On periods modulo p in arithmetic dynamics

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].

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].

Assuming Generalized Riemann’s Hypothesis, Bach proved that the class group C ℓ K \mathcal C\!\ell _{\mathbf {K}} of a number field K \mathbf {K} may be generated using prime ideals whose norm is bounded by 12 log 2 ⁡ Δ K 12\log ^2\Delta _{\mathbf {K}} , and by ( 4 + o ( 1 ) ) log 2 ⁡ Δ K (4+o(1))\log ^2\Delta _{\mathbf {K}} asymptotically, where Δ K \Delta _{\mathbf {K}} is the absolute value of the discriminant of K \mathbf {K} . Under the same assumption, Belabas, Diaz y Diaz and Friedman showed a way to determine a set of prime ideals that generates C ℓ K \mathcal C\!\ell _{\mathbf {K}} and which performs better than Bach’s bound in computations, but which is asymptotically worse. In this paper we show that C ℓ K \mathcal C\!\ell _{\mathbf {K}} is generated by prime ideals whose norm is bounded by the minimum of 4.01 log 2 ⁡ Δ K 4.01\log ^2\Delta _{\mathbf {K}} , 4 ( 1 + ( 2 π e γ ) − N K ) 2 log 2 ⁡ Δ K 4\big (1+\big (2\pi e^{\gamma })^{-N_{\mathbf {K}}}\big )^2\log ^2\Delta _{\mathbf {K}} and 4 ( log ⁡ Δ K + log ⁡ log ⁡ Δ K − ( γ + log ⁡ 2 π ) N K + 1 + ( N K + 1 ) log ⁡ ( 7 log ⁡ Δ K ) log ⁡ Δ K ) 2 4\big (\log \Delta _{\mathbf {K}} +\log \log \Delta _{\mathbf {K}}-(\gamma +\log 2\pi )N_{\mathbf {K}}+1+(N_{\mathbf {K}}+1)\frac {\log (7\log \Delta _{\mathbf {K}})} {\log \Delta _{\mathbf {K}}}\big )^2 . Moreover, we prove explicit upper bounds for the size of the set determined by Belabas, Diaz y Diaz and Friedman’s algorithms, confirming that it has size ≍ ( log ⁡ Δ K log ⁡ log ⁡ Δ K ) 2 \asymp (\log \Delta _{\mathbf {K}}\log \log \Delta _{\mathbf {K}})^2 . In addition, we propose a different algorithm which produces a set of generators which satisfies the above mentioned bounds and in explicit computations turns out to be smaller than log 2 ⁡ Δ K \log ^2\Delta _{\mathbf {K}} except for 7 7 out of the 31292 31292 fields we tested.

The Pólya group [Formula: see text] of an algebraic number field [Formula: see text] is the subgroup of the ideal class group [Formula: see text] generated by the ideal classes of the products of prime ideals of the same norm. If [Formula: see text] is trivial, then the number field [Formula: see text] is said to be a Pólya field. In the first part of this paper, we extend a recent result of ours and prove the existence of infinitely many number fields having Pólya groups as large as possible and those number fields being the compositum of a simplest cubic field and a quintic field. Using a result of Zylinski, we also prove that those fields are of index [Formula: see text]. Next, we produce three new families [Formula: see text], [Formula: see text] and [Formula: see text] of totally real bi-quadratic Pólya fields [Formula: see text] involving prime numbers [Formula: see text] and [Formula: see text] that satisfy certain quadratic residue conditions among themselves. It is worthwhile to note that in each of these fields, exactly five primes ramify in [Formula: see text] and this is the maximum possible number of ramified primes in a bi-quadratic Pólya field over [Formula: see text].

In this paper we shall study some problems which are purely settheoretical in character, but which arose in connection with a problem in the theory of models. The principal results of this paper form an important part of the proofs of the series of results in the theory of models which are announced in [1], [2], [3], and in the appendix of [4]. The present article, however, is entirely free of any notions from the theory of models. It is well-known that, for every infinite cardinal a, there exist nonprincipal prime ideals in the field S(X) of all subsets of a set X of power a, and furthermore there are exactly 221 such prime ideals. The object of this paper is to show that non-principal prime ideals with certain additional properties exist. Let X be a set of power a, and let S( Y) be the set of all non-empty finite subsets of a set Y. If f, g map S.( Y) into the field of subsets of X, we write g > f if, for all a E S.( Y), g(a) -f(a). Let us say that an ideal I in the field S(X) is SJ( Y)-good if, for every monotonic function f on S,(Y) into I, there exists an additive function g > f on S,( Y) into I. If Y is countable, then every ideal I is S.( Y)-good (cf. Corollary 4.2). Our two main results are as follows.

We consider the class of all commutative reduced rings for which there exists a finite subset T ⊂ A such that all projections on quotients by prime ideals of A are surjective when restricted to T. A complete structure theorem is given for this class of rings,and it is studied its relation with other finiteness conditions on the quotients of a ring over its prime ideals.

In this paper, we prove a weak form of the conjecture generalised to algebraic number fields. Given integers satisfying , Stewart and Yu were able to give an exponential bound in terms of the radical over the integers (Stewart and Yu [Math. Ann. 291 (1991), 225–230], Stewart and Yu [Duke Math. J. 108 (2001), no. 1, 169–181]), whereas Győry was able to give an exponential bound in the algebraic number field case for the projective height in terms of the radical for algebraic numbers (Győry [Acta Arith. 133 (2008), 281–295]). We generalise Stewart and Yu's method to give an improvement on Győry's bound for algebraic integers over the Hilbert Class Field of the initial number field K. Given algebraic integers in a number field K satisfying , we give an upper bound for the logarithm of the projective height in terms of norms of prime ideals dividing , where L is the Hilbert Class Field of K. In many cases, this allows us to give a bound in terms of the modified radical as given by Masser (Proc. Amer. Math. Soc. 130 (2002), no. 11, 3141–3150). Furthermore, by employing a recent bound of Győry (Publ. Math. Debrecen 94 (2019), 507–526) on the solutions of S‐unit equations, our estimates imply the upper bound where is an effectively computable constant. Further, given conditions on the largest prime ideal dividing , we obtain a sub‐exponential bound for in terms of the radical. Independently, as a direct application of his bounds on the solutions of S‐unit equations(Győry ([Publ. Math. Debrecen 94 (2019), 507–526]), Győry (Publ. Math. Debrecen 100 (2022), 499–511) also attains results mentioned above, including the above inequality, but over the base field K, as discussed in Section 6. As a consequence of our results, we will give an application to the effective Skolem–Mahler–Lech problem and give an improvement to a result by Lagarias and Soundararajan (J. Théor. Nombres Bordeaux 23 (2011), no. 1, 209–234) on the XYZ conjecture.

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

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.

If a group Γ is generated by a finite subset 5, then one has the gs, where gs(m) is the number of distinct elements of Γ expressible as words of length <m on 5. Roughly speaking, J. Milnor [9] shows that the asymptotic behaviour of gs does not depend on choice of finite generating set S c Γ, and that lower (resp. upper) bounds on the curvature of a riemannian manifold M result in upper (resp. lower) bounds on the growth function of π^M). The types of bounds on the growth function are