Bounding Picard numbers of surfaces using p-adic cohomology

By Timothy G. Abbott, Kiran S. Kedlaya, David Roe. 2006

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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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/804ea5711ce6c49a8bfb2f249172ff34be68c03a