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Title: Practical improvements to the deformation method for point counting
Author: Pancratz, Sebastian Friedrich
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2013
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In this thesis we investigate practical aspects related to point counting problems on algebraic varieties over finite fields. In particular, we present significant improvements to Lauder’s deformation method for smooth projective hypersurfaces, which allow this method to be successfully applied to previously intractable instances. Part I is dedicated to the deformation method, including a complete description of the algorithm but focussing on aspects for which we contribute original improvements. In Chapter 3 we describe the computation of the action of Frobenius on the rigid cohomology space associated to a diagonal hypersurface; in Chapter 4 we develop a method for fast computations in the de Rham cohomology spaces associated to the family, which allows us to compute the Gauss–Manin connection matrix. We conclude this part with a small selection of examples in Chapter 6. In Part II we present an improvement to Lauder’s fibration method. We manage to resolve the bottleneck in previous computation, which is formed by so-called polynomial radix conversions, employing power series inverses and a more efficient implementation. Finally, Part III is dedicated to a comprehensive treatment of the arithmetic in unramified extensions of Qp , which is connected to the previous parts where our computations rely on efficient implementations of p-adic arithmetic. We have made these routines available for others in FLINT as individual modules for p-adic arithmetic.
Supervisor: Lauder, Alan Sponsor: European Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Number theory ; Arithmetic geometry ; point counting ; deformation method