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  1. ; (, Research in number theory)
    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=e2πiz, 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. 
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