On the evaluation of Salié sums

Tóth, Árpád

doi:10.1090/S0002-9939-04-07768-8

On the evaluation of Salié sums
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by Árpád Tóth

Proc. Amer. Math. Soc. 133 (2005), 643-645

DOI: https://doi.org/10.1090/S0002-9939-04-07768-8

Published electronically: October 7, 2004

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Abstract:

The Salié sum $S(m,n;c)$ can be evaluated as the product of a Gauss sum and an exponential sum involving square roots of $mn \bmod {c}$. We give a new proof of this fact that can simultaneously handle a twisted version of these sums that arise in the theory of half-integral weight modular forms.

References

W. Duke, On multiple Salié sums, Proc. Amer. Math. Soc. 114 (1992), no. 3, 623–625. MR 1077785, DOI 10.1090/S0002-9939-1992-1077785-2
Henryk Iwaniec, Fourier coefficients of modular forms of half-integral weight, Invent. Math. 87 (1987), no. 2, 385–401. MR 870736, DOI 10.1007/BF01389423
Salié, Hans. Über die Kloostermanschen Summen $S(u,v;q)$. (German) Math. Z. 34, 91-109, (1931).
Peter Sarnak, Some applications of modular forms, Cambridge Tracts in Mathematics, vol. 99, Cambridge University Press, Cambridge, 1990. MR 1102679, DOI 10.1017/CBO9780511895593
Goro Shimura, On modular forms of half integral weight, Ann. of Math. (2) 97 (1973), 440–481. MR 332663, DOI 10.2307/1970831
Kenneth S. Williams, Note on Salié’s sum, Proc. Amer. Math. Soc. 30 (1971), 393–394. MR 284408, DOI 10.1090/S0002-9939-1971-0284408-5