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. In all instances, the varieties considered had smooth representations in either affine or projective space. In this thesis, two non-smooth situations are introduced: the case of superelliptic curves with singular points that are rational over the field of definition, and the case of nodal projective plane curves. In each case we present, assuming the characteristic is fixed, a polynomial-time algorithm which computes the zeta function the curve, and we provide the results of an implementation in MAGMA. The case of singular superelliptic curves extends a method of Gaudry and Gurel, and the case of nodal projective curves extends a method of Kedlaya, Abbott, and Roe.
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