Counting Sheaves Using Spherical Codes

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 l-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.