70 problems

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…

Breuning's local equivariant epsilon constant conjecture.

Let $K$ be a complete discrete valued field with residue field of characteristic $p>0$, let $L/K$ be a finite Galois extension with Galois group $G$, and suppose that $k_L/k_K$ is…

Knapp's conjecture.

Let $k$ be an algebraically closed field of characteristic $p$, and let $k[[t]]$ be the formal power series ring. Local Oort conjecture. Every cyclic extension of $k[[t]]$ lifts to…

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…

Local-field curve-index conjecture. (a) For all $g$ and $I\mid g-1$, if $K$ admits a totally ramified cyclic extension of degree $I$, then there exists a genus-$g$ curve $C_{/K}$ w…

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 $K$ and $K'$ be one-dimensional local fields in the category $\operatorname{LF}(1)=\operatorname{LF}_0(1)\coprod\operatorname{LF}_p(1)$. Write $\Gamma_K$ and $\Gamma_{K'}$ for…

Let $G$ be an algebraic group over a local field of positive characteristic, and let $U$ be a discrete unipotent subgroup of $G$. A group is residually finite if every nontrivial e…

Let $G$ be a semisimple algebraic group over a local field, and let $\pi$ be a tempered representation of $G$. Consider the measures used in the c-temperedness framework, whose Rad…

Let $X$ be a variety over a local field $K$ with residue field $k$, and let $X_k$ be the special fibre of a regular proper model $\mathcal X$ over $\mathcal O_K$. Write…

Let $F$ be the two-dimensional Lubin–Tate formal group considered in the paper, with coefficient field $U_{h_{1}+h_{2}}$, and let $U_{h_{1}+h_{2}}(\operatorname{Tors}(F))$ denote t…

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…

Generalized density functional-equation conjecture. The density satisfies

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…

Let $K$ be a local field, let $A$ be an elliptic curve over $K$, and let $p$ be the residue characteristic. The group of principal homogeneous spaces over $A$ is known to be dual t…

Fourier self-duality conjecture. The function $g_C$ is its own Fourier transform.

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…

Local Ohno–Nakagawa duality conjecture. The Fourier dual, as a function of $L$, satisfies

Let $E$ be a non-CM elliptic curve. David--Weston finiteness conjecture. There are only finitely many primes $p$ such that … Such primes correspond to local $p$-torsion phenomena,…