Generating subgroups of ray class groups with small prime ideals

The goal of this paper is to prove theorems that elucidate the Cohen–Lenstra–Martinet conjectures for the distributions of class groups of number fields, and further the understanding of their implications. We start by giving a simpler statement of the conjectures. We show that the probabilities that arise are inversely proportional to the number of automorphisms of structures slightly larger than the class groups. We find the moments of the Cohen–Lenstra–Martinet distributions and prove that the distributions are determined by their moments. In order to apply these conjectures to class groups of non-Galois fields, we prove a new theorem on the capitulation kernel (of ideal classes that become trivial in a larger field) to relate the class groups of non-Galois fields to the class groups of Galois fields. We then construct an integral model of the Hecke algebra of a finite group, show that it acts naturally on class groups of non-Galois fields, and prove that the Cohen–Lenstra–Martinet conjectures predict a distribution for class groups of non-Galois fields that involves the inverse of the number of automorphisms of the class group as a Hecke-module.

It is conjectured that within the class group of any number field, for every integer $\ell \geq 1$, the $\ell$-torsion subgroup is very small (in an appropriate sense, relative to the discriminant of the field). In nearly all settings, the full strength of this conjecture remains open, and even partial progress is limited. Significant recent progress toward average versions of the $\ell$-torsion conjecture has crucially relied on counts for number fields, raising interest in how these two types of question relate. In this paper we make explicit the quantitative relationships between the $\ell$-torsion conjecture and other well-known conjectures: the Cohen-Lenstra heuristics, counts for number fields of fixed discriminant, counts for number fields of bounded discriminant (or related invariants), and counts for elliptic curves with fixed conductor. All of these considerations reinforce that we expect the $\ell$-torsion conjecture is true, despite limited progress toward it. Our perspective focuses on the relation between pointwise bounds, averages, and higher moments, and demonstrates the broad utility of the method of moments.

Reflection principles for class groups

We describe classical and recent results concerning the structure of class groups of number fields as modules over the Galois group. When presenting more modern developments, we can only hint at the much broader context and the very powerful general techniques that are involved, but we endeavour to give complete statements or at least examples where feasible. The timeline goes from a classical result proved in 1890 (Stickelberger’s Theorem) to a recent (2020) breakthrough: the proof of the Brumer-Stark conjecture by Dasgupta and Kakde.

LetG be an abelian group of ordern and letR be a commutative ring which admits a homomorphism ℤ[ζ n ] →R, where ζ n is a (complex) primitiven-th root of unity. Given a finiteR[G]-moduleM, we derive a formula relating the order ofM to the product of the orders of the various isotypic componentsM x ofM, where ξ ranges over the group ofR-valued characters ofG. ForG cyclic, we give conditions under which the order ofM is exactly equal to the product of the orders of theM x. To derive these conditions, we build on work of Aljadeff and Ginosar and obtain, in particular, a new criterion for cohomological triviality which improves upon the well-known critetion of T. Nakayama. We also give applications to abelian varieties and to ideal class groups of number fields, obtaining in particular some new class number relations. In an Appendix to the paper, we use étale cohomology to obtain some additional class number relations. Our results also have applications to “non-semisimple” Iwasawa theory, but we do not develop these here. In general, the results of this paper could be used to strengthen a variety of known results involving finiteR[G]-modules whose hypotheses include (an equivalent forn of) the following assumption: “the order ofG is invertible inR”.

We present an algorithm for the computation of logarithmic l- class groups of number fields. Our principal motivation is the effective determination of the l-rank of the wild kernel in the K-theory of number fields.

In a well-known paper, Cohen and Lenstra gave conjectures on class groups of number fields. We give here similar conjectures for Tate-Shafarevitch groups of elliptic curves defined over Q. For such groups (if they are finite), there exists a nondegenerate, alternating, bilinear pairing. We give some properties of such groups and then formulate heuristics which allow us to give precise conjectures.

The purpose of this note is twofold. First, we survey results from Levin (J Theor Nombres Bordeaux 19:485–499, 2007), Gillibert and Levin (Math Res Lett 19:1–14, 2012) and Bilu and Gillibert (Israel J Math, in press) on the construction of large class groups of number fields by specialization of finite covers of curves. Then we give examples of applications of these techniques.

It is well known that the set of all ideals(2) of a ring forms a complete modular lattice with respect to set inclusion. The same is true of the set of all right ideals. Our purpose in this paper is to consider the consequences of imposing certain additional restrictions on these ideal lattices. In particular, we discuss the case in which one or both of these lattices is complemented, and the case in which one or both is distributive. In §1 two strictly latticetheoretic results are noted for the sake of their application to the complemented case. In §2 rings which have a complemented ideal lattice are considered. Such rings are characterized as discrete direct sums of simple rings. The structure space of primitive ideals of such rings is also discussed. In §3 corresponding results are obtained for rings whose lattice of right ideals is complemented. In particular, it is shown that a ring has a complemented right ideal lattice if and only if it is isomorphic with a discrete direct sum of quasi-simple rings. The socle [7](3) and the maximal regular ideal [5] are discussed in connection with such rings. The effect of an identity element is considered in §4. In §5 rings with distributive ideal lattices are considered and still another variant of regularity [20] is introduced. It is shown that a semi-simple ring with a distributive right ideal lattice is isomorphic with a subdirect sum of division rings. In the concluding section a type of ideal, introduced by L. Fuchs [9] in connection with commutative rings with distributive ideal lattice, and which we call strongly irreducible, is considered. Some properties of these ideals, analogous to corresponding ones for prime ideals [19], are developed. Finally, it is observed that a topology may be introduced in the set of all proper strongly irreducible ideals in such a way that the resulting space contains the spaces of prime [19] and primitive [13] ideals as subspaces. I wish to take this opportunity to thank Professor M. F. Smiley for the encouragement and many helpful suggestions he has given me throughout the preparation of this paper. 1. Some lattice-theoretic preliminaries. In this section we state two results of a strictly lattice-theoretic nature with a view toward subsequent applications to rings. Our notation and terminology is that of Birkhoff [3]. In

