Bounding Picard numbers of surfaces using p-adic cohomology

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.

34 pages; v2: refereed version, to appear in proceedings "Arithmetic, Geometry, and Coding Theory (AGCT-10)"

References in corpus (2)

• Counting Points on Hyperelliptic Curves using Monsky-Washnitzer Cohomology
• Counting points on varieties over finite fields of small characteristic

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• The Dwork Family and Hypergeometric Functions
• Variation of Neron-Severi ranks of reductions of K3 surfaces
• Computing zeta functions of generic projective hypersurfaces in larger characteristic
• Improvements to the deformation method for counting points on smooth projective hypersurfaces