共查询到20条相似文献,搜索用时 15 毫秒
1.
Ichiro Miyada 《manuscripta mathematica》1995,88(1):535-540
We show that there exist only finitely many imaginary abelian number fields of type (2,2,...,2) with one class in each genus.
Moreover, if the Generalized Riemann Hypothesis is true, we have exactly 301 such fields, whose degrees are less than or equal
to 23. Finally we give the table of those 301 fields. 相似文献
2.
It is known that there are only finitely many imaginary abelian number fields with class numbers equal to their genus class numbers. Here, we determine all the imaginary cyclic fields of -power degrees with class numbers equal to their genus class numbers.
3.
Stéphane R. Louboutin 《Journal of Number Theory》2009,129(10):2289-2294
J. Cohen, J. Sonn, F. Sairaiji and K. Shimizu proved that there are only finitely many imaginary quadratic number fields K whose Ono invariants OnoK are equal to their class numbers hK. Assuming a Restricted Riemann Hypothesis, namely that the Dedekind zeta functions of imaginary quadratic number fields K have no Siegel zeros, we determine all these K's. There are 114 such K's. We also prove that we are missing at most one such K. M. Ishibashi proved that if OnoK is large enough compared with hK, then the ideal class groups of K is cyclic. We give a short proof and a precision of Ishibashi's result. We prove that there are only finitely many imaginary quadratic number fields K satisfying Ishibashi's sufficient condition. Assuming our Restricted Riemann Hypothesis, we prove that the absolute values dK of their discriminants are less than 2.3⋅109. We determine all these K's with dK?106. There are 76 such K's. We prove that there is at most one such K with dK?1.8⋅1011. 相似文献
4.
Stéphane R. Louboutin 《Journal of Number Theory》2006,121(1):30-39
We prove that there are effectively only finitely many real cubic number fields of a given class number with negative discriminants and ring of algebraic integers generated by an algebraic unit. As an example, we then determine all these cubic number fields of class number one. There are 42 of them. As a byproduct of our approach, we obtain a new proof of Nagell's result according to which a real cubic unit ?>1 of negative discriminant is generally the fundamental unit of the cubic order Z[?]. 相似文献
5.
It is proved that there is no congruence function field of genus 4 over GF(2) which has no prime of degree less than 4 and precisely one prime of degree 4. This shows the nonexistence of function fields of genus 4 with class number one and gives an example of an isogeny class of abelian varieties which contains no jacobian. It is shown that, up to isomorphism, there are two congruence function fields of genus 3 with class number one. It follows that there are seven nonisomorphic function fields of genus different from zero with class number one. Congruence function fields with class number 2 are fully classified. Finally, it is proved that there are eight imaginary quadratic function fields for which the integral closure of K[x] in F has class number 2. 相似文献
6.
J. E. Carter 《Archiv der Mathematik》2003,81(3):266-271
Let K be a number field and let G be a finite abelian group. We call
K a Hilbert-Speiser field of type G if, and only if, every tamely
ramified normal extension L/K with Galois group isomorphic to G has
a normal integral basis. Now let C2 and C3 denote the cyclic
groups of order 2 and 3, respectively. Firstly, we show that among all
imaginary quadratic fields, there are exactly three Hilbert-Speiser
fields of type $C_{2}: \mathbb{Q}(\sqrt {m})$, where $m \in \{-1, -3, -7\}$.
Secondly, we give some necessary and sufficient conditions for a real
quadratic field $K = \mathbb{Q}(\sqrt {m})$ to be a Hilbert-Speiser field of
type C2. These conditions are in terms of the congruence class of m
modulo 4 or 8, the fundamental unit of K, and the class number of K.
Finally, we show that among all quadratic number fields, there are
exactly eight Hilbert-Speiser fields of type $C_{3}: \mathbb{Q}(\sqrt {m})$,
where $m \in \{-11,-3, -2, 2, 5, 17, 41, 89\}$.Received: 2 April 2002 相似文献
7.
Let be an imaginary abelian number field. We know that , the relative class number of , goes to infinity as , the conductor of , approaches infinity, so that there are only finitely many imaginary abelian number fields with given relative class number. First of all, we have found all imaginary abelian number fields with relative class number one: there are exactly 302 such fields. It is known that there are only finitely many CM-fields with cyclic ideal class groups of 2-power orders such that the complex conjugation is the square of some automorphism of . Second, we have proved in this paper that there are exactly 48 such fields.
8.
Ali Mouhib 《Mathematische Nachrichten》2019,292(3):633-639
Let r be a positive integer. Assume Greenberg's conjecture for some totally real number fields, we show that there exists an infinite family of imaginary cyclic number fields F over the field of rational number field , with an elementary 2‐class group of rank equal to r that capitulates in an unramified quadratic extension over F. Also, we give necessary and sufficient conditions for the Galois group of the unramified maximal 2‐extension over F to be abelian. 相似文献
9.
Mikihito Hirabayashi 《Archiv der Mathematik》2008,90(3):223-229
We give a determinant formula for the relative class number of an imaginary abelian number field, which is a generalization
of Newman’s formula of 1970 and of Skula’s of 1981. By our formula we can determine the signs of these determinants, which
these authors did not give.
Received: 17 May 2007 相似文献
10.
Let G be any finite group and any class of fields. By we denote the minimal number of realizations of G as a Galois group over some field from the class . For G abelian and the class of algebraic extensions of ℚ we give an explicit formula for . Similarly we treat the case of an abelian p-group G and the class which is conjectured to be the class of all fields of characteristic ≠p for which the Galois group of the maximal p-extension is finitely generated. For non-abelian groups G we offer a variety of sporadic results.
Received: 27 October 1998 / Revised version: 3 February 1999 相似文献
11.
