Explicit bounds for generators of the class group

We investigate Noetherian maximal orders with enough invertible ideals and their two different divisor class groups. We show that in a Noetherian maximal order R with enough invertible ideals, every height 1 prime ideal P is maximal reflexive and , where P ranges over all height 1 prime ideals of R, and S is a simple Noetherian ring. We show that one of the class groups of R measures, to some extent, the lack of unique factorisation in the ring. We also investigate relations between the class groups of R and the divisor class group of the center of R. Examples are provided to illustrate our results.

Explicit bounds are given on the norms of prime ideals generating arbitrary subgroups of ray class groups of number fields, assuming the Extended Riemann Hypothesis. These are the first explicit bounds for this problem, and are significantly better than previously known asymptotic bounds. Applied to the integers, they express that any subgroup of index $i$ of the multiplicative group of integers modulo $m$ is generated by prime numbers smaller than $16(i\log m)^2$, subject to the Riemann Hypothesis. Two particular consequences relate to mathematical cryptology. Applied to cyclotomic fields, they provide explicit bounds on generators of the relative class group, needed in some previous work on the shortest vector problem on ideal lattices. Applied to Jacobians of hyperelliptic curves, they allow one to derive bounds on the degrees of isogenies required to make their horizontal isogeny graphs connected. Such isogeny graphs are used to study the discrete logarithm problem on said Jacobians.

For a given family (Gi)i∈N of finitely generated abelian groups, we construct a Dedekind domain D having the following properties.(1)Pic(D)≅⨁i∈NGi.(2)For each i∈N, there exists a submonoid Si⊆D• with Pic(DSi)≅Gi.(3)Each class of Pic(D) and of all Pic(DSi) contains infinitely many prime ideals. Furthermore, we study orders as well as sets of lengths in the Dedekind domain D and in all its localizations DSi.

Let A A be a Noetherian Cohen-Macaulay domain, b b , c 1 c_1 , … \dots , c g c_g an A A -sequence, J J = ( b , c 1 , … , c g ) A (b,c_1,\dots ,c_g)A , and B B = A [ J / b ] A[J/b] . Then B B is Cohen-Macaulay, there is a natural one-to-one correspondence between the sets Ass B ⁡ ( B / b B ) \operatorname {Ass}_B(B/bB) and Ass A ⁡ ( A / J ) \operatorname {Ass}_A(A/J) , and each q q ∈ \in Ass A ⁡ ( A / J ) \operatorname {Ass}_A(A/J) has height g + 1 g+1 . If B B does not have unique factorization, then some height-one prime ideals P P of B B are not principal. These primes are identified in terms of J J and P ∩ A P \cap A , and we consider the question of how far from principal they can be. If A A is integrally closed, necessary and sufficient conditions are given for B B to be integrally closed, and sufficient conditions are given for B B to be a UFD or a Krull domain whose class group is torsion, finite, or finite cyclic. It is shown that if P P is a height-one prime ideal of B B , then P ∩ A P \cap A also has height one if and only if b b ∉ \notin P P and thus P ∩ A P \cap A has height one for all but finitely many of the height-one primes P P of B B . If A A has unique factorization, a description is given of whether or not such a prime P P is a principal prime ideal, or has a principal primary ideal, in terms of properties of P ∩ A P \cap A . A similar description is also given for the height-one prime ideals P P of B B with P ∩ A P \cap A of height greater than one, if the prime factors of b b satisfy a mild condition. If A A is a UFD and b b is a power of a prime element, then B B is a Krull domain with torsion class group if and only if J J is primary and integrally closed, and if this holds, then B B has finite cyclic class group. Also, if J J is not primary, then for each height-one prime ideal p p contained in at least one, but not all, prime divisors of J J , it holds that the height-one prime p A [ 1 / b ] ∩ B pA[1/b] \cap B has no principal primary ideals. This applies in particular to the Rees ring R {\mathbf R} = = A [ 1 / t , t J ] A[1/t, tJ] . As an application of these results, it is shown how to construct for any finitely generated abelian group G G , a monoidal transform B B = A [ J / b ] A[J/b] such that A A is a UFD, B B is Cohen-Macaulay and integrally closed, and G G ≅ \cong Cl ⁡ ( B ) \operatorname {Cl}(B) , the divisor class group of B B .

