11 problems

Serre–Oesterlé conjecture. $\tilde{P}(T)$ is a rational function of $T$.

Let ${\mathfrak g}=\operatorname{so}(2n+1)$ and let ${\mathfrak h}=\operatorname{so}(2k+1)\times\operatorname{so}(2n-2k+1)$. Let $Y$ and $Z$ be regular semisimple elements, and sup…

Let ${\mathfrak g}=\operatorname{so}(2n+1)$, and let $\operatorname{strip}(r)$ be the set of all $X\in{\mathfrak g}$ such that $|\alpha(X)|=q^{-r}$ for every root $\alpha$. For…

Let $X_K$ be a smooth connected scheme over $K$, let $x_0,x_1\in X(K)$, and let $F_{x_i}$ denote the corresponding fibre functors on the category under consideration. Write…

Let $F$ be a local field, and let $\phi:\bfM\to\bfN$ be a flat morphism of smooth algebraic varieties defined over $F$, whose fibers are reduced and normal. Say that $\phi$ is geo-…

The Jiang–Rallis integral point-count conjecture. The number of such triples is $p^2-1$. This conjecture gives the remaining finite-field count needed to evaluate one of the sub-in…

Let $S=\operatorname{Spec}\mathcal{O}_F$ where $\mathcal{O}_F$ is a local ring with finite residue field $k_F$. Let $D=K_\mathcal{C}$ and let $L\in\mathcal{P}ic^d(\mathcal{C})$ for…

The degree-five Igusa zeta function conjectures. The following two assertions are conjectured, with (b) stronger than (a): (a) $I_5(s)$ is a power series in $q$ and $t$ with positi…

Borevich–Shafarevich conjecture. $P(T)$ is a rational function of $T$.

Let $F$ be a number field and let $f$ be a nontrivial new eigenclass as above, with associated conjectural abelian variety $A_f/F$ of dimension $d$. Let…

Let $f_n(k)$ denote the number of subrings of ${\mathbb Z}^n$ having additive index $k$. For a real number $B>0$, let $N_n(B)$ denote the number of subrings of ${\mathbb Z}^n$ havi…