46 problems

Let $F$ be a number field, let $\mathfrak{p}$ be a finite place, and let $F_{\mathfrak{p}}$ be the completion of $F$ at $\mathfrak{p}$ with algebraic closure…

Let $E$ be an elliptic curve over ${\mathbf Q}$, and let $d\geq 1$ be fixed. A prime $p$ is a local torsion prime for $E$ if there exists an extension $K/{\mathbf Q}_p$ of degree a…

IFG field curve-index conjecture. If $K$ is an IFG field and $I\mid 2g-2$, then there exists a curve $C_{/K}$ of genus $g$ with $I(C)=I$.

Let $K=\mathbb{Q}_q$ be unramified over $\mathbb{Q}_p$, let $\overline{k}$ be an algebraic closure of its residue field, and let $P=\operatorname{Gal}(K^s/K^\mathrm{tr})$ be the wi…

Let $K=\mathbb{Q}_q$ be unramified over $\mathbb{Q}_p$, with maximal unramified extension $K^\mathrm{nr}$ and separable algebraic closure $K^s$. Let…

Hesselholt's descent conjecture. For all positive integers $q$, this canonical map is an isomorphism of pro-abelian groups, and the higher continuous cohomology groups of the pro-…

Let $G$ be the group under consideration, let $\widehat{G}$ denote its associated central extension, and let $\Pi$ be an irreducible cuspidal representation of $G$. The functor…

Kato's residue conjecture. The map $\Delta_j$ is an isomorphism for all $r\geq 1$ and $j\geq 0$. This conjecture concerns comparison of Kato homology on a regular proper flat model…

Let $K$ be a finite extension of $\mathbf{Q}_p$, with ring of integers $\mathcal{O}_K$ and maximal ideal $\mathfrak{m}_K$. Let $P,U\in T\cdot\mathcal{O}_K[[T]]$ satisfy … with…

Let $n$ be a positive integer and let $\sigma$ be a splitting type of degree $n$. For a prime $p$, let $\rho(n,\sigma;p)$ be the Haar measure of the $p$-adic polynomials of degree…

Local Ohno–Nakagawa box conjecture. The lattices $\mathcal V_{\tau,\tau^{n-1}f}$ and $\mathcal V_{2\tau^{-1},(2\tau^{-1})^{n-1}f}$ are naturally dual with duality constant…

Let $E/\mathbb{Q}$ be an elliptic curve without complex multiplication. A prime $p$ is a local torsion prime when $E$ has good reduction at $p$ and $E(\mathbb{Q}_p)[p]\neq 0$. Davi…

Local conjecture. Under these hypotheses,

Let $n$ be a positive integer, and let $\Sigma=(\Sigma_2,\Sigma_3,\Sigma_5,\ldots)$ be an acceptable collection of local specifications, where each $\Sigma_p$ is a set of…

Let $K$ be a local field with $\operatorname{char}K\neq2$, let $n$ be odd, and let $\tau$ be a nonzero parameter. Let $\mathcal V_{\tau,\tau^{n-1}f}$ and…

Let $K$ be a $2$-adic local field and let $0\leq t\leq s\leq e=v_K(2)$. Let $C_1\subseteq C_2$ be orders in a cubic $K$-algebra $R$ with $C_2/C_1\cong\mathcal{O}_K/\pi^{s-t}$ cycli…

Let $k$ be the residue field of a discrete valuation ring $R$ of characteristic zero, and let $k[[z_2]]/k[[x]]$ be a cyclic $\Gamma$-extension with Galois sub-extension…

Let $K$ be a number field, let $v$ be a finite place of $K$, and let $L/K$ be an algebraic extension. For an extension $w$ of $v$ to $L$, write $L_w$ for the corresponding completi…

Local root-number formula. The local root number of the Jacobian satisfies

Let $K$ be a local field with residue-field cardinality $q$, let $L$ be a lattice in $K^d$, and let $V$ be the tropicalization of a full-dimensional Gaussian in $K^d$ with lattice…

Sign conjecture. With the above notations, $G(Q)$ is $\pm1$.

Let $F$ be a non-Archimedean local field with residue field of cardinality $q$, let $K/F$ be a quadratic étale extension, and let $D$ be a division algebra over $F$ of invariant…

Let $F$ be a non-archimedean local field of characteristic not $2$, let $E/F$ be a quadratic extension, let $D$ be a central division $F$-algebra of dimension $d^2$, and let…

Let $K'/K$ be a geometric, purely inseparable, untwisted extension of geometric discrete valuation fields with finite exponent. Write $e$ for its ramification index, $f$ for its du…

The equivariant local Iwasawa main conjecture. There exists $\zeta^{(j)}_{L_\infty/K}\in K_1(\mathcal{Q}^c(\mathcal{G}))$ such that