In this work we consider two problems – one concerns finding fundamental units in totally real cubic field extensions of and the other concerns total escape of mass of compact orbits of the diagonal group in the space of 3-dimensional lattices. While these problems seem unrelated, we manage to construct sequences of examples of cubic rings\compact orbits for which we can compute the unit group (fundamental units) on the one hand and show full escape of mass on the other.
Totally real extensions and integral units
Given a finite extension , and its integer ring , it is usually a very hard and interesting question to find its unit group . For example, in the case where , finding integral units is equivalent to solving Pell’s equation . The structure of is well known by Dirichlet’s unit theorem. Indeed, let be all the distinct embeddings of in (so that ) , where are real embeddings and are conjugate nonreal embeddings. Then Dirichlet’s unit theorem tell us that where is the group of roots of unity in . A system of integral units which span together with is called a system of fundamental units, and in general it is very hard to find such a system.
In the following we shall concentrate on totally real extensions, i.e. and . A more geometric interpretation of the problem above can be described as follows. Letting we define
It is well known that under this map is a lattice in and the discriminant of is equal to . If , then ,
and in particular if , then is in the stabilizer of in the group of all diagonal matrices. For simplicity, we shall identify between lattices which are reflections of one another through the principal hyperplanes, hence is invariant under . The other direction is true as well, i.e. the stabilizer of in the positive diagonal group is exactly . Moreover, for we have that , i.e. is in the group of determinant 1 diagonal matrices with nonnegative entries.
In order to study the phenomenon mentioned above we similarly define
so that is a unit if and only if , i.e. . With this notation, Dirichlet’s unit theorem states that is a lattice in (and its covolume is the regulator of up to a constant which depends on ). Moreover, it corresponds exactly to the stabilizer of in the diagonal group .
We call this lattice the Dirichlet unit shape of and we denote it by .
A similar construction can be done in a more general case. Recall that is called a full module if and and it is called an order if is a ring. Its stabilizer is defined to be , which is always a ring of rank in (i.e. it is an order). We then get that is a lattice in , and its Dirichlet shape (shape of the -orbit) is . We remark that while is an -dimensional lattice, its Dirichlet shape is an -dimensional lattice.
We shall denote the space of Dirichlet unit shapes of -dimensional full modules by . Since is not canonically isomorphic to , we shall only consider these lattices up to isometry, i.e. .
Compact -orbits and invariant measures
We now consider our second problem of escape of mass.
Once we fix a totally real expansion and a full module , we denote by the covolume normalization of in the space of unimodular lattices . Since is a lattice in , we obtain that is a compact -orbit. The other direction is true as well – any compact -orbit in contains a point of the form as above. More over, for any arising from full modules, we have that if and only if are full modules in the same field extension and for some . In particular, if is an integral ideal, then exactly means that are in the same class in the class group. Thus, in a sense, the study of -dimensional full modules up to principal modules (e.g. principal ideals) is equivalent to study compact -orbits in .
Any compact -orbit as above supports a unique -invariant probability measure . Given a sequence of compact -orbits we can ask what can be said about a weak limit of , if such a limit exists. More generally, we can consider limits of measures of the form where is a sequence of finite families of compact -orbits. One natural choice of such families, is to take the (finite) families which correspond to class groups for a sequence of fields (or more generally sequence of orders). In dimension 2, it is known that this sequence equidistributes, i.e. it converges to the -invariant measure on . This result implies Linnik’s theorem which states that the rational solutions to as equidistribute when projected on the unit sphere (see  for details). The same equidistribution result is known also in dimension 3 (see ), and ergodicity arguments shows that “most” choices of sequences where are compact -orbits coming from some fixed sequence of orders equidistribute. Given this statement, is there a sequence of distinct orbits for which doesn’t equidistribute? In particular, is there such a sequence which has full (or partial) escape of mass, i.e. (resp. some partial weak limit is not a probability measure)?
In this work we specialize to the case of totally real cubic extensions, in which case the Dirichlet unit shapes are 2-dimensional lattice. Given an irreducible cubic monic integral polynomial , and one of its root , we denote .
Definition: We say that a pair of nonzero integers is a mutually cubic root pair if they satisfy the congruence conditions and . A sequence is called a mutually cubic root sequence if is a mutually cubic root pair for all and as .
Theorem: Given a mutually cubic root pair , there exists a sequence of polynomials such that for all big enough is generated by . Moreover exhibit full escape of mass and the sequence converges to the triangles lattice (the lattice where is a primitive 3rd root of unity).
Remark: There are many cubic root pairs. For example is such a pair for any . Less trivial pairs are .
Theorem: Given a mutually cubic root sequence , let . Then contains a curve which depends on . In addition, the set of all possible limits for mutually cubic root sequences is infinite.
The mutually cubic root sequence that we found correspond to the curves in in the figure below.
- Let be the set of all possible limits of for mutually cubic root sequences . We can show that this set is infinite (and with infinitely many accumulation points). Is it true that has a nonempty interior? A positive solution to this problem will imply that has a nonempty interior.
- Below are images generated in sage which show the Dirichlet unit shapes (in ) of all integer rings of totally real cubic extensions with discriminant bounded by . In particular we note that it seems that is dense in the space of 2-dimensional lattices (up to rotation), and moreover it seems that these points equidistribute with respect to the discriminant.
- O. David and U. Shapira, “Dirichlet shapes of unit lattices and escape of mass“, International Mathematics Research Notices (January 2017): rnw324.
DOI: 10.1093/imrn/rnw324, arXiv: arXiv:1607.04048 [math.NT]
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