TY - JOUR

T1 - On the nonexistence of automorphic eigenfunctions of exponential growth on SL(3 , Z) \ SL(3 , R) / SO(3 , R)

AU - Miller, Stephen D.

AU - Trinh, Tien

N1 - Funding Information:
Funding was provided by National Science Foundation (Grant Nos. DMS-1500562 and DMS-1801417).
Funding Information:
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 was provided by National Science Foundation (Grant Nos. DMS-1500562 and DMS-1801417).
Publisher Copyright:
© 2019, Springer Nature Switzerland AG.

PY - 2019/12/1

Y1 - 2019/12/1

N2 - It is well-known that there are automorphic eigenfunctions on SL(2 , Z) \ SL(2 , R) / SO(2 , R) —such as the classical j-function—that have exponential growth and have exponentially growing Fourier coefficients (e.g., negative powers of q= e2 π i z, or an I-Bessel function). We show that this phenomenon does not occur on the quotient SL(3 , Z) \ SL(3 , R) / SO(3 , 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 ≥ 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.

AB - It is well-known that there are automorphic eigenfunctions on SL(2 , Z) \ SL(2 , R) / SO(2 , R) —such as the classical j-function—that have exponential growth and have exponentially growing Fourier coefficients (e.g., negative powers of q= e2 π i z, or an I-Bessel function). We show that this phenomenon does not occur on the quotient SL(3 , Z) \ SL(3 , R) / SO(3 , 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 ≥ 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.

KW - Automorphic forms

KW - Exponential growth

KW - Miatello-Wallach conjecture

KW - Moderate growth

KW - Whittaker

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U2 - 10.1007/s40993-019-0168-8

DO - 10.1007/s40993-019-0168-8

M3 - Article

AN - SCOPUS:85073939624

SN - 2363-9555

VL - 5

JO - Research in Number Theory

JF - Research in Number Theory

IS - 4

M1 - 31

ER -