5 problems

Loeser's generalized monodromy conjecture. For almost all finite places $\mathfrak P$, if $H$ is a polar hyperplane in $\mathcal P(Z_\phi(F_{\mathfrak P},\chi;s))$, with $\phi$ res…

The degree-five Igusa zeta function conjectures. The following two assertions are conjectured, with (b) stronger than (a): (a) $I_5(s)$ is a power series in $q$ and $t$ with positi…

Let $L$ be a non-Archimedean local field over $\operatorname{O}_K$, let $f\in\operatorname{O}_K[x_1,\ldots,x_n]$ be a nonconstant polynomial, let $\Phi$ be a Schwartz–Bruhat functi…

Let $f\in\operatorname{O}_K[x_1,\ldots,x_n]$ be a nonconstant polynomial, let $Z$ be a subscheme of $\mathbb A^n_{\operatorname{O}_K}$, and let $L$ be a non-Archimedean local field…

Let $f$ be a non-constant polynomial in $\mathbb{Z}[x_1,\ldots,x_n]$, for some $n\geq 1$, with $f(\mathbf{0})=0$. For every prime number $p$, let $Z_{f,p}(s)$ be Igusa's $p$-adic z…