7 problems

Let $f$ be a singular complex analytic germ at $0\in\mathbb{C}^{n}$ with $f(0)=0$, and let $\varphi$ be a Schwarz function on $\mathbb{C}^{n}$ with sufficiently small support and…

Local zeta-function conjecture. These values of the constants give the correct local zeta functions of the family.

Let $f$ be a polynomial over a number field, and let the associated non-archimedean local zeta functions have poles in the complex variable $s$. Let the Bernstein–Sato polynomial o…

Fix $N$ and let $Z_{\mathbb{R}}^{(N)}(\boldsymbol{s})$ and $Z_{\mathbb{Q}_p}^{(N)}(\boldsymbol{s})$ denote the archimedean and $p$-adic local zeta functions, respectively, regarded…

Let $A_{\mathbb{Q}_p}^{(N)}(\boldsymbol{s})$ denote the tree-level $p$-adic Koba–Nielsen string amplitudes, and let $A_{\mathrm{DL}}^{(N)}(\boldsymbol{s})$ denote their Denef–Loese…

Let $f$ be a polynomial, and let $\varphi$ be a test function whose support is sufficiently small. Gel'fand's conjecture. The integrals $Z_f(\varphi)(s)$ can be analytically contin…

Let $F$ be a number field, let $K$ be a completion of $F$, and let $Z_{\Phi}(s,\chi,f_l,V_{l-1},K)$ be the local zeta function associated with the complex non-degenerate polynomial…