6 problems

Let $p$ be a prime and let $f\in\mathbb{Z}[x_1,\dots,x_d]$. The poles of the Igusa zeta function encode the asymptotic behavior of the numbers of solutions of $f\equiv 0\pmod{p^k}$…

Let $F:(\mathbb{C}^{n+1},\mathbf{0})\to(\mathbb{C},\mathbf{0})$ define a germ of an isolated hypersurface singularity. For $x\in F^{-1}(0)$, let $\mathcal{F}_{F,x}$ be the Milnor f…

Let $f$ be a polynomial, and let the complexification of $f$ have monodromy eigenvalues. Consider the Igusa zeta function of $f$ with respect to a character, and let the order of t…

Topological multivariable strong monodromy conjecture. The polar locus of $Z_{F}^{\operatorname{top}}(S)$ is contained in $V(B_{F})$.

Let $F:(\mathbb{C}^{n+1},\mathbf{0})\to(\mathbb{C},\mathbf{0})$ define a germ of an isolated hypersurface singularity. For $x\in F^{-1}(0)$, let $\mathcal{F}_{F,x}$ be the Milnor f…

Monodromy conjecture. If $s_0\in\mathbb C$ is a pole of $Z_{f,\operatorname{top}}(s)$, then $e^{2\pi i s_0}$ is an eigenvalue of $T_f$ and $s_0$ is a zero of $b_f(s)$.