Related papers: An extension of Kedlaya's algorithm for hyperellip…

We introduce a new algorithm to compute the zeta function of a curve over a finite field. This method extends Kedlaya's algorithm to a very general class of curves using a map to the projective line. We develop all the necessary bounds,…

Kedlaya's algorithm (Kedlaya, J. Ramanujan Math. Soc 16, 2001) can be used to count the points of arbitrary hyperelliptic curves over finite fields of characteristic p, where p is an odd prime. The algorithm uses the cohomology of a p-adic…

In recent algorithms that use deformation in order to compute the number of points on varieties over a finite field, certain differential equations of matrices over p-adic fields emerge. We present a novel strategy to solve this kind of…

In 2003, Kedlaya gave an algorithm to compute the zeta function associated to a hyperelliptic curve over a finite field, by computing the rigid cohomology of the curve. Edixhoven remarked that it is actually possible to compute the…

Let C/Q be a curve of genus three, given as a double cover of a plane conic. Such a curve is hyperelliptic over the algebraic closure of Q, but may not have a hyperelliptic model of the usual form over Q. We describe an algorithm that…

We develop algorithms to compute two versions of the motivic Hilbert zeta function for curve singularities: the classical version, applicable to singularities with a monomial valuation semigroup or to singular curves defined by…

We introduce a new algorithm to compute the zeta function of a curve over a finite field. This method extends previous work of ours to all curves for which a good lift to characteristic zero is known. We develop all the necessary bounds,…

We describe an algorithm to compute the zeta-function of a proper, smooth curve over a finite field, when the curve is given together with some auxiliary data. Our method is based on computing the matrix of the action of a semi-linear…

We present a new method for computing the zeta function of an algebraic curve over a finite field. The algorithm relies on a trace formula of Harvey to count points on a plane model of the curve. The zeta function of the curve is then…

We describe an algorithm to compute the zeta function of a cyclic cover of the projective line over a finite field of characteristic $p$ that runs in time $p^{1/2 + o(1)}$. We confirm its practicality and effectiveness by reporting on the…

In algebraic geometry, superspecial curves are important research objects. While the number of superspecial genus-3 curves in characteristic $p$ is known, the number of hyperelliptic ones among them has not been determined even for small…

We present new algorithms for computing zeta functions of algebraic varieties over finite fields. In particular, let X be an arithmetic scheme (scheme of finite type over Z), and for a prime p let zeta_{X_p}(s) be the local factor of its…

Curves over finite fields are of great importance in cryptography and coding theory. Through studying their zeta-functions, we would be able to find out vital arithmetic and geometric information about them and their Jacobians, including…

We assign an arbitrary density matrix to a weighted graph and associate to it a graph zeta function that is both a generalization of the Ihara zeta function and a special case of the edge zeta function. We show that a recently developed…

Physically meaningful periodic solutions to certain integrable partial differential equations are given in terms of multi-dimensional theta functions associated to real Riemann surfaces. Typical analytical problems in the numerical…

We propose a Las Vegas probabilistic algorithm to compute the zeta function of a genus-3 hyperelliptic curve defined over a finite field $\mathbb F_q$, with explicit real multiplication by an order $\mathbb Z[\eta]$ in a totally real cubic…

In this paper, we give an overview of the various general methods in computing the zeta function of an algebraic variety defined over a finite field, with an emphasis on computing the reduction modulo $p^m$ of the zeta function of a…

We present a deterministic algorithm that computes the zeta function of a nonsupersingular elliptic curve E over a finite field with p^n elements in time quasi-quadratic in n. An older algorithm having the same time complexity uses the…

We present a method for computing the zeta function of a smooth projective variety over a finite field which proceeds by induction on the dimension. We have implemented our approach for some surfaces using the Magma programming language,…

Let $X$ be a smooth projective hypersurface over a finite field $k$ of characteristic $p$. We address the problem of practically computing the zeta function $Z(X,T)$ of $X$ (equivalently, the point counts $\#X(\mathbb{F}_q)$, where $q =…