Geodesic
The rationality of Stark-Heegner points over genus fields of real quadratic fields
Massimo Bertolini1 ; Henri Darmon2
1Dipartimento di Matematica<br/>Università degli Studi di Milano<br/>Via Saldini, 50<br/>20133 Milano<br/>Italy
2Department of Mathematics<br/>McGill University<br/>805 Shebrooke Street West<br/>Montreal, QC H3A 2K6<br/>Canada
Annals of mathematics, Tome 170 (2009) no. 1, pp. 343-369
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We study the algebraicity of Stark-Heegner points on a modular elliptic curve $E$. These objects are $p$-adic points on $E$ given by the values of certain $p$-adic integrals, but they are conjecturally defined over ring class fields of a real quadratic field $K$. The present article gives some evidence for this algebraicity conjecture by showing that linear combinations of Stark-Heegner points weighted by certain genus characters of $K$ are defined over the predicted quadratic extensions of $K$. The non-vanishing of these combinations is also related to the appropriate twisted Hasse-Weil $L$-series of $E$ over $K$, in the spirit of the Gross-Zagier formula for classical Heegner points.

DOI : 10.4007/annals.2009.170.343

Massimo Bertolini  1   ; Henri Darmon  2

1 Dipartimento di Matematica<br/>Università degli Studi di Milano<br/>Via Saldini, 50<br/>20133 Milano<br/>Italy
2 Department of Mathematics<br/>McGill University<br/>805 Shebrooke Street West<br/>Montreal, QC  H3A 2K6<br/>Canada 
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Massimo Bertolini; Henri Darmon. The rationality of Stark-Heegner points over genus fields of real quadratic fields. Annals of mathematics, Tome 170 (2009) no. 1, pp. 343-369. doi: 10.4007/annals.2009.170.343

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