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Related Experiment Videos

Maass cusp forms.

J M Deshouillers1, H Iwaniec, R S Phillips

  • 1Université de Bordeaux I, F-33405 Talence, France.

Proceedings of the National Academy of Sciences of the United States of America
|June 1, 1985
PubMed
Summary
This summary is machine-generated.

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Researchers found that the scattering matrix for a generic group is unexpectedly of order 2. This suggests that such groups likely possess very few Maass cusp forms, impacting number theory research.

Area of Science:

  • Number Theory
  • Mathematical Physics

Background:

  • The study of scattering matrices and their properties is crucial in understanding the behavior of systems in various mathematical and physical contexts.
  • The properties of groups like Gamma subset of SL(2, R) are fundamental in areas such as spectral theory and automorphic forms.

Purpose of the Study:

  • To investigate the order of the scattering matrix varphi(s) for generic Gamma subset of SL(2, R) under standard assumptions.
  • To explore the implications of the scattering matrix order on the number of Maass cusp forms for cofinite Gamma.

Main Methods:

  • The research relies on standard mathematical assumptions, including extended Riemann hypotheses.
  • Analysis of the scattering matrix varphi(s) for generic Gamma subset of SL(2, R).

Related Experiment Videos

Main Results:

  • The scattering matrix varphi(s) for generic Gamma subset of SL(2, R) is shown to be unexpectedly of order 2.
  • This finding leads to a conjecture regarding the scarcity of Maass cusp forms for generic cofinite Gamma.

Conclusions:

  • The order of the scattering matrix provides significant insight into the structure of associated mathematical objects.
  • The conjecture suggests a potentially sparse distribution of Maass cusp forms for a broad class of groups.