Infinite class field towers of quadratic fields.
Infinite Hilbert 2-class field tower of quadratic number fields
Integer Valued polynomials over a number field.
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.
Invariants of the ideal class group and the Hasse norm theorem.
Iterated Ring Class Fields and the 168-Tesselation.
Iterated Ring Class Fields and the Icosahedron.
K2-analogs of Hasse's norm theorems.
Kapitulation der Idealklassen und Einheitenstruktur in Zahlkörpern.
Kapitulationsproblem und Knotentheorie.
Konstruktion von Klassenkörpern und Potenzrestkriterien für quadratische Einheiten.
Konstruktion von Potenzganzheitsbasen in Strahlklassenkörpern über imaginär-quadratischen Zahlkörpern.
Lien algébrique entre les deux facteurs de la formule analytique du nombre de classes dans les corps abéliens
Logarithme -adique et corps de classes
Logarithme p-adique, p-ramification abélienne et K ...
Lois de réciprocité et lois de décomposition.
Lower powers of elliptic units
In the previous paper [Sch2] it has been shown that ray class fields over quadratic imaginary number fields can be generated by simple products of singular values of the Klein form defined below. In the present article the second named author has constructed more general products that are contained in ray class fields thereby correcting Theorem 2 of [Sch2]. An algorithm for the computation of the algebraic equations of the numbers in Theorem 1 of this paper has been implemented in a KASH program...
Making sense of capitulation: reciprocal primes
Let ℓ be a rational prime, K be a number field that contains a primitive ℓth root of unity, L an abelian extension of K whose degree over K, [L:K], is divisible by ℓ, a prime ideal of K whose ideal class has order ℓ in the ideal class group of K, and any generator of the principal ideal . We will call a prime ideal of K ’reciprocal to ’ if its Frobenius element generates for every choice of . We then show that becomes principal in L if and only if every reciprocal prime is not a norm inside...
Maximal unramified extensions of imaginary quadratic number fields of small conductors, II
In the previous paper [15], we determined the structure of the Galois groups of the maximal unramified extensions of imaginary quadratic number fields of conductors under the Generalized Riemann Hypothesis (GRH) except for 23 fields (these are of conductors ) and give a table of . We update the table (under GRH). For 19 exceptional fields of them, we determine . In particular, for , we obtain , the fourth Hilbert class field of . This is the first example of a number field whose...
Maximal unramified extensions of imaginary quadratic number fields of small conductors
We determine the structures of the Galois groups Gal of the maximal unramified extensions of imaginary quadratic number fields of conductors under the Generalized Riemann Hypothesis). For all such , is , the Hilbert class field of , the second Hilbert class field of , or the third Hilbert class field of . The use of Odlyzko’s discriminant bounds and information on the structure of class groups obtained by using the action of Galois groups on class groups is essential. We also use class...