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Published on: June 19, 2010
Maass cusp forms
J M Deshouillers1, H Iwaniec, R S Phillips
1Université de Bordeaux I, F-33405 Talence, France.
Summary
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).
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.
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