The Weil bound for generalized Kloosterman sums of half-integral weight

Let L be an even lattice of odd rank with discriminant group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>/</m:mo> <m:mi>L</m:mi> </m:mrow> </m:math> {L^{}/L} , and let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>α</m:mi> <m:mo>,</m:mo> <m:mi>β</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>/</m:mo> <m:mi>L</m:mi> </m:mrow> </m:mrow> </m:math> {α,β∈ L^{}/L} . We prove the Weil bound for the Kloosterman sums <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>S</m:mi> <m:mrow> <m:mi>α</m:mi> <m:mo>,</m:mo> <m:mi>β</m:mi> </m:mrow> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>m</m:mi> <m:mo>,</m:mo> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mi>c</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {Sα,β(m,n,c)} of half-integral weight for the Weil Representation attached to L . We obtain this bound by proving an identity that relates a divisor sum of Kloosterman sums to a sparse exponential sum. This identity generalizes Kohnen’s identity for plus space Kloosterman sums with the theta multiplier system.