [3], [4] and F. Kurihara [5].
Integer Valued polynomials over a number field.
Integral Points on Certain Elliptic Curves
Interpolation of hypergeometric ratios in a global field of positive characteristic
For each global field of positive characteristic we exhibit many examples of two-variable algebraic functions possessing properties consistent with a conjectural refinement of the Stark conjecture in the function field case recently proposed by the author. All the examples are Coleman units. We obtain our results by studying rank one shtukas in which both zero and pole are generic, i. e., shtukas not associated to any Drinfeld module.
Interprétation factorielle du nombre de classes dans les ordres des corps quadratiques
Invarianten des total reellen Körpers siebten Grades mit Minimaldiskriminante
Invariants and coinvariants of semilocal units modulo elliptic units
Let be a prime number, and let be an imaginary quadratic number field in which decomposes into two primes and . Let be the unique -extension of which is unramified outside of , and let be a finite extension of , abelian over . Let be the projective limit of principal semi-local units modulo elliptic units. We prove that the various modules of invariants and coinvariants of are finite. Our approach uses distributions and the -adic -function, as defined in [5].
Invariants of number fields related to central embedding problems.
Inverse zero-sum problems in finite Abelian p-groups
We study the minimal number of elements of maximal order occurring in a zero-sumfree sequence over a finite Abelian p-group. For this purpose, and in the general context of finite Abelian groups, we introduce a new number, for which lower and upper bounds are proved in the case of finite Abelian p-groups. Among other consequences, our method implies that, if we denote by exp(G) the exponent of the finite Abelian p-group G considered, every zero-sumfree sequence S with maximal possible length over...
Kapitulation der Idealklassen und Einheitenstruktur in Zahlkörpern.
Kuroda's class number relation