Class numbers of ray class fields of imaginary quadratic fields
نویسندگان
چکیده
منابع مشابه
Class numbers of ray class fields of imaginary quadratic fields
Let K be an imaginary quadratic field with class number one and let p ⊂ OK be a degree one prime ideal of norm p not dividing 6dK . In this paper we generalize an algorithm of Schoof to compute the class numbers of ray class fields Kp heuristically. We achieve this by using elliptic units analytically constructed by Stark and the Galois action on them given by Shimura’s reciprocity law. We have...
متن کاملClass numbers of imaginary quadratic fields
The classical class number problem of Gauss asks for a classification of all imaginary quadratic fields with a given class number N . The first complete results were for N = 1 by Heegner, Baker, and Stark. After the work of Goldfeld and Gross-Zagier, the task was a finite decision problem for any N . Indeed, after Oesterlé handled N = 3, in 1985 Serre wrote, “No doubt the same method will work ...
متن کاملNormal Bases of Ray Class Fields over Imaginary Quadratic Fields
We first develop a criterion to determine normal bases (Theorem 2.4), and by making use of necessary lemmas which were refined from [3] we further prove that singular values of certain Siegel functions form normal bases of ray class fields over all imaginary quadratic fields other than Q( √−1) and Q( √−3) (Theorem 4.5 and Remark 4.6). This result would be an answer for the Lang-Schertz conjectu...
متن کاملIndivisibility of class numbers of imaginary quadratic fields
We quantify a recent theorem of Wiles on class numbers of imaginary quadratic fields by proving an estimate for the number of negative fundamental discriminants down to −X whose class numbers are indivisible by a given prime and whose imaginary quadratic fields satisfy any given set of local conditions. This estimate matches the best results in the direction of the Cohen–Lenstra heuristics for ...
متن کاملOn Congruent Primes and Class Numbers of Imaginary Quadratic Fields
We consider the problem of determining whether a given prime p is a congruent number. We present an easily computed criterion that allows us to conclude that certain primes for which congruency was previously undecided, are in fact not congruent. As a result, we get additional information on the possible sizes of Tate-Shafarevich groups of the associated elliptic curves. We also present a relat...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Mathematics of Computation
سال: 2011
ISSN: 0025-5718
DOI: 10.1090/s0025-5718-2010-02413-5