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We give a new characterisation of elliptic curves of Shimura type in terms commuting families of Frobenius lifts and also strengthen an old principal ideal theorem for ray class fields. These two results combined yield the existence of global minimal models for elliptic curves of Shimura type, generalising a result of Gross. Along the way we also prove a handful of small but new results regarding elliptic curves with complex multiplication.

Let $P$ be a finitely generated ideal of a commutative ring $R$. Krull's Principal Ideal Theorem states that if $R$ is Noetherian and $P$ is minimal over a principal ideal of $R$, then $P$ has height at most one. Straightforward examples show that this assertion fails if $R$ is not Noetherian. We consider what can be asserted in the non-Noetherian case in place of Krull's theorem., 15 pages, to appear in Math. Proc. Cambridge Philos. Soc

In this paper, new probability estimates are derived for ideal lattice codes from totally real number fields using ideal class Dedekind zeta functions. In contrast to previous work on the subject, it is not assumed that the ideal in question is principal. In particular, it is shown that the corresponding inverse norm sum depends not only on the regulator and discriminant of the number field, but also on the values of the ideal class Dedekind zeta functions. Along the way, we derive an estimate of the number of elements in a given ideal with a certain algebraic norm within a finite hypercube. We provide several examples which measure the accuracy and predictive ability of our theorems., 24 pages. Extends our earlier arxiv submission arxiv.1303.3475

We shall consider the maximal cyclotomic extension of a totally real finite algebraic number field and its ideal class group. We shall investigate the structure of the ideal class group with the action of the cyclotomic Galois group.

Using class field theory I give an example of a function field of genus 4 with class number one over the finite field F 2 . In a previous paper (see [2, Section 2] ) a proof of the nonexistence of such a function field is given. This counterexample shows that the proof in [2] is wrong and so the list of algebraic function fields with class number one given in [2] should admit one more example.

As for Scholz' inequalities s ≤ r ≤ s + 1 with respect to the 3-rank of the ideal class group of quadratic fields, we give a criterion to be r = s + 1. From this, we give a new family of imaginary quadratic fields whose ideal class groups have 3-rank at least two.

In this paper, we study the notion of norm-Euclidean ideal class, which was defined by Lenstra [Euclidean ideal classes, Astérisque 61 (1979) 121–131]. Using a slight modification of an algorithm determining among other properties the Euclidean minimum described in [P. Lezowski, Computation of the Euclidean minimum of algebraic number fields, preprint (2011); http://hal.archives-ouvertes.fr/hal-00632997/en/], we give new examples of number fields with norm-Euclidean ideal classes. Extending the results of Cioffari [The Euclidean condition in pure cubic and complex quartic fields, Math. Comput. 33 (1979) 389–398], we also establish the complete list of pure cubic number fields which admit a norm-Euclidean ideal class. Finally, we show that [Formula: see text], which is known to admit a non-principal Euclidean ideal class, has no norm-Euclidean ideal class.

Let 𝑇 be a commutative ring, 𝐼 a prime ideal of 𝑇 , 𝐷 a subring of 𝑇 / 𝐼 , and 𝑅 the pullback 𝑇 × 𝑇 / 𝐼 𝐷 . Ascent and descent results are given for the transfer of the 𝑛 -PIT and GPIT (generalized principal ideal theorem) properties between 𝑇 and 𝑅 . As a consequence, it follows that if 𝐼 is a maximal ideal of both 𝑇 and 𝑅 , then 𝑅 satisfies 𝑛 -PIT (resp., GPIT) if and only if 𝑇 satisfies 𝑛 -PIT (resp., GPIT).

Genus theory belongs to algebraic number theory and, in very broad terms, deals with the part of the ideal class group of a number field that is ‘easy to compute’. Historically, the importance of genus theory stems from the fact that it was the essential algebraic ingredient in the derivation of the classical reciprocity laws – from Gaus’s second proof over Kummer’s contributions up to Takagi’s ‘general’ reciprocity law for p-th power residues. The central theorem in genus theory is the principal genus theorem, which is hard to describe in just one sentence – readers not familiar with genus theory might want to glance into Section 2 before reading on. In modern terms, the principal genus theorem for abelian extensions k/Q describes the splitting of prime ideals of k in the genus field kgen of k, which by definition is the maximal unramified extension of k that is abelian over Q. In this note we outline the development of the principal genus theorem from its conception in the context of binary quadratic forms by Gaus (with hindsight, traces of genus theory can be found in the work of Euler on quadratic forms and idoneal numbers) to its modern formulation within the framework of class field theory. It is somewhat remarkable that, although the theorem itself is classical, the name ‘principal ideal theorem’ (‘Hauptgeschlechtssatz’ in German) was not used in the 19th century, and it seems that it was coined by Hasse in his Bericht [28] and adopted immediately by the abstract algebra group around Noether. It is even more remarkable that Gaus doesn’t bother to formulate the principal genus theorem except in passing: after observing in [25, §247] that duplicated classes (classes of forms composed with themselves) lie in the principal genus, the converse (namely the principal genus theorem) is stated for the first time in §261: si itaque omnes classes generis principalis ex duplicatione alicuius classis provenire possunt (quod revera semper locum habere in sequentibus demonstrabitur), . . . 1 The actual statement of the principal genus theorem is somewhat hidden in [25, §286], where Gaus formulates the following Problem. Given a binary form F = (A,B,C) of determinant D belonging to a principal genus: to find a binary form f from whose duplication we get the form F . It strikes us as odd that Gaus didn’t formulate this central result properly; yet he knew exactly what he was doing [25, §287]

