On a theorem of Bertolini-Darmon on the rationality of Stark-Heegner points over genus fields of real quadratic fields

Mok, Chung Pang

doi:10.1090/tran/8254

Trans. Amer. Math. Soc.

2020

DOI: 10.1090/tran/8254

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On a theorem of Bertolini-Darmon on the rationality of Stark-Heegner points over genus fields of real quadratic fields

Chung Pang Mok

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2022

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“…. This section merely collects together the basic notations and principal results of [Da01], [BD09], [Mo17] and [LMY17]. Exact references for the analogous results needed to cover the more general setting of (1.2) are given along the way.…”

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confidence: 99%

Stark-Heegner points and diagonal classes

Darmon¹,

Rotger²

2022

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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 GK on them. The main conjectures of [Da01] 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) = 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 Lp(f /K, ψ) does not vanish at the point associated to (E/K, ψ).Résumé. -Les points de Stark-Heegner généralisent les points de Heegner quand un corps quadratique réel K remplace le corps quadratique imaginaire de la théorie de la multiplication complexe. La démarche p-adique qui les sous-tend fait que ces points sur une courbe elliptique E sont à priori locaux, définis sur sur une extension finie de Qp. On conjecture qu'ils sont de nature globale, qu'ils appartiennent aux groupes de Mordell-Weil de E sur certains corps de classe de K, et qu'ils satisfont une loi de réciprocité de Shimura décrivant l'action de GK := Gal( K/K). Les conjectures de [Da01] prédisent ainsi qu'une combinaison linéaire de points de Stark-Heegner pondérée par les valeurs d'un caractère ψ de GK appartient au sous-espace propre correspondant du groupe Mordell-Weil de E sur le corps de classe découpé par ψ, et qu'elle est non triviale si et seulement si L ′ (E/K, ψ, 1) = 0. On démontre que cette combinaison linéaire provient tout au moins d'une classe globale dans la ψ-partie du pro-p groupe de Selmer de E, et qu'elle est nontriviale lorsque la dérivée première d'une certaine fonction L p-adique associée à E ne s'annule pas en ψ.

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confidence: 99%

Stark-Heegner points and diagonal classes

Darmon¹,

Rotger²

2022

Preprint

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On a theorem of Bertolini-Darmon on the rationality of Stark-Heegner points over genus fields of real quadratic fields

Cited by 1 publication

References 29 publications

Stark-Heegner points and diagonal classes

Stark-Heegner points and diagonal classes

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