STNB2017, (31st edition)
The rationality of traces of Stark--Heegner points
Presenters
Marc Masdeu
Abstract
Let $E/\mathbb{Q}$ be an elliptic curve over the rationals, and let $K/\mathbb{Q}$ be a real quadratic field for which $L(E/K,s)$ vanishes to odd order at $s=1$. Let $p$ be a prime of multiplicative reduction. Darmon attached to each order $O$ of $K$ a set of points $P_i$ in $E(K_p)$ (as many as the class number of $O$), and conjectured that each of them was in fact defined over the ring class field of $O$. The goal of this talk is to define the Stark--Heegner points, and discuss (a particular case of) the main result of [BD09], which heavily uses [BD07a] to prove, under the additional assumption that E has \emph{split} multiplicative reduction at $p$, that the sum of these points is defined over $K$ (in particular, it is algebraic), and that it is non-torsion if and only if $L'(E/K,1)$ is nonzero.
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