Let K be a quadratic imaginary number field, f ∈ N and let O f be the order of conductor f in K. We consider the singular values of the Kleinian normalization φ of the Weierstrass σ-function belonging to an arbitrary proper ideal of O f . The factorization of these singular values goes back to K. Ramachandra, R. Schertz and W. Bley. But the factorization formula in [W. Bley, Konstruktion von Ganzheitsbasen in abelschen Korpererweiterungen von imaginar-quadratischen Zahlkorpern, Dissertation, Universitat Augsburg, 1991] is very implicit and not easy to handle in view of many practical applications. In this paper we provide an explicit factorization formula and give different tools to control this factorization. As an immediate application we prove the generalized principal ideal theorem in the ring class field situation.

We shall discuss the local triviality in the ideal class group of the basic $\mathbf Z_p$-extension over an imaginary quadratic field and prove, in particular, a result which implies that such triviality distributes with natural density $1$.

It has been shown by Madden that there are only finitely many quadratic extensions of k(x), k a finite field, in which the ideal class group has exponent two and the infinity place of k(x) ramifies. We give a characterization of such fields that allow us to find a full list of all such field extensions for future reference. In doing so we correct some errors in earlier published literature., To appear in the Journal of Number Theory

Global class field theory is a major achievement of algebraic number theory, based on the Artin reciprocity map and the existence theorem. The author works out the consequences and the practical use of these results by giving detailed studies and illustrations of classical subjects (classes, idèles, ray class fields, symbols, reciprocity laws, Hasse's principles, Grunwald--Wang's theorem, Hilbert's towers,...). He also proves some new or less-known results (reflection theorem, structure of the abelian closure of a number field) and puts emphasis on the invariant T_p, of abelian p-ramification, which is related to important Galois cohomology properties and p-adic conjectures. This book, intermediary between the classical literature published in the sixties and recent computational one, gives much material in an elementary way, and is suitable for students, researchers, and all those who are fascinated by this theory. In the corrected 2nd printing 2005, the author improves some mathematical and bibliographical details and adds a few pages about rank computations for the general reflection theorem; then he gives an arithmetical interpretation for usual class groups, and applies this to the Spiegelungssatz for quadratic fields and for the p-th cyclotomic field regarding the Kummer--Vandiver conjecture in a probabilistic point of view. ----------------------------------------------------------------------------------------------------------------------------------------http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1019.11032&format=complete --------------------------------------------------------------------------------------------------------------------------------- FOR A COMPLETE VIEW OF MY PUBLICATIONS, PLEASE LOOK AT MY HOME PAGE: http://www.researchgate.net/profile/Georges_Gras

Let $F$ be a real quadratic field and $\mathfrak{p}$ a prime ideal of degree 2. We construct a quadratic extension of the Hilbert class field in the ray class field of conductor $\mathfrak{p}$.

For some algebraic number fields $F$ with only one fundamental unit, we give a lower bound of the extension degree of the ray class field of conductor a rational prime $p$ over the Hilbert class field of $F$.

Let $p$ and $q$ be distinct primes such that $p \equiv q \pmod{4}$ and consider the quadratic field $K = \mathbf{Q}(\sqrt{pq})$. In this paper, we shall investigate the class group and determine the exact power of 2 dividing the class number of $K$ using the theory of ideals and a theorem on the solvability of $ax^2 + by^2 = z^2$.

