14 problems

Let $k$ be a field of characteristic zero, let $X=\mathbb A^d_{\mathbb C}$, let $h\in\mathbb C[x_1,\ldots,x_d]$ satisfy $h(\mathbf 0)=0$, and let $Z_{DL}(h,T)$ be the local Denef–L…

Let $k$ be a field, let $X$ be a variety over $k$, and let ${\mathcal M}_k$ denote the relevant localization of the Grothendieck ring of varieties. Let $Z_{\rm mot}$ be the motivic…

Let $f \in \operatorname{\mathbb{C}}[x_1,\ldots,x_n] \setminus \mathbb{C}$ with $f(0)=0$. Write $Z_f^{\operatorname{mot}}(s)$ and $Z_{f,0}^{\operatorname{mot}}(s)$ for the global a…

Let $X$ be an $n$-dimensional smooth complex irreducible variety and let $f:X\to\mathbb A^1$ be a regular function with non-empty zero locus. Let $\alpha$ be a remarkable number of…

Let $X$ be a smooth irreducible complex variety and let $\mathcal I$ be a coherent ideal sheaf of $X$ with a non-empty zero locus. Let $Z_{\mathcal I}^{\mathrm{mot}}(T)$ be its mot…

Monodromy Conjecture. If $s_0$ is a pole of $Z_{f,W}^{\mathrm{mot}}(s)$, then $(s-s_0)\mid b_f(s)$, and $e^{s_0\cdot 2\pi \sqrt{-1}}$ is an eigenvalue of a local monodromy at some…

Let $R^{(2,2k)}$ be the germ of the plane curve singularity $y(y-x^k)=0$, let $R^{(2,2k)\oplus N}$ denote the direct sum of $N$ copies of this singularity, and let…

Let $R^{(2,2k)}$ be the germ of the plane curve singularity $y(y-x^k)=0$, and let $\widehat{N\!Z}_{R^{(2,2k)}}(t)$ denote its motivic Cohen–Lenstra zeta function. Here $k$ is a pos…

Let $f: X \longrightarrow \mathbb{C}$ be a non-constant regular function and let $x \in X$. Denote by $Z_{\textnormal{mot}, x}(f; s)$ the local motivic zeta function and by…

The exact motivic formula conjecture.

Let $X$ be a reduced curve over $\mathbb{F}_q$, and let $\widetilde X$ be its normalization. Let $\#_q$ denote the point-count homomorphism from the completed Grothendieck ring to…

Let $f$ define a hypersurface, let $x$ be a point in its domain, and let $Z_{f,x}(T)$ be the local motivic zeta function. Write ${\mathbb L}$ for the Lefschetz motive and let…

Finite-group generalization conjecture. The motivic zeta function $\zeta_{(C,\sigma)}(t)$ is rational.

Let $k$ be a field of characteristic zero, let $X$ be the relevant $k$-variety, let $f:X\to\mathbb{A}^1_k$ be the morphism, let $x$ be a closed point with residue field $k_x$, and…