ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

3. Number Theory and Arithmetic Algebraic Geometry

1991 MS Classification: 11 Miatello, Roberto, Pacharoni, Inés, U. Córdoba, Argentina, Bruggeman, Roelof, U. Utrecht, Estimates for Kloosterman sums for totally real number fields For any number field F, there are Kloosterman sums

S(r, r′; c) = ∑_{d mod(c)} e^{2πiTr _{F/Q}((rd+r′a)/c)},

where c ≠ 0, c ∈ O_{F}, and r, r′ ∈ F − {0} satisfy T r_{F/Q}(ry) ∈ Z, for all y ∈ O_{F}. These sums satisfy a bound of Weil-Salié type (see Bruggeman-Miatello, Duke M. J., 1995). If F is totally real of degree d, denote the different embeddings σ_{1}, …, σ_{d} : F → R. We prove, for any d ≥ 2, ε > 0:

∑_{c∈O_{F} , 0<|c^{σ_{j}} |≤X} N_{F/Q}(c)^{−1}S(r, r′; c) ≪ X^{(d−1)/2+ε} as X → ∞, (1)

in the absence of eigenvalues with exceptional coordinates. We also give the influence of the parameters r and r′. Our proof uses a sum formula of Kuznetsov type that includes the contribution of non-trivial K-types.