17 problems

Let $K$ be any number field and let $E$ be any elliptic curve over $K$. Write $K^{\operatorname{ab}}$ for the maximal abelian extension of $K$. Conjectural strengthenings. The foll…

Amoroso–Zannier's conjecture. There exists a constant $c(K)>0$ such that

Let $K$ be a number field, and let $A/K$ be an abelian variety. The discreteness conjecture for abelian varieties. 1. $A(K^{\rm ab})$ has the discreteness property. 2. If $A$ does…

Rubin–Stark conjecture. The Rubin–Stark element satisfies

Let $K$ be a number field, let $K^{\mathrm{ab}}$ be its maximal abelian extension, let $f:{\mathbb P}^1\to{\mathbb P}^1$ be a polynomial map of degree $d\geq2$ defined over $K$, an…

Pole-order conjecture. The series $D_{\Q,G}(s)$ should have a meromorphic continuation to

Let $E$ be an elliptic curve over $\mathbb{Q}$, and let $X$ be the set of even characters of $\operatorname{Gal}(\mathbb{Q}^{\operatorname{ab}}/\mathbb{Q})$. Character-occurrence f…

Let $E$ be an elliptic curve over $\mathbb{Q}$, and let $\mathbf L\subset\mathbb{Q}^{\operatorname{ab}}$ be a real abelian field containing only finitely many extensions of…

Let $\mathbb{Q}^{\operatorname{ab}}$ denote the maximal abelian extension of $\mathbb{Q}$, and let $\mathcal{O}_{\mathbf L}$ denote the ring of integers of a subfield $\mathbf L$.…

Let $L/K$ be an abelian CM extension of number fields, with $K$ totally real, and let $G=\operatorname{Gal}(L/K)$. For finite sets of places $S,T$ of $K$, assume $S$ contains all r…

Let $E$ be an elliptic curve defined over a number field $K$. The field $K^{\mathrm{ab}}$ is the compositum of all abelian extensions of $K$, and Property $\mathrm{(N)}$ is the pro…

Let $E$ be the elliptic curve under consideration, and let $F/\mathbb{Q}$ be an infinite real abelian extension containing only finitely many subfields of degree $2$, $3$, or $5$ o…

Let $E$ be an elliptic curve over $\mathbb{Q}$ and let $F/\mathbb{Q}$ be a real abelian extension containing only finitely many subfields of degree $2$, $3$, or $5$ over…

Let $K$ be a finite abelian extension of a totally real field $F$ containing a CM-subfield, and let $K_{\mathrm{CM}}$ be its maximal CM-subfield. Put $G:=\operatorname{Gal}(K/F)$,…

A part of Stark's conjecture. For every $\sigma\in G$,

Twisted-trace conjecture. The following two assertions hold: 1.…

Let $K$ be a global field, and let $L/K$ be an abelian extension of degree $n$. Write $\operatorname{Dec}(L/K)$ for the set of places of $K$ that split completely in $L$. Balazard'…