9 problems

Let $A$ be an abelian variety defined over a number field, and let $A_{\mathrm{tors}}$ denote its torsion subgroup. S. David's conjecture. The field $\mathbb{Q}(A_{\mathrm{tors}})$…

Let $\mathbb{Q}^{\mathrm{ab}}$ be the maximal abelian extension of $\mathbb{Q}$, equivalently the field obtained by adjoining all torsion points of the multiplicative group. A fiel…

Let $f\in S_k(\Gamma_1(N))$ be a normalized eigenform, and let $\rho_{f,v}$ denote its Galois representation at a non-Archimedean place $v$ and $\widehat{\rho}_f$ the product of th…

Let $K$ be a number field and let $E$ be an elliptic curve over $K$. Write $E_{\text{tor}}$ for the torsion subgroup of $E(\overline{\mathbb{Q}})$, and let $K(E_{\text{tor}})$ be t…

Let $k\geq 2$, let $f\in S_k(\Gamma_0(N))$ be a normalized eigenform, let ${\mathbb{K}}_f$ be the number field generated by its Hecke eigenvalues, and let $\mathcal{O}_f$ be its ri…

Product conjecture. If $E$ and $F$ have the Bogomolov property, then the group product $E^*F^*$ also has the Bogomolov property.

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…

Let $A$ be an abelian variety defined over a number field $K$, let $\mathcal{L}$ be a symmetric ample line bundle on $A/K$, and let $L/K$ be a finite extension. For the Néron–Tate…

Soit $A$ une variété abélienne définie sur un corps de nombres $K$. Pour un sous-corps $L\subset\overline{K}$, on dit que $L$ a la propriété de Bogomolov relativement à $A$ si, pou…