Related papers: Counting points on surfaces in polynomial time

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…

In this manuscript, we give effective methods for computing the zeta function of maximal orders on surfaces.

We introduce a new algorithm for computing the periods of a smooth complex projective hypersurface. The algorithm intertwine with a new method for computing an explicit basis of the singular homology of the hypersurface. It is based on…

We give an explicit formula for the motivic zeta function in terms of a log smooth model. It generalizes the classical formulas for snc-models, but it gives rise to much fewer candidate poles, in general. As an illustration, we explain how…

In this paper, we give an approach to the zeta values of a (proper regular) arithmetic scheme X at the integers r>=d:=dim(X), using \'etale cohomology of X with Q_p(r) and Z_p(r)-coefficients.

We exhibit a quantum algorithm for determining the zeta function of a genus g curve over a finite field F_q, which is polynomial in g and log(q). This amounts to giving an algorithm to produce provably random elements of the class group of…

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…

In the present paper, we construct an algorithm for the evaluation of real Riemann zeta function $\zeta(s)$ for all real $s$, $s>1$, in polynomial time and linear space on Turing machines in Ko-Friedman model. The algorithms is based on a…

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,…

In this paper, we improve the algorithms of Lauder-Wan \cite{LW} and Harvey \cite{Ha} to compute the zeta function of a system of $m$ polynomial equations in $n$ variables over the finite field $\FF_q$ of $q$ elements, for $m$ large. The…

Let $C$ be a smooth plane quartic curve over $\mathbb{Q}$. Costa, Harvey and Sutherland provide an algorithm with an implementation, improving Harvey's average polynomial-time algorithm, to compute the $\bmod \ p$ reduction of the numerator…

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 =…

We compute integral moments of partial sums of the Riemann zeta function on the critical line and obtain an expression for the leading coefficient as a product of the standard arithmetic factor and a geometric factor. The geometric factor…

We calculate the discrete moments of the characteristic polynomial of a random unitary matrix, evaluated a small distance away from an eigenangle. Such results allow us to make conjectures about similar moments for the Riemann zeta…

I give a formula for the zeta function of a projective toric hypersurface over a finite field and estimate its Newton polygon. As an application this formula allows us to compute the exact number of rational points on the families of…

Let $k,p\in \mathbb{N}$ with $p$ prime and let $f\in\mathbb{Z}[x_1,x_2]$ be a bivariate polynomial with degree $d$ and all coefficients of absolute value at most $p^k$. Suppose also that $f$ is variable separated, i.e., $f=g_1+g_2$ for…

We show that the recent conjecture of the first-named author for the special value at $s=1$ of the zeta function of an arithmetic surface is equivalent to the Birch-Swinnerton-Dyer conjecture for the Jacobian of the generic fibre.

In this paper we present a p-adic algorithm to compute the zeta function of a nondegenerate curve over a finite field using Monsky-Washnitzer cohomology. The paper vastly generalizes previous work since all known cases, e.g. hyperelliptic,…

We introduce a new method which enables us to calculate the coefficients of the poles of local zeta functions very precisely and prove some explicit formulas. Some vanishing theorems for the candidate poles of local zeta functions will be…

We give an algorithm to compute the zeta function of the Fano surface of lines of a smooth cubic threefold $F$ into $\mathbb{P}^4$ defined over a finite field. We obtain some examples of Fano surfaces with supersingular reduction.