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Research in Number Theory

, 5:31 | Cite as

On the nonexistence of automorphic eigenfunctions of exponential growth on \(SL(3,{\mathbb {Z}})\backslash SL(3,{\mathbb {R}})/SO(3,{\mathbb {R}})\)

  • Stephen D. MillerEmail author
  • Tien Trinh
Research
  • 12 Downloads

Abstract

It is well-known that there are automorphic eigenfunctions on \(SL(2,{\mathbb {Z}})\backslash SL(2,{\mathbb {R}})/SO(2,{\mathbb {R}})\)—such as the classical j-function—that have exponential growth and have exponentially growing Fourier coefficients (e.g., negative powers of \(q=e^{2\pi i z}\), or an I-Bessel function). We show that this phenomenon does not occur on the quotient \(SL(3,{\mathbb {Z}})\backslash SL(3,{\mathbb {R}})/SO(3,{\mathbb {R}})\) and eigenvalues in general position (a removable technical assumption). More precisely, if such an automorphic eigenfunction has at most exponential growth, it cannot have non-decaying Whittaker functions in its Fourier expansion. This confirms part of a conjecture of Miatello and Wallach, who assert all automorphic eigenfunctions on this quotient (among other rank \(\ge 2\) examples) always have moderate growth. We additionally confirm their conjecture under certain natural hypotheses, such as the absolute convergence of the eigenfunction’s Fourier expansion.

Keywords

Automorphic forms Moderate growth Exponential growth Whittaker Miatello-Wallach conjecture 

Notes

Acknowledgements

The authors would like to thank Nolan Wallach for his generous discussions and advice, out of which key ideas emerged. The authors would also like to thank Daniel Bump, Bill Casselman, Dorian Goldfeld, Peter Sarnak, Wilfried Schmid, Eric Stade, Nicolas Templier, Akshay Venkatesh, and Gregg Zuckerman for their guidance on various aspects of growth estimates.

Funding

Funding was provided by National Science Foundation (Grant Nos. DMS-1500562 and DMS-1801417).

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Rutgers UniversityPiscatawayUSA
  2. 2.Hanoi National University of EducationHanoiVietnam

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