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\(\ell\)-adic GKZ hypergeometric sheaves and exponential sums. (English) Zbl 1368.14032
Summary: To a torus action on a complex vector space, Gelfand, Kapranov and Zelevinsky [I. M. Gel’fand et al., Funct. Anal. Appl. 23, No. 2, 94–106 (1989; Zbl 0721.33006); translation from Funkts. Anal. Prilozh. 23, No. 2, 12–26 (1989)] introduced a system of differential equations, called the GKZ hypergeometric system. Its solutions are GKZ hypergeometric functions. We study the \(\ell\)-adic counterpart of the GKZ hypergeometric system, which we call the \(\ell\)-adic GKZ hypergeometric sheaf. It is an object in the derived category of \(\ell\)-adic sheaves on the affine space over a finite field. Traces of Frobenius on stalks of this object at rational points of the affine space define the hypergeometric functions over the finite field introduced by I. M. Gelfand and M. I. Graev [Dokl. Math. 64, No. 3, 732–737 (2001; Zbl 1059.33019); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 381, No. 6, 732–737 (2001)]. We prove that the \(\ell\)-adic GKZ hypergeometric sheaf is perverse, calculate its rank, and prove that it is irreducible under the non-resonance condition. We also study the weight filtration of the GKZ hypergeometric sheaf, determine its lisse locus, and apply our result to the study of weights of twisted exponential sums.

14F20 Étale and other Grothendieck topologies and (co)homologies
11T23 Exponential sums
14G15 Finite ground fields in algebraic geometry
33C70 Other hypergeometric functions and integrals in several variables
11L03 Trigonometric and exponential sums (general theory)
Full Text: DOI
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