Related papers: Global Igusa zeta function and $K$-equivalence

Let $f:X\rightarrow Y$ be an algebraic fibre space between normal projective varieties and $F$ be a general fibre of $f$. We prove an Iitaka-type inequality $\kappa(X,-K_X)\leq \kappa(F,-K_F)+\kappa(Y,-K_Y)$ under some mild conditions. We…

For every smooth complex projective variety $W$ of dimension $d$ and nonnegative Kodaira dimension, we show the existence of a universal constant $m$ depending only on $d$ and two natural invariants of the very general fibres of an Iitaka…

In this paper we will prove a uniformity result for the Iitaka fibration $f:X \rightarrow Y$, provided that the generic fiber has a good minimal model and the variation of $f$ is zero or that $\kappa(X)=\rm{dim}(X)-1$.

In the paper, we shall establish the existence of a meromorphic continuation of the Global Zeta Function $\zeta(f,\chi)$ of a Global Number Field $K$ and also deduce the functional equation for the same, using different properties of the…

The motivic zeta function of a smooth and proper $\mathbb{C}((t))$-variety $X$ with trivial canonical bundle is a rational function with coefficients in an appropriate Grothendieck ring of complex varieties, which measures how $X$…

In this paper we classify isogeny classes of global $\mathsf{G}$-shtukas over a smooth projective curve $C/\mathbb{F}_q$ (or equivalently $\sigma$-conjugacy classes in $\mathsf{G}(\mathsf{F} \otimes_{\mathbb{F}_q} \overline{\mathbb{F}_q})$…

We generalize the Ihara-Selberg zeta function to hypergraphs in a natural way. Hashimoto's factorization results for biregular bipartite graphs apply, leading to exact factorizations. For $(d,r)$-regular hypergraphs, we show that a modified…

Let $f:X\to Y$ be a fibration from a smooth projective 3-fold to a smooth projective curve, over an algebraically closed field $k$ of characteristic $p >5$. We prove that if the generic fiber $X_{\eta}$ has big canonical divisor…

In this paper, we prove the rationality of Igusa's local zeta functions of semiquasihomogeneous polynomials with coefficients in a non-archimedean local field K. The proof of this result is based on Igusa's stationary phase formula and some…

We study pluricanonical systems on smooth projective varieties of positive Kodaira dimension, following the approach of Hacon-McKernan, Takayama and Tsuji succesfully used in the case of varieties of general type. We prove a uniformity…

We prove that if $X$ is a smooth projective variety of dimension greater than 1 over a field $K$ of characteristic zero such that $\operatorname{Pic}(X_{\bar{K}}) = \mathbb{Z}$ and $X_{\bar{K}}$ is simply connected, then the natural map…

Given a fibration $f: X \to Y$ with normal general fibre $X_y$, over a field of any characteristic, we establish the Iitaka-type inequality $\kappa(X,-K_X) \leq \kappa(X_y,-K_{X_y})+\kappa(Y,-K_Y)$ whenever the $\mathbb{Q}$-linear series…

Let $X$ be a real prehomogeneous vector space under a reductive group $G$, such that $X$ is an absolutely spherical $G$-variety with affine open orbit. We define local zeta integrals that involve the integration of Schwartz-Bruhat functions…

We consider pro-isomorphic zeta functions of the groups $\Gamma(\mathcal{O}_K)$, where $\Gamma$ is a unipotent group scheme defined over $\mathbb{Z}$ and $K$ varies over all number fields. Under certain conditions, we show that these…

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $X$ be a normal projective surface over $k$ with canonical singularities whose anti-canonical divisor is nef and big. We prove that $X$ is globally $F$-regular except for…

In this paper we give a description of hypersurfaces with trivial ring $AK(S)$, introduced by the second author as following. Let $X$ be an affine variety and let $G(X)$ be the group generated by all $\Bbb {C}^+$-actions on $X$. Then…

We generalize the notion of the auto-Igusa zeta function to formal deformations of algebraic spaces. By incorporating data from all algebraic transformations of local coordinates, this function can be viewed as a generalization of the…

Given a fibration over a perfect field of positive characteristic, we study an Iitaka-type inequality for the anticanonical divisors. We conclude that it holds when the source of the fibration is a threefold or when the target is a curve,…

We explain the proof, obtained in collaboration with Chenyang Xu, of a 1999 conjecture of Veys about poles of maximal order of Igusa zeta functions. The proof technique is based on the Minimal Model Program in birational geometry, but the…

Let $X$ denote a connected $(q+1)$-regular undirected graph of finite order $n$. The graph $X$ is called Ramanujan whenever $$ |\lambda|\leq 2q^{\frac{1}{2}} $$ for all nontrivial eigenvalues $\lambda$ of $X$. We consider the variant…