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Related Experiment Video

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On the bounded generation of arithmetic SL2.

Bruce W Jordan1, Yevgeny Zaytman2

  • 1Department of Mathematics, Baruch College, The City University of New York, New York, NY 10010-5526; bruce.jordan@baruch.cuny.edu.

Proceedings of the National Academy of Sciences of the United States of America
|September 6, 2019
PubMed
Summary
This summary is machine-generated.

Researchers reduced the number of elementary matrices needed to represent matrices in SL2(O) for number fields. This advancement in algebraic number theory offers a tighter bound for matrix decomposition in rings of S-integers.

Keywords:
SL2bounded generation

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Area of Science:

  • Algebraic Number Theory
  • Matrix Theory
  • Group Theory

Background:

  • The ring of S-integers (π’ͺ) in a number field (K) is a key object in algebraic number theory.
  • Understanding the structure of the general linear group SL2(π’ͺ) is crucial for number theory research.
  • Previous work established an upper bound of 9 elementary matrices for SL2(π’ͺ) decomposition.

Purpose of the Study:

  • To refine the upper bound on the number of elementary matrices required to represent any matrix in SL2(π’ͺ).
  • To explore the impact of specific number field properties (real embeddings, finite places) on this bound.
  • To investigate the effect of the generalized Riemann hypothesis on matrix decomposition length.

Main Methods:

  • Utilizing techniques from algebraic number theory and group theory.
  • Analyzing the structure of the group of units in the ring of S-integers.
  • Applying matrix decomposition methods within SL2(π’ͺ).

Main Results:

  • Established an improved upper bound of 8 elementary matrices for SL2(π’ͺ) under specific hypotheses (real embedding or finite place in S).
  • Demonstrated that under a generalized Riemann hypothesis, the bound can be further reduced to 5, 6, or 7 elementary matrices depending on field properties.
  • Provided concrete numerical bounds for matrix decomposition in SL2(π’ͺ).

Conclusions:

  • The number of elementary matrices needed for SL2(π’ͺ) decomposition is sensitive to the properties of the number field and the set of S-integers.
  • The generalized Riemann hypothesis significantly tightens the bounds for matrix decomposition.
  • This research contributes to a deeper understanding of the structure of SL2(π’ͺ) in algebraic number theory.