This chapter develops tools for a more penetrating study of algebraic number theory than was possible in Chapter V and concludes by formulating two of the main three theorems of Chapter V in the modern setting of “adeles” and “ideles” commonly used in the subject.Sections 1–5 introduce discrete valuations, absolute values, and completions for fields, always paying attention to implications for number fields and for certain kinds of function fields. Section 1 contains a prototype for all these notions in the construction of the fieldQp of p-adic numbers formed out of the rationals. Discrete valuations in Section 2 are a generalization of the order-of-vanishing function about a point in the theory of one complex variable. Absolute values in Section 3 are real-valued multiplicative functions that give a metric on a field, and the pair consisting of a field and an absolute value is called a valued field. Inequivalent absolute values have a certain independence property that is captured by the Weak Approximation Theorem. Completions in Section 4 are functions mapping valued fields into their metric-space completions. Section 5 concerns Hensel’s Lemma, which in its simplest form allows one to lift roots of polynomials over finite prime fields Fp to roots of corresponding polynomials over p-adic fields Qp.Section 6 contains the main theorem for investigating the fundamental question of how prime ideals split in extensions. Let K be a finite separable extension of a field F, let R be a Dedekind domain with field of fractions F, and let T be the integral closure of R in K. The question concerns the factorization of an ideal pT in T when p is a nonzero prime ideal in R. If Fp denotes the completion of F with respect to p, the theorem explains how the tensor product K ⊗F Fp splits uniquely as a direct sum of completions of valued fields. The theorem in effect reduces the question of the splitting of pT in T to the splitting of Fp in a complete field in which only one of the prime factors of pT plays a role.Section 7 is a brief aside mentioning additional conclusions one can draw when the extension K/F is a Galois extension.Section 8 applies the main theorem of Section 6 to an analysis of the different of K/F and ultimately to the absolute discriminant of a number field. With the new sharp tools developed in the present chapter, including a Strong Approximation Theorem that is proved in Section 8, a complete proof is given for the Dedekind Discriminant Theorem; only a partial proof had been accessible in Chapter V.Sections 9–10 specialize to the case of number fields and to function fields that are finite separable extensions of Fq (X), where Fq is a finite field. The adele ring and the idele group are introduced for each of these kinds of fields, and it is shown how the original field embeds discretely in the adeles and how the multiplicative group embeds discretely in the ideles. The main theorems are compactness theorems about the quotient of the adeles by the embedded field and about the quotient of the normalized ideles by the embedded multiplicative group. Proofs are given only for number fields. In the first case the compactness encodes the Strong Approximation Theorem of Section 8 and the Artin product formula of Section 9. In the second case the compactness encodes both the finiteness of the class number and the Dirichlet Unit Theorem.KeywordsPrime IdealValuation RingBasic AlgebraDedekind DomainFractional IdealThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

For every prime integer p, and for every number field K defined by a p-regular polynomial, the form of the factorization of the principal ideal pℤK into prime ideals of ℤK is given. To illustrate the potential applications of this factorization, we derive from this result an explicit description of the factorization of pℤK, where K is a quartic number field defined by an irreducible polynomial X4 + aX + b ∈ ℤ[X].

On periods modulo p in arithmetic dynamics

Based on Newton polygon techniques, for every prime integer p, a p-integral basis of ℤK, and the factorization of the principal ideal pℤK into prime ideals of ℤK are given, where K is a quintic number field defined by an irreducible trinomial X5 + aX2 + b ∈ ℤ[X].

Further results are flowing like a stream. By early 2018 the principal works which have formed the core of this book had collectively received over 210 citations registered on MathSciNet. In this chapter we summarize some of these results, depending also on the list of Andrew Granville. The material covered is but an indication of the fundamental nature of the breakthroughs which have been described, and much more is expected in the future. This chapter gives a description of the results but without proofs. The topics include bounded gaps for primes described by affine forms, clusters of primes in intervals, gaps between almost primes, clusters of primes in intervals, the set of limit points of the sequence of normalized consecutive prime differences, Artin’s primitive root conjecture, arithmetic progressions of primes with a fixed common difference, prime ideals in the ring of integers of number fields, irreducible polynomials, the coefficients of a class of modular forms including Ramanujan’s tau-function form, quadratic twists of a class of elliptic curves including the congruent number elliptic curve, and results obtained assuming the Elliott–Halberstam and generalized Elliott–Halberstam conjectures. The results often take the form of establishing bounded gaps for primes of a particular form.

