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On sums of \(\mathit{SL}(3,\mathbb{Z})\) Kloosterman sums

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Abstract

We show that sums of the \(\mathit{SL}(3,\mathbb{Z})\) long element Kloosterman sum against a smooth weight function have cancelation due to the variation in argument of the Kloosterman sums, when each modulus is at least the square root of the other. Our main tool is Li’s generalization of the Kuznetsov formula on \(\mathit{SL}(3,\mathbb{R})\), which has to date been prohibitively difficult to apply. We first obtain analytic expressions for the weight functions on the Kloosterman sum side by converting them to Mellin–Barnes integral form. This allows us to relax the conditions on the test function and to produce a partial inversion formula suitable for studying sums of the long-element \(\mathit{SL}(3,\mathbb{Z})\) Kloosterman sums.

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Acknowledgements

The author would like to thank Prof. W. Duke for his guidance along the way, Prof. X. Li for a copy of her “Kloostermania” notes and accompanying advice in addition to some timely warnings, Prof. V. Blomer for the Kim–Sarnak result and other helpful comments, and Prof. P. Sarnak for bringing the Goldfeld–Sarnak result to his attention and some history on the Kloostermania.

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  1. University of California, 520 Portolla Plaza, Los Angeles, CA, 90095, USA

    Jack Buttcane

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  1. Jack Buttcane
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Correspondence to Jack Buttcane.

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During the time of this research, the author was supported by National Science Foundation DMS-10-01527 and VIGRE grants.

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Buttcane, J. On sums of \(\mathit{SL}(3,\mathbb{Z})\) Kloosterman sums. Ramanujan J 32, 371–419 (2013). https://doi.org/10.1007/s11139-013-9488-9

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