291 problems

Let $n_1,n_2$ be non-negative integers, let $c(n_1,n_2)$ denote the leading-order coefficient in the mixed second-moment asymptotic for the relevant family of quadratic Dirichlet…

Let $q=p^a$ be a prime power, let $\mathbb{F}_q[T]$ be the polynomial ring over the finite field $\mathbb{F}_q$, and let $\mathfrak{M}_2(2g)$ denote the set of primitive quadratic…

Periods in this setting have algebraic relations, and motivic relations provide the corresponding algebraic framework. Grothendieck's period conjecture. The relations between perio…

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…

An optimal recursive tower is a recursive tower of algebraic function fields attaining the Drinfeld–Vlăduț bound. Elkies' conjecture. Every optimal recursive tower is modular in an…

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

Let $\mathbb{K}$ be an arbitrary field and $P(t) \in \mathbb{K}[t]$ an irreducible polynomial over $\mathbb{K}$. For $\Gamma(t) \in \mathbb{K}((t^{-1}))$, write…

Thakur's basis conjecture. The values $\zeta_{A}(\mathfrak{s})$ for $\mathfrak{s} \in \mathcal{I}^{\mathrm{T}}$ form a basis of $\mathcal{Z}_{k}$.

Let $\mathbb{F}_q(t)$ be the rational function field over the finite field $\mathbb{F}_q$, and consider cubic extensions $K/\mathbb{F}_q(t)$ counted by discriminant. Secondary-term…

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…

Common-zero conjecture. Any such element has at least $4$ zeros in common with each of the elements $x_1-\alpha$ and $x_2-\beta$. This anticipated common-zero property is used to c…

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…

Feuerverger–Martin's conjecture. There should exist a bias factor $F_n(e_1,\dots,e_r)$, defined as a linear combination of the quantities $G_n(e_j)$, such that, for ordered $r$-tup…

Splitting conjecture. For any $k\geq 0$,

Andrade–Jung–Shamesaldeen moment conjecture. For any $k\in\mathbb{N}$,

Scully's isotropy-index conjecture. One has $k\geq 0$, and

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

Refined Birch–Swinnerton-Dyer conjecture. The Birch–Swinnerton-Dyer conjecture holds, $\sha(J_X)$ is finite, and

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…

Nonnegativity conjecture. The residue scalar product of a divisor with itself is nonnegative:

Let $I$ be the augmentation ideal of $\mathbb{Z}[\Gamma]$, let $epsilon$ be the element supplied by the Rubin–Stark conjecture, and let $Phi\in\iota(Lambda^n U^*)$. Let…

Let $\mathrm{Gamma}$ be the Galois group in the setup, let $U$ be the relevant group of $S$-units, and let $\mathcal{R}_{eta}$ be the regulator map associated with…

Let $p$ be a prime. Consider elliptic curves, or absolutely simple abelian varieties of any dimension, defined over ${\mathbb F}_p(t)$, and the tower of function fields…