28 problems

Let $L/K$ be a finite weakly ramified Galois extension of number fields, with $G=\operatorname{Gal}(L/K)$, ring of integers $\mathcal{O}_L$, and different $\mathcal{D}_{L/K}$. When…

Chinburg's $\Omega(3)$ conjecture modulo the kernel group.

Let $K/\mathbb{Q}$ be Galois of degree $n$ with group $G$, let $\eta\in K^\times$ generate a multiplicative $\mathbb{Z}[G]$-module of $\mathbb{Z}$-rank $n$, and let…

Taylor's conjecture.

Generalized projective Euler characteristic conjecture. There exists an integer $Q(d)$ such that, whenever $G$ is finite and ${\rm gcd}(\#G,Q(d))=1$, one has

The higher-dimensional inverse Galois conjecture. The elements $t_{d+1}(X/Y)$, as $X\to Y$ ranges over all such torsors, generate

Let $E/\mathbb Q$ be an elliptic curve of rank $1$, let $\mathcal F_b$ be the twist family specified by the local data $b$, and write…

Let $H/F$ be a CM extension with $F$ totally real, let $S$ contain $S_\infty\cup S_{\operatorname{ram}}(H/F)$, and let $V^\theta$ be the $\mathbb{Z}[G]$-module occurring in the Rit…

Let $L/K$ be a weakly ramified Galois extension of number fields of odd degree, with $G=\operatorname{Gal}(L/K)$. Let $\mathfrak{a}_{L/K}$ and $\mathfrak{c}_{L/K}$ be the canonical…

Let $L/K$ be a weakly ramified Galois extension of number fields for which the square root of the inverse different $\mathcal{A}_{L/K}$ exists, and set $G=\operatorname{Gal}(L/K)$.…

Localized Arakelov class group conjecture. The two displayed equalities hold for every such finite Galois extension and set $P$.

The Arakelov class group conjecture.

Let $L/K$ be a weakly ramified odd-degree Galois extension of number fields. Let $\mathfrak{a}_{L/K}$ and $\mathfrak{c}_{L/K}$ be the global elements defined in the source. Global…

Let $L/K$ be a finite Galois extension of number fields with Galois group $G$, let $r>1$ be an integer, let $p\ne2$ be a prime, and let $S$ contain…

Let $L/K$ be a finite Galois extension of number fields, let $S$ be the relevant finite set of places of $K$, let $p$ be an odd prime, and let $r\ne0$ be an integer. Schneider's co…

Let $L/K$ be a finite Galois extension of number fields with arbitrary Galois group $G$, let $r>1$ be an integer, and let $p$ be an odd prime. Let $\mathcal J_r^S$ be the canonical…

Let $L/K$ be a finite Galois extension of number fields with Galois group $G$, let $r>1$ be an integer, and let $p$ be an odd prime. Let $\mathcal J_r^S$ be the canonical fractiona…

Let $K/\mathbb{Q}$ be Galois of degree $n$ with group $G$, and let $\eta\in K^\times$ generate a multiplicative $\mathbb{Z}[G]$-module $F$ of $\mathbb{Z}$-rank $n$. An element of…

Let $K/\mathbb{Q}$ be Galois, let $G=\operatorname{Gal}(K/\mathbb{Q})$, and let $F$ be the multiplicative $\mathbb{Z}[G]$-module generated by $\eta$. Suppose that $F$ is monogenic.…

Let $K$ be a finite extension of ${\mathbb Q}_p$, let $N/K$ be a finite Galois extension with Galois group $G$, and retain the notation $\tilde R_{N/K}$ and…

Let $K$ be a finite extension of ${\mathbb Q}_p$, let $N/K$ be a finite Galois extension with Galois group $G$, and let $T$ be a ${\mathbb Z}_p[G_K]$-lattice such that…

Chinburg's conjecture. If $G$ has no irreducible symplectic representation, then

Integrality conjecture. $E(G,X)$ should be describable within $K_0(\bar{\mathbb Q}_p[G])_{\mathbb Q}$ in terms of global geometric invariants of $\bar{X}$, in a way that generalise…

Let $G$ be a finite group and let $N/E$ be a tame Galois extension of number fields with Galois group $G$. Write $4\mathcal O_N4$ for the ring of integers of $N$, let…

Cyclic-generator conjecture. There exists $\omega\in U_{S,1}(K)$ such that $\tau(\omega)=\omega$ and the set