In this paper abelian function fields are restricted to the subfields of cyclotomic function fields. For any abelian function field K/k with conductor an irreducible polynomial over a finite field of odd characteristic, we give a calculating formula of the relative divisor class number of K. And using the given calculating formula we obtain a criterion for checking whether or not the relative divisor class number is divisible by the characteristic of k. 相似文献
12.
The set S consisting of those positive integers n which are uniquely expressible in the form n = a2 + b2 + c2, , is considered. Since n ∈ S if and only if 4n ∈ S, we may restrict attention to those n not divisible by 4. Classical formulas and the theorem that there are only finitely many imaginary quadratic fields with given class number imply that there are only finitely many n ∈ S with n = 0 (mod 4). More specifically, from the existing knowledge of all the imaginary quadratic fields with odd discriminant and class number 1 or 2 it is readily deduced that there are precisely twelve positive integers n such that n ∈ S and n ≡ 3 (mod 8). To determine those n ∈ S such that n ≡ 1, 2, 5, 6 (mod 8) requires the determination of the imaginary quadratic fields with even discriminant and class number 1, 2, or 4. While the latter information is known empirically, it has not been proved that the known list of 33 such fields is complete. If it is complete, then our arguments show that there are exactly 21 positive integers n such that n ∈ S and n ≡ 1, 2, 5, 6 (mod 8). 相似文献
13.
Shin Nakano 《Journal of Number Theory》2009,129(12):2943-2951
We give a family of quintic cyclic fields with even class number parametrized by rational points on an elliptic curve associated with Emma Lehmer's quintic polynomial. Further, we use the arithmetic of elliptic curves and the Chebotarev density theorem to show that there are infinitely many such fields. 相似文献
14.
Sara Arias‐de‐Reyna Wojciech Gajda Sebastian Petersen 《Mathematische Nachrichten》2013,286(13):1269-1286
In this paper we prove the Geyer‐Jarden conjecture on the torsion part of the Mordell‐Weil group for a large class of abelian varieties defined over finitely generated fields of arbitrary characteristic. The class consists of all abelian varieties with big monodromy, i.e., such that the image of Galois representation on ?‐torsion points, for almost all primes ?, contains the full symplectic group. 相似文献
15.
Sun-Mi Park 《Journal of Number Theory》2007,125(1):59-84
It is known that if we assume the Generalized Riemann Hypothesis, then any normal CM-field with relative class number one is of degree less than or equal to 96. All normal CM-fields of degree less than 48 with class number one are known. In addition, for normal CM-fields of degree 48 the class number one problem is partially solved. In this paper we will show that under the Generalized Riemann Hypothesis there is no more normal CM-fields with class number one except for the possible fields of degrees 64 or 96. 相似文献
16.
Yves Aubry 《Journal of Number Theory》2008,128(7):2053-2062
We show that, for any finite field Fq, there exist infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is a separable polynomial. As pointed out by Anglès, this is a necessary condition for the existence, for any finite field Fq, of infinitely many real function fields over Fq with ideal class number one (the so-called Gauss conjecture for function fields). We also show conditionally the existence of infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is an irreducible polynomial. 相似文献
17.
Cristian Virdol 《Journal of Number Theory》2011,131(4):681-684
In this paper we prove that if the Birch and Swinnerton-Dyer conjecture holds for abelian varieties attached to Hilbert newforms of parallel weight 2 with trivial central character, then the Birch and Swinnerton-Dyer conjecture holds for abelian varieties attached to Hilbert newforms of parallel weight 2 with trivial central character regarded over arbitrary totally real number fields. 相似文献
18.
Yoonjin Lee 《Journal of Number Theory》2007,122(2):408-414
The Scholz theorem in function fields states that the l-rank difference between the class groups of an imaginary quadratic function field and its associated real quadratic function field is either 0 or 1 for some prime l. Furthermore, Leopoldt's Spiegelungssatz (= the Reflection theorem) in function fields yields a comparison between the m-rank of some subgroup of the class group of an imaginary cyclic function field L1 and the m-rank of some subgroup of the class group of its associated real cyclic function field L2 for some prime number m; then their m-ranks also equal or differ by 1. In this paper we find an explicit necessary condition for their m-ranks (respectively l-ranks) to be the same in the case of cyclic function fields (respectively quadratic function fields). In particular, in the case of quadratic function fields, if l does not divide the regulator of L2, then their l-ranks are the same, equivalently if their l-ranks differ by 1, then l divides the regulator of L2. 相似文献
19.
F. Blanchet-SadriRobert Merca? William SeveraSean Simmons Dimin Xu 《Journal of Combinatorial Theory, Series A》2012,119(1):257-270
Erd?s raised the question whether there exist infinite abelian square-free words over a given alphabet, that is, words in which no two adjacent subwords are permutations of each other. It can easily be checked that no such word exists over a three-letter alphabet. However, infinite abelian square-free words have been constructed over alphabets of sizes as small as four. In this paper, we investigate the problem of avoiding abelian squares in partial words, or sequences that may contain some holes. In particular, we give lower and upper bounds for the number of letters needed to construct infinite abelian square-free partial words with finitely or infinitely many holes. Several of our constructions are based on iterating morphisms. In the case of one hole, we prove that the minimal alphabet size is four, while in the case of more than one hole, we prove that it is five. We also investigate the number of partial words of length n with a fixed number of holes over a five-letter alphabet that avoid abelian squares and show that this number grows exponentially with n. 相似文献
20.
In this paper, we study the image of l-adic representations coming from Tate module of an abelian variety defined over a number field. We treat abelian varieties with complex and real multiplications. We verify the Mumford-Tate conjecture for a new class of abelian varieties with real multiplication. 相似文献