Bounding Picard numbers of surfaces using p-adic cohomology

TL;DR

This paper presents a computational method using p-adic cohomology to bound the Picard number of surfaces over finite fields, with implementations and examples demonstrating its effectiveness.

Contribution

The paper introduces a new computational procedure for bounding Picard numbers of surfaces over finite fields using p-adic cohomology, including an implementation in Magma.

Findings

Successfully bounded Picard numbers for various surfaces.

Implemented the method in Magma and provided explicit examples.

Identified surfaces with Picard number 1 and 2 over finite fields.

Abstract

Motivated by an application to LDPC (low density parity check) algebraic geometry codes described by Voloch and Zarzar, we describe a computational procedure for establishing an upper bound on the arithmetic or geometric Picard number of a smooth projective surface over a finite field, by computing the Frobenius action on p-adic cohomology to a small degree of p-adic accuracy. We have implemented this procedure in Magma; using this implementation, we exhibit several examples, such as smooth quartics over F_2 and F_3 with arithmetic Picard number 1, and a smooth quintic over F_2 with geometric Picard number 1. We also produce some examples of smooth quartics with geometric Picard number 2, which by a construction of van Luijk also have trivial geometric automorphism group.