Using symmetric algebras we simplify (and slightly strengthen) the Bruns-Eisenbud-Evans "generalized principal ideal theorem" on the height of order ideals of non-minimal generators in a module. We also obtain a simple proof and an extension of a result by Kwieci\'nski, which estimates the height of certain Fitting ideals of modules having an equidimensional symmetric algebra., Comment: 6 pages, LaTeX

Let N be a finitely generated module over a Noetherian local ring (R,m). We give criteria for the height of the order ideal N^*(x) of an element x \in N to be bounded by the rank of N. The Generalized Principal Ideal Theorem of Bruns, Eisenbud and Evans says that this inequality always holds if x \in mN, but if x is a minimal generator the question is more subtle; for example, when R is graded and N represents a vector bundle on Proj R, the question is connected with the vanishing of the top chern class. We show that the inequality does hold if the hypothesis x \in mN becomes true after first extending scalars to some local domain and then factoring out torsion. We also give conditions under which it holds in terms of residual intersections and integral closures of modules. We derive information about order ideals that leads to bounds on the heights of trace ideals of modules--even in circumstances where we do not have the expected bounds for the heights of the order ideals., 20 pages

It is shown that a semigroup algebra K [ S ] which is a principal left ideal ring is a finitely generated PI-algebra of Gelfand–Kirillov dimension at most 1. A complete description of principal (left and right) ideal rings K [ S ], and of the underlying semigroups S , is obtained. Semiprime principal left ideal rings K [ S ] are shown to be principal right ideal rings and a description of this class of rings follows.

In Bautista-Ancona and Diaz-Vargas (2006) [B-D] a characterization and complete listing is given of the imaginary quadratic extensions K of k ( x ) , where k is a finite field, in which the ideal class group has exponent two and the infinite prime of k ( x ) ramifies. The objective of this work is to give a characterization and list of these kind of extensions but now considering the case in which the infinite prime of k ( x ) is inert in K . Thus, we get all the imaginary quadratic extensions of k ( x ) , in which the ideal class group has exponent two.

The relations between the conjecture about the minus part of the ideal class group of the cyclotomic field Q(μp) (which is a consequence of the Vandiver conjecture) and the structures of the Hecke algebra and of the Eisenstein ideal of level 1 modular forms are studied.

Let k be an imaginary quadratic number field with C k , 2 , the 2-Sylow subgroup of its ideal class group, isomorphic to Z /2 Z × Z /2 Z × Z /2 Z . By the use of various versions of the Kuroda class number formula, we improve significantly upon our previous lower bound for | C k 1 , 2 |, the 2-class number of the Hilbert 2-class field of k .

We prove that a large class of triangular UHF algebras are primitive. We use two avenues to obtain our results: a direct approach in which we explicitly construct a faithful, algebraically irreducible representation of the algebra on a separable Hilbert space, as well as an indirect, algebraic approach which utilizes the prime ideal structure of the algebra. Using these results, we completely characterize the primitive ideal spaces of all lexicographic algebras: An ideal there is primitive if and only if it is closed prime. Specializing on the algebrasA( Q , ν), we obtain a complete classification of their algebraic isomorphisms and epimorphisms through the use of a new invariant involving the primitive ideal space. Finally, we characterize the primitive ideal spaces of Z -analytic and order-preserving algebras, and obtain information about their epimorphisms.

Cataloged from PDF version of article. The Hilbert ideal is the ideal generated by positive degree invariant polynomials of a finite group. For a cyclic group of prime order p, we show that the image of the transfer lie in the ideal generated by invariants of degree at most p - 1. Consequently we show that the Hilbert ideal corresponding to an indecomposable representation is generated by polynomials of degree at most p, confirming a conjecture of Harm Derksen and Gregor Kemper for this case. © 2007 Elsevier Inc. All rights reserved.

The classical genus theory of Gauss has been extended by Hilbert from the quadratic field over the rational field to the Kummer field over the cyclotomic field as a precise analog. Still, many problems involving the computation of pure cubic unit and class number are related to the rational and not the cubic cyclotomic field. It is, therefore, desirable to have (at least partially) a genus theory of cubic characters and an analog of such things as Hilbert's Theorem 90, ambiguous (principal) classes, and class group invariants divisible by 3 and 9,-all based on the rational field. This study shows a strong similarity between pure cubic and quadratic fields.

We prove that, for each prime p dividing n , every infinite class of Witt equivalent number fields of degree n ⩾2 contains a field with the p -rank of the ideal class group exceeding any given number.

In this paper we construct, for any integers m and n, and 2 ⩽ g ⩽ m - 1 , infinitely many function fields K of degree m over F ( T ) such that the prime at infinity splits into exactly g primes in K and the ideal class group of K contains a subgroup isomorphic to ( Z / n Z ) m - g . This extends previous results of the author and Lee for the cases g = 1 and g = m .

