Kloosterman Sum - Lifting of Kloosterman Sums

Lifting of Kloosterman Sums

Although the Kloosterman sums may not be calculated in general they may be "lifted" to algebraic number fields, which often yields more convenient formulas. Let be a squarefree integer with . Assume that for any prime factor p of m we have . Then for all integers a,b coprime to m we have

K(a,b; m) = (-1)^{\Omega(m)} \sum_{v,w\ \text{mod}\ m,\, v^2-\tau w^2\equiv ab\ \text{mod}\ m} e^{4\pi i v/m}.

Here is the number of prime factors of m counting multiplicity.The sum on the right can be reinterpreted as a sum over algebraic integers in the field . This formula is due to Yangbo Ye, inspired by Don Zagier and extending the work of Hervé Jacquet and Ye on the relative trace formula for . Indeed, much more general exponential sums can be lifted.

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