The exponent three class group problem for some real cyclic cubic number fields

Let {Km}m≥4 be the family of non-normal totally real cubic number fields as- sociated with the Q-irreducible cubic polynomials Pm(x) = x 3 − mx 2 − (m + 1)x − 1, m ≥ 4. We determine all these Km's with class numbers hm ≤ 3: there are 14 such Km's. Assuming the Generalized Riemann hypothesis for all the real quadratic number fields, we also prove that the exponents em of the ideal class groups of these Km go to infinity with m and we determine all these Km's with ideal class groups of exponents em ≤ 3: there are 6 such Km with ideal class groups of exponent 2, and 6 such Km with ideal class groups of exponent 3.

Given any family of cubic fields defined by local conditions at finitely many primes, we determine the mean number of 2-torsion elements in the class groups and narrow class groups of these cubic fields when ordered by their absolute discriminants. For an order $\cal O$ in a cubic field, we study the three groups: $\rm Cl_2(\cal O)$, the group of ideal classes of $\cal O$ of order 2; $\rm Cl^+_2(\cal O)$, the group of narrow ideal classes of $\cal O$ of order 2; and ${\cal I}_2(\cal O)$, the group of ideals of $\cal O$ of order 2. We prove that the mean value of the difference $|\rm Cl_2({\cal O})|-\frac14|{\cal I}_2(\cal O)|$ is always equal to $1$, whether one averages over the maximal orders in real cubic fields, over all orders in real cubic fields, or indeed over any family of real cubic orders defined by local conditions. For the narrow class group, we prove that the mean value of the difference $|\rm Cl^+_2({\cal O})|-|{\cal I}_2(\cal O)|$ is equal to $1$ for any such family. For any family of complex cubic orders defined by local conditions, we prove similarly that the mean value of the difference $|\rm Cl_2(\mathcal O)|-\frac12|{\cal I}_2(\cal O)|$ is always equal to $1$, independent of the family. The determination of these mean numbers allows us to prove a number of further results as by-products. Most notably, we prove---in stark contrast to the case of quadratic fields---that: 1) a positive proportion of cubic fields have odd class number; 2) a positive proportion of real cubic fields have isomorphic 2-torsion in the class group and the narrow class group; and 3) a positive proportion of real cubic fields contain units of mixed real signature. We also show that a positive proportion of real cubic fields have narrow class group strictly larger than the class group, and thus a positive proportion of real cubic fields do not possess units of every possible real signature.

In this paper the authors provide strong evidence for a conjecture of Chinburg in the context of abelian extensions N/K of quadratic imaginary fields K with group G. For abelian groups G, the conjecture specialises to the statement that Chinburg's third invariant Ω(N/K, 3)=0 ∈ Cl(Z[G]). We prove (Theorem 1) that if K has class number 1 (and some technical conditions on the conductor of N/K and its roots of unity hold) then Ω(N/K, 3) lies in the kernel group D(Z[G]). For a special class of genus extensions of prime power degree we prove (Theorem 2) the full conjecture, that Ω(N/K, 3)=0. The proof technique for Theorems 1 and 2 uses Kolyvagin's Euler systems of units applied to certain explicit groups of elliptic units, which essentially determines the difference in Galois structure between units and ideal class groups over the maximal order MG ∈ Q[G]. This is made precise by using Hecke factorisation and working in the factorisability defect class group of the category of finitely generated Z[G]-modules, which is isomorphic to Cl(MG). For Theorem 2, the fact that the class number of N is coprime to [N:K] in such genus extensions N/K allows one to lift these results to Cl(Z[G]), after using cohomological calculations to check that (essentially) an explicit representative for a Tate sequence defining Ω (N/K, 3) has the correct extension class (which is related to the fundamental class of N/K). 1991 Mathematics Subject Classification: 11R33.

Let F be a cubic cyclic field with t (⩾ 2) ramified primes. For a finite abelian group G, let r3(G) be the 3-rank of G. If 3 does not ramify in F, then it is proved that t−1 ⩽ r3(K2OF) ⩽ 2t. Furthermore, if t is fixed, for any s satisfying t−1 ⩽ s ⩽ 2t−1, there is always a cubic cyclic field F with exactly t ramified primes such that r3(K2OF) = s. It is also proved that the densities for 3-ranks of tame kernels of cyclic cubic number fields satisfy a Cohen-Lenstra type formula $$d_{\infty ,r} = 3^{ - r^2 } {{\prod\limits_{k = 1}^\infty {\left( {1 - 3^{ - k} } \right)} } \mathord{\left/ {\vphantom {{\prod\limits_{k = 1}^\infty {\left( {1 - 3^{ - k} } \right)} } {\prod\limits_{k = 1}^r {\left( {1 - 3^{ - k} } \right)} }}} \right. \kern-\nulldelimiterspace} {\prod\limits_{k = 1}^r {\left( {1 - 3^{ - k} } \right)} }}^2 .$$ This suggests that the Cohen-Lenstra conjecture for ideal class groups can be extended to the tame kernels of cyclic cubic number fields.

