3 problems

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…

Smooth-coefficient conjecture. The function $H(x;t)$ satisfies the following properties:

Let $p$ be an $\mathbb F_q$-point of a reduced curve $X$. Write $\widehat Z_{X,p}(x)$ for the associated Cohen–Lenstra series, and let $(xq^{-1};q^{-1})_\infty$ denote the $q$-Poch…