## New Research In

### Physical Sciences

### Social Sciences

#### Featured Portals

#### Articles by Topic

### Biological Sciences

#### Featured Portals

#### Articles by Topic

- Agricultural Sciences
- Anthropology
- Applied Biological Sciences
- Biochemistry
- Biophysics and Computational Biology
- Cell Biology
- Developmental Biology
- Ecology
- Environmental Sciences
- Evolution
- Genetics
- Immunology and Inflammation
- Medical Sciences
- Microbiology
- Neuroscience
- Pharmacology
- Physiology
- Plant Biology
- Population Biology
- Psychological and Cognitive Sciences
- Sustainability Science
- Systems Biology

# On the bounded generation of arithmetic SL_{2}

Edited by Kenneth A. Ribet, University of California, Berkeley, CA, and approved July 30, 2019 (received for review May 3, 2019)

## Significance

Let 𝒪 be the ring of integers in a number field with a finite number of prime ideals inverted. Whether every

## Abstract

Let *K* be a number field and *S* be a finite set of primes of *K* containing the archimedean valuations. Let 𝒪 be the ring of *S*-integers in *K*. Morgan, Rapinchuck, and Sury [A. V. Morgan *et al*., *Algebra Number Theory* 12, 1949–1974 (2018)] have proved that if the group of units _{2}(𝒪) is a product of at most 9 elementary matrices. We prove that under the additional hypothesis that *K* has at least 1 real embedding or *S* contains a finite place we can get a product of at most 8 elementary matrices. If we assume a suitable generalized Riemann hypothesis, then every matrix in SL_{2}(𝒪) is the product of at most 5 elementary matrices if *K* has at least 1 real embedding, the product of at most 6 elementary matrices if *S* contains a finite place, and the product of at most 7 elementary matrices in general.

## 1. Introduction

Let K be a number field and S be a finite set of primes of K containing the archimedean valuations. Denote by

Consider the case where K is the field of rational numbers Q. Taking

The key difference between Z and

### Theorem 1.1 (Morgan, Rapinchuk, and Sury)

*Assume that the group of units* *is infinite. Then every matrix in* *can be written as a product of at most* 9 *elementary matrices with the first one lower triangular.*The lower triangular assertion follows from their proof (ref. 3, equation 21 and following text).

Here we prove 2 theorems on a matrix

### Theorem 1.2

*Suppose that* S *contains a finite place or suppose that the group of units* *is infinite and* K *has at least 1 real embedding. Then* *can be written as the product of at most 8 elementary matrices with the first one lower triangular.*

### Theorem 1.3

*Assume that the group of units* *is infinite and assume the Generalized Riemann Hypothesis* 3.7*. Then* *can be written as the product of at most 5 elementary matrices if* K *has at least 1 real embedding*, *the product of at most 6 elementary matrices if* S *contains a finite place*, *and the product of at most 7 elementary matrices in general with the first one lower triangular in each case.*

We give diophantine applications of *Theorems 1.2* and *1.3* in ref. 4. These applications require us to know that the first matrix in our factorization into elementary matrices can be taken to be lower triangular. Hence we keep track of this here, whereas it is not a concern in ref. 3.

## 2. Theorem 1.2

### A. Reducing the First Row of a Matrix A ∈ S L 2 ( 𝒪 )

Following ref. 3, section 4, let**1** is

The following succinct notation using only the first rows of matrices is convenient:

#### Definition 2.1:

For

If there exist

### B. The Proof of Theorem 1.2

First we need the following *Lemma 2.3*, which requires a definition.

### Definition 2.2 (ref. 3, section 3.1):

A prime q of the number field K lying above the rational prime q is Q-split if

### Lemma 2.3 (cf. ref. 3, lemma 4.4).

*Suppose K has at least real embedding or S contains a finite place and (a,b)**. Then there exists**and infinitely many* Q*-split prime principal ideals*q*of*𝒪*with a generator**such that for any**we have*.

### Proof:

Let v be either a real place of K or a finite place in S. To simplify subsequent notation we use the convention that the valuation of an element

Let

For a prime w of K, denote by

Observe that by the reciprocity law

### Proof of Theorem 1.2:

Suppose S contains a finite place or *Lemma 2.3* instead of ref. 3, lemma 4.4. Thus we do not need to use ref. 3, lemma 4.3 and we end up showing

## 3. Theorem 1.3

### A. Division Chains

### Definition 3.1 (cf. ref. 5, section 2):

Let **2**. A division chain of length k starting with

### Remark 3.2:

The division chains of *Definition 3.1* are closely related to the row reductions of *Definition 2.1.* The division chain in Eq. **6** of length k starting with

### Lemma 3.3.

*We have* *for* *if and only if there exists a terminating division chain of length* 2 *starting with*

### B. Terminating Division Chains of Length 2

Consider the matrix*Lemma 3.3*, we have

### Proposition 3.4.

*for some*

### Proof:

Multiplying matrices verifies that

### Theorem 3.5.

*Let* A *be as in* Eq. **7** *and assume there is a terminating division chain of length* 2 *starting with* *. Then* A *can be written as a product of at most* 4 *elementary matrices with the first one lower triangular.*

### Proof:

From *Proposition 3.4* we have**8** becomes

□

## C. General Matrices in S L 2 ( 𝒪 ) .

### Theorem 3.6.

*Let* A *be as in* Eq. **7***. If there exists a terminating division chain of length* *starting at**then* A *can be written as the product of at most* *elementary matrices with the first one lower triangular.*

### Proof:

We proceed by induction on k. The *Theorem 3.5*.

Suppose k is odd. Then by the definition of a terminating division chain there exists

The k even case is handled similarly; only switch the roles of a and b as well as multiply by

Note that this construction is similar to that used in ref. 5, corollary 2.3 except ours is more efficient, so we end up with

### D. The Generalized Riemann Hypothesis and the Proof of Theorem 1.3

The relevant Riemann hypothesis is most clearly stated in ref. 6, theorem 3.1.

### Riemann Hypothesis 3.7.

*The* ζ *function of* *satisfies the Riemann hypothesis for all integers*

### Proof of Theorem 1.3:

Let 𝒪 be the S-integers in K and **2**. Assume *Hypothesis 3.7*. Then by ref. 5, theorem 2.2 there is a terminating division chain of length 5 starting with *Theorem 3.6*. □

Morgan, Rapinchuk, and Sury (ref. 3, proposition 5.1) show that if *Theorem 1.3* assuming *Hypothesis 3.7* would be strict.

## Acknowledgments

We sincerely thank the referee for corrections and improvements to the paper.

## Footnotes

↵

^{1}B.W.J. and Y.Z. contributed equally to this work.- ↵
^{2}To whom correspondence may be addressed. Email: bruce.jordan{at}baruch.cuny.edu.

Author contributions: B.W.J. and Y.Z. performed research and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Published under the PNAS license.

## References

- ↵
- ↵
- L. N. Vaseršteĭn

_{2}over Dedekind rings of arithmetic type. Mat. Sb. (N.S.) 89, 313–322 (1972). - ↵
- A. V. Morgan,
- A. S. Rapinchuk,
- B. Sury

_{2}over rings of*S*-integers with infinitely many units. Algebra Number Theory 12, 1949–1974 (2018). - ↵
- B. W. Jordan,
- Y. Zaytman

- ↵
- G. Cooke,
- P. J. Weinberger

_{2}. Comm. Algebra 3, 481–524 (1975). - ↵
- H. W. Lenstra Jr

_{2}

## Citation Manager Formats

## Sign up for Article Alerts

## Article Classifications

- Physical Sciences
- Mathematics