Related papers: On some curve singularity invariants and reduction…

Using toric modifications and some compatibility we compute the local $p$-adic zeta function of a plane curve singularity. Thanks to the compatibility, we can work over the analytic change of variables formula for $p$-adic integrals, hence…

Curve singularities are classical objects of study in algebraic geometry. The key player in their combinatorial structure is the {\it value semigroup}, or its compactification, the {\it value semiring}. One natural problem is to explicitly…

The purpose of this paper is to extend the symmetry of maximals of the ring of a germ of reducible plane curve proved by Delgado to a relation between the relative maximals of a fractional ideal and the absolute maximals of its dual for any…

The special uniformity of zeta functions claims that pure non-abelian zeta functions coincide with group zeta functions associated to the special linear groups. Naturally associated are three aspects, namely, the analytic, arithmetic, and…

We compute the $\delta$-invariant of a curve singularity parameterized by generic sparse polynomials. We apply this to describe topological types of generic singularities of sparse resultants and ``algebraic knot diagrams'' (i.e. generic…

A possible connection between quantum computing and Zeta functions of finite field equations is described. Inspired by the 'spectral approach' to the Riemann conjecture, the assumption is that the zeroes of such Zeta functions correspond to…

The appearance of multiple zeta values in anomalous dimensions and $\beta$-functions of renormalizable quantum field theories has given evidence towards a motivic interpretation of these renormalization group functions. In this paper we…

This article deals with applications of Voronin's universality theorem for the Riemann zeta-function $\zeta$. Among other results we prove that every plane smooth curve appears up to a small error in the curve generated by the values…

To count bundles on curves, we study zetas of elliptic curves and their zeros. There are two types, i.e., the pure non-abelian zetas defined using moduli spaces of semi-stable bundles, and the group zetas defined for special linear groups.…

In this short survey we give a description of the theta functions of algebraic curves, half-integer theta-nulls, and the fundamental theta functions. We describe how to determine such fundamental theta functions and describe the components…

The theta characteristics on a Riemann surface are permuted by the induced action of the automorphism group, with the orbit structure being important for the geometry of the curve and associated manifolds. We describe two new methods for…

We study the behaviour near s=1/2 of zeta functions of varieties over finite fields F_q with q a square. The main result is an Euler-characteristic formula for the square of the special value at s=1/2. The Euler-characteristic is…

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 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…

To an ideal in $\mathbb{C}[x,y]$ one can associate a topological zeta function. This is an extension of the topological zeta function associated to one polynomial. But in this case we use a principalization of the ideal instead of an…

For a root of unity $\zeta$ of odd prime order, we restrict coefficients of non-semisimple quantum representations of mapping class groups associated with the small quantum group $\mathfrak{u}_\zeta \mathfrak{sl}_2$ from $\mathbb{Q}(\zeta)$…

We compute the equivariant zeta function for bundles over infinite graphs and for infinite covers. In particular, we give a ``transfer formula'' for the zeta function of infinite graph covers. Also, when the infinite cover is given as a…

The motivic Hilbert zeta function of a variety is the generating function for classes in the Grothendieck ring of varieties of Hilbert schemes of points of the variety. In this paper, the motivic Hilbert zeta function of a reduced curve is…

The zeta function of a motive over a finite field is multiplicative with respect to the direct sum of motives. It has beautiful analytic properties, as were predicted by the Weil conjectures. There is also a multiplicative zeta function,…

Using analytic torsion associated to stable bundles, we introduce zeta functions for compact Riemann surfaces. To justify the well-definedness, we analyze the degenerations of analytic torsions at the boundaries of the moduli spaces, the…