Stark and Terras introduced the edge zeta function of a finite graph in 1996. The edge zeta function is the reciprocal of a polynomial in twice as many variables as edges in the graph and can be computed in polynomial time. We look at graph properties which we can determine using the edge zeta function. In particular, the edge zeta function is enough to deduce the clique number, the number of H...

Motivated by arithmetic applications, we introduce the notion of a partial zeta function which generalizes the classical zeta function of an algebraic variety defined over a finite field. We then explain two approaches to the general structural properties of the partial zeta function in the direction of the Weil type conjectures. The first approach, using an inductive fibred variety point of vi...

Let F be an algebraic number field. Recently, L. Weng introduced zeta functions of rank n associated to F as a generalization of the Iwasawa-Tate zeta integral from the Arakelov geometric point of view. In the article, we call them the Weng zeta function of rank n. In the case of rank one, the Weng zeta function coincide with the Dedekind zeta function of F . The background and precise definiti...

All plane curves of degree less than 7 with coefficients in F2 are examined for curves with a large number of Fq rational points on their smooth model, for q = 2,m = 3, 4, ..., 11. Known lower bounds are improved, and new curves are found meeting or close to Serre’s, Lauter’s, and Ihara’s upper bounds for the maximal number of Fq rational points on a curve of genus g.

In [Iha86b], Ihara constructs a universal cocycle Gal ( Q/Q ) −→ Zp[[t0, t1, t∞]]/ ((t0 + 1)(t1 + 1)(t∞ + 1)− 1) arising from the action of Gal ( Q/Q ) on certain quotients of the Jacobians of the Fermat curves

In this paper, we show how to determine several properties of a finite graph G from its Ihara zeta function ZG(u). If G is connected and has minimal degree at least 2, we show how to calculate the number of vertices of G. To do so we use a result of Bass, and in the case that G is nonbipartite, we give an elementary proof of Bass’ result. We further show how to determine whether G is regular, a...

To a polynomial f over a p{adic eld K and a character of the group of units of the valuation ring of K one associates Igusa's local zeta function Z(s; f;), which is a meromorphic function on C. Several theorems and conjectures relate the poles of Z(s; f;) to the monodromy of f; the so{called holomorphy conjecture states roughly that if the order of does not divide the order of any eigenvalue of...

In this manuscript we show that function fields K|k with td(K|k) > dim(k)+1 can be recovered from their maximal pro-` abelian-by-central Galois theory, where dim(k) is the Kronecker dimension. This extends the main result of Pop [P5] beyond the case where k is an algebraic closure of a finite field. We also show how this implies the pro-` form of Ihara’s question / Oda-Matsumoto conjecture I/OM...