Stark-Heegner points and diagonal classes

Abstract

Stark-Heegner points are conjectural substitutes for Heegner points when the imaginary quadratic field of the theory of complex multiplication is replaced by a real quadratic field $K$. They are constructed analytically as local points on elliptic curves with multiplicative reduction at a prime $p$ that remains inert in $K$, but are conjectured to be rational over ring class fields of $K$ and to satisfy a Shimura reciprocity law describing the action of $G_K$ on them. The main conjectures of \cite{darmon-hpxh} predict that any linear combination of Stark-Heegner points weighted by the values of a ring class character $ψ$ of $K$ should belong to the corresponding piece of the Mordell-Weil group over the associated ring class field, and should be non-trivial when $L'(E/K,ψ,1) \ne 0$. Building on the results on families of diagonal classes described in the remaining contributions to this volume, this note explains how such linear combinations arise from global classes in the idoneous pro-$p$ Selmer group, and are non-trivial when the first derivative of a weight-variable $p$-adic $L$-function associated to the Hida family passing through $f$ does not vanish at the point associated to $(E/K,ψ)$.