Noether lattices were introduced by Dilworth [5]. Simply stated, a Noether lattice is a complete, modular, commutative multiplicative lattice satisfying the ascending chain condition in which every element is the join of principal elements. The resultant setting not only admits natural formulations of the fundamental definitions and results of classical ideal theory, but is simultaneously rich enough to yield, for example, the Noether decomposition theorems, the Intersection Theorem, and the Principal Ideal Theorem [Sj. The key to the richness of the setting lies, of course, in the definition of a principal element. An element E is said to be principal if it satisfies the two identities BE A A = (B A (A : E))E and (B v AE) : E = (B : E) v A, for all A and B. These identities are easily verified for the principal ideals of a commutative ring. In [4], Bogart obtained the surprising result that a regular local Noether lattice with exactly n < 3 rank 1 primes is isomorphic to RL, , the Noether sublattice of k[x, ,..., x,] (k a field) consisting of all ideals which are generated by monomials in x1 , xs ,..., x, . He conjectured the validity of the result for all n, but noted that, due to the heavy dependence of the case n = 3 on the observation that in a regular local Noether lattice of dimension 2, all rank 1 primes are principal, the general case might be closely tied to a generalization of the Auslander-Buchsbaum theorem that regular local rings are UFDs. In [2], Anderson conjectured that any local Noether lattice with exactly n rank 1 primes which is a n-domain (i.e., every principal element is a product of primes, or, equivalently in this case, every rank 1 prime is principal) is isomorphic to RL, , and therefore regular. In this paper we establish the validity of both conjectures. While our results do not yield a generalization of the Auslander-Buchsbaum theorem as hoped, they do at least show that any counterexample to the generalization will involve an infinite number of primes.

For the function field K of hyperelliptic curves over Q we define a subgroup of the ideal class group called the group of Z-primitive ideals. We then show that there are homomorphisms from this subgroup to ideal class groups of certain quadratic number fields.

Let R be an atomic integral domain. Suppose that H is a nonempty subset of irreducible elements of R, u is a unit of R, and α1,..., αn, β1,..., βm are irreducible elements of R such that (1) α1... αn = u · β1... βm. Set H α = {i|αi ∈ H } and H β = {j|βj ∈ H }. H is a factorization set (F-set) of R if for any equality involving irreducibles of the form (1), | H α| ≠ 0 implies that | H β| ≠ 0. H is a half-factorial set (HF-set) if any equality of the form (1) implies that | H α| = | H β|. In this paper, we explore in detail the structure of the F-sets and HF-sets of an atomic integral domain R. If P is a nonzero prime ideal of R and J (R) the set of irreducible elements of R, then set H P = P ∩ J (R). We show that if F is an F-set of R, then there exists a nonempty set X of nonzero prime ideals of R such that F = ∪ P ∈ X H P. We define an F-set to be minimal if it contains no proper subsets which are F-sets. We then show that in R every F-set can be written as a union of minimal F-sets if and only if R satisfies the Principal Ideal Theorem. If, in addition, every height-one prime ideal of R is the radical of a principal ideal, then this representation is unique. We study the structure of the HF-sets of R and concentrate on the case where R is a Krull domain with torsion divisor class group. For such a domain R, we show that if H is an HF-set of R and P is a nonprincipal prime ideal of R with H P ⊆ H , then H Q ⊆ H for each prime ideal Q of R in the same divisor class as P. We also show that if R is a Dedekind domain with prime class number p ≥ 2 and P (R) is the set of prime elements of R, then H an HF-set of R implies that either H ∪ P (R) = J (R) or H ⊆ P (R).

For quadratic number fields F with 2∈NF/Q(F∗), we set up the relations between the class of dyadic ideal in narrow class group and the element of order 4 in the tame kernel, and determine the 2n(n>2)-rank of the tame kernel of some quadratic number fields and their Tate kernels.

Let ej denote the number of 2j-invariants of the narrow ideal class group of a quadratic field Q(D1/2). Pumplün has stated a criterion for e2 > 0. The equivalence of this criterion to one arising from the Redei-Reichardt theorem is shown to be a graph-theoretic result, for which a direct proof is given.

The generalized Lagrange algorithm is a number geometric generalization of Lagrange's continued fraction method for computing fundamental unit and class number of real quadratic number fields. This algorithm yields a system of fundamental units and the class number of an arbitrary algebraic number field by means of computing cycles of reduced ideals. In this paper we prove that the cardinality of a cycle of reduced ideals in an ideal class of an order of an algebraic number field is O(R), where R is the regulator of this order, and where the O-constant only depends on the degree of the field. We also give a lower bound on this cardinality.

Let p be a fixed prime integer, and G a finite p-group. For a subgroup H of G, we denote the centralizer of H in G by CG(H). The commutator subgroup of G is denoted by [G, G].

In a previous paper on this subject, the author gave a new proof of the theorem of "analytic independence of systems of parameters". The methods used can be applied to prove a converse result. Hence: An ideal of a Noetherian ring is of the principal class if, and only if, it is generated by an analytically independent set.