Counting sheaves using spherical codes

TL;DR

This paper establishes explicit polynomial upper bounds on the number of certain sheaves over finite fields using spherical codes and the Riemann Hypothesis, and applies these bounds to limitations on approximating functions by sheaf trace functions.

Contribution

It introduces a novel method combining spherical codes and the Riemann Hypothesis to bound sheaf counts and demonstrates limitations on function approximation by sheaves with small complexity.

Findings

Polynomial bounds on sheaf isomorphism classes

Limitations on approximating functions with small-sheaf trace functions

Application of spherical codes to algebraic geometry problems

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 can not usually be approximated by short linear combinations of trace functions of sheaves with small complexity.