## Departmental PhD Thesis Exam – Robert Burko

Everyone is welcome to attend. Refreshments will be served in the Math Lounge before the exam.

Wednesday, November 20, 2013
2:00 p.m.
BA 6183, 40 St George St.

Ph.D. Candidate: Robert Burko

Thesis Title: Computing the Zeta Function of Two Classes of Singular Curves

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Motivated by applications to cryptography, for over a decade mathematicians have successfully used $p$-adic cohomological methods to compute the zeta functions of various classes of varieties defined over finite fields of order $q$ in an amount of time polynomial in $\log q$ (assuming small characteristic).  In all instances, the varieties considered had smooth representations in either affine or projective space.

In this thesis, we extend a method of K. Kedlaya to two non-smooth situations: the case of superelliptic curves with singular points that are rational over the defining field, and the case of nodal projective plane curves.  We describe an algorithm that computes the zeta function of these curves and runs in polynomial time.  We also demonstrate its validity by showing the result of an implementation of each algorithm.

The method involves computing the matrix of the Frobenius automorphism on the cohomology groups of Monsky and Washnitzer up to a certain amount of $p$-adic accuracy.  Estimates on the amount of accuracy needed are drawn from the theory of crystalline cohomology introduced by Grothendieck and developed by Berthelot.

The thesis can be found here Burko_thesis