We give a necessary condition for the ideal class group of a CM‐field to be of exponent at most two. This condition enables us to drastically reduce the amount of relative class number computation for determination of the CM ‐ fields of some types (e. g. the imaginary cyclic non ‐quadratic number fields of 2 ‐ power degrees) whose ideal class groups are of exponents at most two. We also give a necessary condition for some quartic non ‐ CM ‐ fields to have class number one.

In this paper, we consider a family of cubic fields $\{K_m\}_{m\geq4}$ associated to the irreducible cubic polynomials $P_m(x)=x^3-mx^2-(m+1)x-1,\,\,\,(m\geq4).$ We prove that there are infinitely many $\{K_m\}_{m\geq4}$'s whose class numbers are divisible by a given integer n. From this, we find that there are infinitely many non-normal totally real cubic fields with class number divisible by any given integer n.

On the Birch-Tate conjecture for cyclic number fields

In this paper the author describes the construction of a table of totally real cubic number fields. Each field is distinguished by the coefficients of a generating polynomial, the index of this polynomial over the field and the discriminant of the field. The class number and a fundamental pair of units is also given.

Let K be a number field, let A be an abelian variety defined over K and let $$K_\infty /K$$ be a uniform p-adic Lie extension. We compare several arithmetic invariants of Iwasawa modules of ideal class groups on the one side and fine Selmer groups of abelian varieties on the other side. If $$K_\infty $$ contains sufficiently many p-power torsion points of A, then we can compare the ranks and the Iwasawa $$\mu $$ -invariants of these modules over the Iwasawa algebra. In several special cases (e.g. multiple $$\mathbb {Z}_p$$ -extensions), we can also prove relations between suitably generalised Iwasawa $$\lambda $$ -invariants of the two types of Iwasawa modules. In the literature, two different kinds of generalised Iwasawa $$\lambda $$ -invariants have been introduced for ideal class groups and Selmer groups. We define analogues of both concepts for fine Selmer groups and compare the resulting invariants. In order to obtain some of our main results, we prove new asymptotic formulas for the growth of ideal class groups and fine Selmer groups in multiple $$\mathbb {Z}_p$$ -extensions.

The class group of a number field K measures how far its ring of integers is from having unique factorization into irreducible elements. It is the quotient of the group of all fractional ideals of K by the subgroups of principal fractional ideals. It is well known from class field theory that the ideal class group is also the Galois group of the maximal unramified abelian extension of K.

Hilbert Class Fields of Real Biquadratic Fields

Ternary quadratic forms over number fields with small class number

We describe an algorithm for finding the coefficients of $F(X)$ modulo powers of $p$, where $p\neq 2$ is a prime number and $F(X)$ is the power series associated to the zeta function of Kubota and Leopoldt. We next calculate the 5-adic and 7-adic ${\it\lambda}$-invariants attached to those cubic extensions $K/\mathbb{Q}$ with cyclic Galois group ${\mathcal{A}}_{3}$ (up to field discriminant ${<}10^{7}$), and also tabulate the class number of $K(e^{2{\it\pi}i/p})$ for $p=5$ and $p=7$. If the ${\it\lambda}$-invariant is greater than zero, we then determine all the zeros for the corresponding branches of the $p$-adic $L$-function and deduce ${\rm\Lambda}$-monogeneity for the class group tower over the cyclotomic $\mathbb{Z}_{p}$-extension of $K$.Supplementary materials are available with this article.

Class Numbers and Iwasawa Invariants of Certain Totally Real Number Fields

The problem of limited tagged training samples and unknown the number of classes is challenging for the classification of remote sensing scenes. This paper presents a new GMRF self-supervised algorithm for SAR image. We add a GOF process in the process of estimating GMM parameters by EM algorithm, which can not only dynamically select the best number of significant classes but also provides an initial feature parameter to calculate the MRF minimum energy. After the iterative region label and region growth cycle, iteration is combined with the Mll context model to obtain the best mark of each region. Since the initial feature parameter selection of the MRF is not random, the operation efficiency is also improved while reducing the number of iteration cycles of the algorithm. The experiment validates that our design not only solves the problem of manual input of the number of classes but also provide the better output result graph in terms of detail maintenance than the expert interpretation of the truth map in the unsupervised image classification process, and we hope that it could support operation and meet the real-time requirements.