16 problems

Let $(L/K,S,T)$ be a basic triple, let $G=\operatorname{Gal}(L/K)$, and let $r\geq 0$ be an integer. Let $U_{S,T}$ be the group of $S$-units of $L$ that are congruent to $1$ at all…

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

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 $L/K$ 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 $G=\operatorname{Gal}(L/K)$.…

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.

Chinburg's $\Omega(3)$ 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 $K$ 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…

Taylor's conjecture.

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

Let $K/k$ be an abelian extension of number fields with Galois group $G$, and let $S$ contain the places ramified in $K/k$. For each odd prime $\ell$, let $\mathcal{J}^r(K/k)$ be t…