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.
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Notes
Strictly speaking, formula (1.2) needs to be slightly adjusted when \(\nu =0\), by multiplying the second term by \(\log (y)\).
See [11, p. 415] for a precise statement, which includes some reducibility conditions to rule out products of automorphic functions on rank one groups.
The particular choice of the constants \(\frac{\sqrt{3}}{2}\) and \(\frac{3}{4}=(\frac{\sqrt{3}}{2})^2\) in (1.5) comes from the fact that region described by the inequalities contains a fundamental domain for \(SL(3,{\mathbb {Z}})\), but is not essential to the statement.
The existence of such a \(\gamma \) is equivalent to that of a relatively prime pair of integers (c, d) for which \(c^2y_1^2+d^2\) lies in the interval \((\frac{1}{4}|\frac{k}{\ell }|^{2/3},4|\frac{k}{\ell }|^{2/3})\); since \(|k|\ge |\ell |\) and \(y_1<1\), one can take \((c,d)=(0,1)\) when \(|\frac{k}{\ell }|<8\) and \((c,d)=(1,\lceil |\frac{k}{\ell }|^{1/3}\rceil )\) when \(|\frac{k}{\ell }|\ge 8\).
Of course the pointwise estimate \([P^{m,n}F](g)\rightarrow 0\) as \(m^2+n^2\rightarrow \infty \) holds by the Riemann-Lebesgue Lemma. We shall not require this fact, but remark that the tension between this decay and the growth of Whittaker functions appears to be a fundamental reason behind the truth of the Miatello–Wallach conjecture.
This is readily deduced from \( \left( {\begin{matrix}{1} &{} {\theta _\gamma } \\ {0} &{} {1} \end{matrix}}\right) \left( {\begin{matrix}{(c^2+d^2)^{-1}} &{} {0} \\ {0} &{} {c^2+d^2} \end{matrix}}\right) \left( {\begin{matrix}{1} &{} {0} \\ {\theta _g} &{} {1} \end{matrix}}\right) = \gamma \gamma ^t = \left( {\begin{matrix}{a^2+b^2} &{} {ac+bd} \\ {ac+bd} &{} {c^2+d^2} \end{matrix}}\right) \).
Suppose to the contrary that \(\theta _\gamma ={\textstyle {\frac{ac+bd}{c^2+d^2}}}\in \pm \frac{1}{4}+{\mathbb {Z}}\); it is clear c cannot be 0, for then \(d=\pm 1\). By adding integral multiplies of (c, d) to (a, b) (which does not change the coset representative of \( \left( {\begin{matrix}{a} &{} {b} \\ {c} &{} {d} \end{matrix}}\right) \in \varGamma _{\infty }^{(2)}\backslash \varGamma ^{(2)}\)), we may assume \({\textstyle {\frac{ac+bd}{c^2+d^2}}}=\frac{a}{c}-\frac{d}{c(c^2+d^2)}=\pm \frac{1}{4}\), or equivalently that \(4d=(c^2+d^2)(4a\mp c)\). However, \(c^2+d^2>4|d|\) if \(|d|>4\), in which case it certainly cannot divide 4|d|. Neither can \(c^2+9\) divide 12 (when \(|d|=3\)), nor \(c^2+4\) divide 8 (when \(|d|=2\)), since c must be odd when d is even. Is it trivial to see there are no solutions when \(|d|\le 1\) and \(c\ne 0\).
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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.
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Funding was provided by National Science Foundation (Grant Nos. DMS-1500562 and DMS-1801417).
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Miller, S.D., Trinh, T. On the nonexistence of automorphic eigenfunctions of exponential growth on \(SL(3,{\mathbb {Z}})\backslash SL(3,{\mathbb {R}})/SO(3,{\mathbb {R}})\). Res. number theory 5, 31 (2019). https://doi.org/10.1007/s40993-019-0168-8
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DOI: https://doi.org/10.1007/s40993-019-0168-8