We show that finding small solutions to random modular linear equations is at least as hard as approximating several lattice problems in the worst case within a factor almost linear in the dimension of the lattice. The lattice problems we consider are the shortest vector problem, the shortest independent vectors problem, the covering radius problem, and the guaranteed distance decoding problem (a variant of the well‐known closest vector problem). The approximation factor we obtain is $n \log^{O(1)} n$ for all four problems. This greatly improves on all previous work on the subject starting from Ajtai’s seminal paper [Generating hard instances of lattice problems, in Complexity of Computations and Proofs, Quad. Mat. 13, Dept. Math., Seconda Univ. Napoli, Caserta, Italy, 2004, pp. 1–32] up to the strongest previously known results by Micciancio [SIAM J. Comput., 34 (2004), pp. 118–169]. Our results also bring us closer to the limit where the problems are no longer known to be in NP intersect coNP. Our main tools are Gaussian measures on lattices and the high‐dimensional Fourier transform. We start by defining a new lattice parameter which determines the amount of Gaussian noise that one has to add to a lattice in order to get close to a uniform distribution. In addition to yielding quantitatively much stronger results, the use of this parameter allows us to simplify many of the complications in previous work. Our technical contributions are twofold. First, we show tight connections between this new parameter and existing lattice parameters. One such important connection is between this parameter and the length of the shortest set of linearly independent vectors. Second, we prove that the distribution that one obtains after adding Gaussian noise to the lattice has the following interesting property: the distribution of the noise vector when conditioning on the final value behaves in many respects like the original Gaussian noise vector. In particular, its moments remain essentially unchanged.

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.

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.

The security of public-key cryptography relies on well-studied hard problems, problems for which we do not have efficient algorithms. Factorization and discrete logarithm are the two most known and used hard problems. Unfortunately, they can be easily solved on a quantum computer by Shor's algorithm. Also, the research area of cryptography demands for crypto-diversity which says that we should offer a range of hard problems for public-key cryptography. If one hard problem proves to be easy, we should be able to provide alternative solutions. Some of the candidates for post-quantum hard problems, i.e. problems which are believed to be hard even on a quantum computer, are the Learning Parity with Noise (LPN), the Learning with Errors (LWE) and the Shortest Vector Problem (SVP). A thorough study of these problems is needed in order to assess their hardness. In this thesis we focus on the algorithmic study of LPN. LPN is a hard problem that is attractive, as it is believed to be post-quantum resistant and suitable for lightweight devices. In practice, it has been employed in several encryption schemes and authentication protocols. At the beginning of this thesis, we take a look at the existing LPN solving algorithms. We provide the theoretical analysis that assesses their complexity. We compare the theoretical results with practice by implementing these algorithms. We study the efficiency of all LPN solving algorithms which allow us to provide secure parameters that can be used in practice. We push further the state of the art by improving the existing algorithms with the help of two new frameworks. In the first framework, we split an LPN solving algorithm into atomic steps. We study their complexity, how they impact the other steps and we construct an algorithm that optimises their use. Given an LPN instance that is characterized by the noise level and the secret size, our algorithm provides the steps to follow in order to solve the instance with optimal complexity. In this way, we can assess if an LPN instance provides the security we require and we show what are the secure instances for the applications that rely on LPN. The second framework handles problems that can be decomposed into steps of equal complexity. Here, we assume that we have an adversary that has access to a finite or infinite number of instances of the same problem. The goal of the adversary is to succeed in just one instance as soon as possible. Our framework provides the strategy that achieves this. We characterize an LPN solving algorithm in this framework and show that we can improve its complexity in the scenario where the adversary is restricted. We show that other problems, like password guessing, can be modeled in the same framework.

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.

Diophantine equations are a popular and active area of research in number theory. In this paper we consider Mordell equations, which are of the form $y^2=x^3+d$, where $d$ is a (given) nonzero integer number and all solutions in integers $x$ and $y$ have to be determined. One non-elementary approach for this problem is the resolution via descent and class groups. Along these lines we formalized in Lean 3 the resolution of Mordell equations for several instances of $d<0$. In order to achieve this, we needed to formalize several other theories from number theory that are interesting on their own as well, such as ideal norms, quadratic fields and rings, and explicit computations of the class number. Moreover we introduced new computational tactics in order to carry out efficiently computations in quadratic rings and beyond.