Mathematical Research Letters

Volume 20 (2013)

Number 2

Counting sheaves using spherical codes

Pages: 305 – 323

DOI: http://dx.doi.org/10.4310/MRL.2013.v20.n2.a8

Authors

Étienne Fouvry (Laboratoire de Mathématique, Université Paris Sud, Orsay, France)

Emmanuel Kowalski (D-MATH, ETH Zürich, Switzerland)

Philippe Michel (EPFL/SB/IMB/TAN, Lausanne, Switzerland)

Abstract

Using the Riemann Hypothesis over finite fields and bounds for the size of spherical codes, we give explicit upper bounds, of polynomial size with respect to the size of the field, for the number of geometric isomorphism classes of geometrically irreducible $\ell$-adic middle-extension sheaves on a curve over a finite field, which are pointwise pure of weight $0$ and have bounded ramification and rank. As an application, we show that “random” functions defined on a finite field cannot usually be approximated by short linear combinations of trace functions of sheaves with small complexity.

Keywords

Lisse $\ell$-adic sheaves, trace functions, spherical codes, Riemann Hypothesis over finite fields

2010 Mathematics Subject Classification

11G20, 11T23, 94B60, 94B65

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