Bounding Picard numbers of surfaces using p-adic cohomology

2006/01/20 by Timothy G. Abbott, Abbott, Timothy G., Kiran S. Kedlaya +3 · 1 citation
Mathematics · #14C22 #14F30 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.math/0601508

openalex publication_date 2006/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 F₂ and F₃ with arithmetic Picard number 1, and a smooth quintic over F₂ 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.

Citations

• Counting points on varieties over finite fields of small characteristic
• The Criteria for Algebraic Equivalence and the Torsion Group

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