191 problems

Let $q$ be fixed, let $a\in\mathbb{F}_q[t]\setminus\{0\}$, and let $M=M(n)$ be in the truncation range specified for the truncated Möbius function. For a monic polynomial $f$, writ…

Let $K$ be a global function field of characteristic $p$, let $f\in K[x]$ have degree $d\ge 2$, and let $a\in K$. For $n\ge 1$, let $K_n$ be the splitting field over $K$ of…

For every prime power $q$ and every primitive imaginary quadratic Dirichlet character $\chi$ of $\mathbb{F}_q(T)$, one has $L\left(\tfrac{1}{2},\chi\right)\neq 0$.

Automorphism-group conjecture.

Let $A$ be the coefficient ring and let … be a Drinfeld module. For each prime polynomial $P$, let $g_{P,\phi}$ denote the auxiliary polynomial associated with $P$ and $\phi$. Prim…

Let $X$ be a smooth projective surface over a field $k$, let $Y$ be a smooth projective curve over $k$, and let $f:X\to Y$ be a semistable fibration whose smooth generic fiber…

Matzat's conjecture. The absolute differential Galois group of $F$ is the free proalgebraic group on a set of cardinality $|F|$.

Let $G$ be a nontrivial finite group. Write $d(G)$ for the minimal number of generators of $G$, $G^{\mathrm{ab}}$ for its abelianization, and $p(G^{\mathrm{ab}})$ for the subgroup…

Nagao's conjecture. The rank $r$ of $\mathcal{E}(\mathbb{Q}(T))$ satisfies

Assume the setup in which $G$ is simply connected and almost simple, $S$ is a set of places with at least two elements, and $T=\emptyset$. For each positive integer $m$, let…

Let $\mathcal{F}_q$ be a finite field, let $\mathcal{F}_q((t^{-1}))$ be the field of formal Laurent series, and let $|\Theta(t)|=q^{\deg(\Theta(t))}$ be the norm. For an irreducibl…

Let $K$ be a function field or a number field, let $C$ be a smooth projective curve of genus $g\geq 2$ over $\overline{K}$, let $J_C$ be its Jacobian, and fix a degree-one divisor…

Let $\mathcal F$ be a family of $L$-functions over a function field $\mathbb F_q[x]$. For each parameter $D\in\mathcal F$, let $\Lambda_D$ denote the associated De Bruijn–Newman co…

Hassett–Tschinkel's weak approximation conjecture. Proper rationally connected varieties defined over $F$ satisfy weak approximation.

Period–Index Conjecture for Function Fields. For every $\alpha\in\operatorname{Br}(k(X))$,

Let $E$ be an Anderson $A$-module over a field $R$, and let $b(E)$ be an exponent such that the residue-in-$\tau$ pairing factors through $R^{1/q^{b(E)}}$. Let $w(E)$ range over th…

Let $K=\mathbb{C}(B)$ be the function field of a smooth projective curve $B$ over $\mathbb{C}$. Let $X$ be a projective variety over $K$ that does not contain any possibly singular…

For an integer $n$ written as $n=\delta+h(q-1)$ by Euclidean division by $q-1$, define … and let $P_n(x)$ be … Let $\zeta_C^*(-n)$ be the first nonzero coefficient in the expansion…

Let $Z(x,n)$ be written uniquely as … in $A[\![x-1]\!]$, with $h_n\geq 0$ and $\zeta_C^*(n)\in A$. Let $\operatorname{Cl}(\underline{A}(n))$ denote the class module of the Carlitz…

Banks–Martin conjecture analogue. For each $q\geq 5$, the inequalities

Failure of the local-global principle. The local-global principle for isotropy of quadratic forms fails to hold in dimension $2^n$ over $K$.

Let $K$ be an algebraically closed field and let $F$ be an extension field of $K$. Let $S$ be a finite-dimensional $K$-subspace of $F$ such that $K\subset S$. Define the combinator…

Let $\overline{k}$ be a fixed algebraic closure of $k$, let $k^{\rm sep}$ be its separable closure, and let $G=\operatorname{Gal}(k^{\rm sep}/k)$. Let $\rho:G\to\mathbf C_\infty^*$…

Let $L(x,y)=\prod_i(1-\beta_i^{(y)}/x)$, where the nonzero $\beta_i^{(y)}$ are the zeroes of $L(x,y)$, and let $\lambda_i^{(y)}:=1/\beta_i^{(y)}$ be the reciprocal zeroes. The char…

Let $L(s)$, with $s=(x,y)\in S_\infty$, be a characteristic-$p$ $L$-series arising from arithmetic, and let $\mathbf K_L$ be the finite extension of $\mathbf K_{\mathbf V}$ generat…