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# 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)

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## Significance

Let πͺ 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 πͺ be the ring of *S*-integers in *K*. Morgan, Rapinchuck, and Sury [A. V. Morgan *et al*., *Algebra Number Theory* 12, 1949β1974 (2018)] have proved that if the group of units _{2}(πͺ) 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}(πͺ) 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.

## Footnotes

β΅

^{1}B.W.J. and Y.Z. contributed equally to this work.- β΅
^{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

- β΅
- β΅
- L. N. VaserΕ‘teΔn

_{2}over Dedekind rings of arithmetic type. Mat. Sb. (N.S.) 89, 313β322 (1972). - β΅
- A. V. Morgan,
- A. S. Rapinchuk,
- B. Sury

_{2}over rings of*S*-integers with infinitely many units. Algebra Number Theory 12, 1949β1974 (2018). - β΅
- B. W. Jordan,
- Y. Zaytman

- β΅
- G. Cooke,
- P. J. Weinberger

_{2}. Comm. Algebra 3, 481β524 (1975). - β΅
- H. W. Lenstra Jr

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_{2}

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