6 problems

Let $f$ be a rigid $K$-analytic function in arbitrary dimension $n>1$, and let $Z_{f,\chi}(s)$ be its Igusa $p$-adic zeta function for a character $\chi$. A general bound shows tha…

Let $F$ be an arbitrary number field and let $f\in F[x_1,\dots,x_n]\setminus F$. For a non-archimedean completion $K$ of $F$, a character $\varkappa$ of $\mathcal{O}_K^\times$ has…

Let $\Bbbk$ be a number field, let $f$ be a polynomial over $\Bbbk$, and let $Z_\mathfrak p(s)$ denote its local $\mathfrak p$-adic zeta function. For a pole $\alpha$ of…

Let $p$ be an odd prime. Let $\mathcal{M}_p$ be the maximal abelian pro-$p$ extension of $\mathbb{Q}(\mu_{p^{\infty}})$ unramified outside $p$, and let $M_p$ be the maximal abelian…

Let $f$ be a polynomial, let the poles of its $p$-adic zeta function be considered, and let $b_f(s)$ be the Bernstein polynomial of $f$. Stronger form of Igusa's conjecture. The re…

Let $L$ be a number field and let $f\in L[\widetilde{x}]$, with $X=\mathbb{C}^d$ and special fiber $X_s=\{f=0\}$. For almost all finite places $\mathfrak{P}$, let…