Fractional ideals and integration with respect to the generalised Euler characteristic

Let $b$ be a fractional ideal of a one-dimensional Cohen-Macaulay local ring $O$ containing a perfect field $k$. This paper is devoted to the study some $O$-modules associated with $b$. In addition, different motivic Poincar\'e series are introduced by considering ideal filtrations associated with $b$; the corresponding functional equations of these Poincar\'e series are also described

The universal zeta function for curve singularities and its relation with global zeta functions

The purpose of this note is to give a brief overview on zeta functions of curve singularities and to provide some evidences on how these and global zeta functions associated to singular algebraic curves over perfect fields relate to each other.Comment: Survey on the "universal zeta function" defined for curve singularities by W. Z\'u\~niga and the author in their paper "Motivic zeta functions for curve singularities" [Nagoya Math. J. 198 (2010), 47-75]; a poster of it was presented in the "Workshop on Positivity and Valuations" held at the Centre de Recerca Matem\`atica, Barcelona, in 2016 February 22nd-26t

Functions and differentials on the non-split Cartan modular curve of level 11

The genus 4 modular curve Xns(11) attached to a non-split Cartan group of level 11 admits a model defined over Q. We compute generators for its function field in terms of Siegel modular functions. We also show that its Jacobian is isomorphic over Q to the new part of the Jacobian of the classical modular curve X0(121)Postprint (author's final draft

Hilbert depth of graded modules over polynomial rings in two variables

In this article we mainly consider the positively Z-graded polynomial ring R=F[X,Y] over an arbitrary field F and Hilbert series of finitely generated graded R-modules. The central result is an arithmetic criterion for such a series to be the Hilbert series of some R-module of positive depth. In the generic case, that is, the degrees of X and Y being coprime, this criterion can be formulated in terms of the numerical semigroup generated by those degrees.Comment: 28 page