Various bounds on the absolute values of $L$-functions of number fields at $s=1$ and on residues at $s=1$ of Dedekind zeta functions of a number field $\mathbb {L}$ are known. Also, better bounds depending on the splitting behavior of given prime ideals o

Let K be a fixed number field, assumed to be Galois over ℚ. Let r and f be fixed integers with f positive. Given an elliptic curve E, defined over K, we consider the problem of counting the number of degree f prime ideals of K with trace of Frobenius equal to r. Except in the case f = 2, we show that ‘on average,’ the number of such prime ideals with norm less than or equal to x satisfies an asymptotic identity that is in accordance with standard heuristics. This work is related to the classical Lang–Trotter conjecture and extends the work of several authors.

This paper introduces a novel framework for constructing algebraic lattices based on Construction-D, leveraging nested linear codes and prime ideals from algebraic number fields. We focus on the application of these lattices in block-fading (BF) channels, which are characterized by piecewise-constant fading across blocks of transmitted symbols. This approach results in a semi-systematic generator matrix, providing a structured foundation for high -dimensional lattice design for BF channels. The proposed Construction-D lattices exhibit the full diversity property, making them highly effective for error performance improvement. To address this, we develop an efficient decoding algorithm designed specifically for full-diversity Construction-D lattices. Simulations indicate that the proposed lattices notably enhance error performance compared to full-diversity Construction-A lattices, especially in high-diversity scenarios. Interestingly, the anticipated performance gain from increasing the number of nested code levels was only observed in the two-level case. We believe that this is due to the amplified impact of error propagation during successive cancellation at higher levels. These findings highlight the promise of Construction-D lattices as an effective and practical coding strategy for enhancing communication reliability in BF channels.

On a class of ideals in an algebraic number field

We show that there are finitely many imaginary quadratic number fields for which the class group has exponent 5. Indeed there are finitely many with exponent at most 6. The proof is based on a method of Pierce [Pierce L. B.: A bound for the 3-part of class numbers of quadratic fields by means of the square sieve. Forum Math. 18 (2006), 677–698]. The problem is reduced to one of counting integral points on a certain affine surface. This is tackled using the author's “square-sieve”, in conjunction with estimates for exponential sums. The latter are derived using the q-analogue of van der Corput's method.

In this paper a probabilistic algorithm is exhibited that factors any positive integer n n into prime factors in expected time at most L n [ 1 2 , 1 + o ( 1 ) ] {L_n}[\tfrac {1}{2},1 + o(1)] for n → ∞ n \to \infty , where L x [ a , b ] = exp ( b ( log ⁡ x ) a ( log log ⁡ x ) 1 − a ) {L_x}[a,b] = {\text {exp}}(b{(\log x)^a}{({\text {log}}\log x)^{1 - a}}) . Many practical factoring algorithms, including the quadratic sieve and the elliptic curve method, are conjectured to have an expected running time that satisfies the same bound, but this is the first algorithm for which the bound can be rigorously proved. Nevertheless, this does not close the gap between rigorously established time bounds and merely conjectural ones for factoring algorithms. This is due to the advent of a new factoring algorithm, the number field sieve, which is conjectured to factor any positive integer n n in time L n [ 1 3 , O ( 1 ) ] {L_n}[\tfrac {1}{3},O(1)] . The algorithm analyzed in this paper is a variant of the class group relations method, which makes use of class groups of binary quadratic forms of negative discriminant. This algorithm was first suggested by Seysen, and later improved by A. K. Lenstra, who showed that the algorithm runs in expected time at most L n [ 1 2 , 1 + o ( 1 ) ] {L_n}[\tfrac {1}{2},1 + o(1)] if one assumes the generalized Riemann hypothesis. The main device for removing the use of the generalized Riemann hypothesis from the proof is the use of multipliers. In addition a character sum estimate for algebraic number fields is used, with an explicit dependence on possible exceptional zeros of the corresponding L L -functions. Another factoring algorithm using class groups that has been proposed is the random class groups method. It is shown that there is a fairly large set of numbers that this algorithm cannot be expected to factor as efficiently as had previously been thought.