Elliptic units of cyclic unramified extensions of complex quadratic fields

Elliptic units of cyclic unramified extensions of complex quadratic fields

Unramified Quaternion Extensions of Quadratic Number Fields

The first mathematician who studied quaternion extensions (H8-extensions for short) was Dedekind [6]; he gave Q( √ (2 + √ 2)(3 + √ 6) ) as an example. The question whether given quadratic or biquadratic number fields can be embedded in a quaternion extension was extensively studied by Rosenblüth [32], Reichardt [31], Witt [36], and Damey and Martinet [5]; see Ledet [19] and the surveys [15] and...

Unramified Alternating Extensions of Quadratic Fields

We exhibit, for each n ≥ 5, infinitely many quadratic number fields admitting unramified degree n extensions with prescribed signature whose normal closures have Galois group An. This generalizes a result of Uchida and Yamamoto, which did not include the ability to restrict the signature, and a result of Yamamura, which was the case n = 5. It is a folk conjecture that for n ≥ 5, all but finitel...

Elliptic units for real quadratic fields

1. A review of the classical setting 2. Elliptic units for real quadratic fields 2.1. p-adic measures 2.2. Double integrals 2.3. Splitting a two-cocycle 2.4. The main conjecture 2.5. Modular symbols and Dedekind sums 2.6. Measures and the Bruhat-Tits tree 2.7. Indefinite integrals 2.8. The action of complex conjugation and of Up 3. Special values of zeta functions 3.1. The zeta function 3.2. Va...

UNRAMIFIED EXTENSIONS AND GEOMETRIC Zp-EXTENSIONS OF GLOBAL FUNCTION FIELDS

We study on finite unramified extensions of global function fields (function fields of one valuable over a finite field). We show two results. One is an extension of Perret’s result about the ideal class group problem. Another is a construction of a geometric Zp-extension which has a certain property.