Mathematical Research Letters
Volume 20 (2013)
Counting sheaves using spherical codes
Pages: 305 – 323
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.
Lisse $\ell$-adic sheaves, trace functions, spherical codes, Riemann Hypothesis over finite fields
2010 Mathematics Subject Classification
11G20, 11T23, 94B60, 94B65