Circular Units of Real Abelian Fields with Four Ramified Primes
نویسنده
چکیده
Abstract. In this paper we study the groups of circular numbers and circular units in Sinnott’s sense in real abelian fields with exactly four ramified primes under certain conditions. More specifically, we construct Z-bases for them in five special infinite families of cases. We also derive some results about the corresponding module of relations (in one family of cases, we show that the module of Ennola relations is cyclic). The paper is based upon the thesis [6], which builds upon the results of the paper [2].
منابع مشابه
Twisted Zeta Functions of Quaternion Orders
Given an abelian Galois extension K/F of number fields, a quaternion algebra A over F that is ramified at all infinite primes, and a character χ of the Galois group of K over F , we consider the twist of the zeta function of A by the character χ. We show that such twisted zeta functions provide a factorization of the zeta function of A(K) = A ⊗F K. Also, the quotient of the zeta function for A(...
متن کاملDiophantine Definability and Decidability in the Extensions of Degree 2 of Totally Real Fields
We investigate Diophantine definability and decidability over some subrings of algebraic numbers contained in quadratic extensions of totally real algebraic extensions of Q. Among other results we prove the following. The big subring definability and undecidability results previously shown by the author to hold over totally complex extensions of degree 2 of totally real number fields, are shown...
متن کاملMath 254a: Class Field Theory Iii
1. The main theorems revisited We now state in more detail the main theorems of class field theory. However, we need two more preliminary definitions: Definition 1.1. Given an extension L/K of number fields, and a fractional ideal I of L, we define the fractional ideal N L/K (I) of K as follows: write I = i q ei i , let p i = q i ∩ O K , and let N L/K (I) = i p eifi i. Definition 1.2. Let σ : K...
متن کاملA Gross-Zagier formula for quaternion algebras over totally real fields
We prove a higher dimensional generalization of Gross and Zagier’s theorem on the factorization of differences of singular moduli. Their result is proved by giving a counting formula for the number of isomorphisms between elliptic curves with complex multiplication by two different imaginary quadratic fields K and K′, when the curves are reduced modulo a supersingular prime and its powers. Equi...
متن کاملCompanion Forms over Totally Real Fields
We show that if F is a totally real field in which p splits completely and f is a mod p Hilbert modular form with parallel weight 2 < k < p, which is ordinary at all primes dividing p and has tamely ramified Galois representation at all primes dividing p, then there is a “companion form” of parallel weight k′ := p + 1 − k. This work generalises results of Gross and Coleman–Voloch for modular fo...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2017