Darmon points on elliptic curves over totally real fields
Let F be a number field and let E be an elliptic curve over F of conductor an ideal N of F . We assume throughout that F is totally real: in that case, it is known, under minor hypotheses, that there is a newform f of weight 2 on Γ0(N) over F whose L-function coincides with that of E (see, e.g., [Zha01] or [Dar04, §7.4]). We fix a quadratic extension K/F . When K is totally complex, a classical theory produces a family of Heegner points of E, defined over ring class fields of K. The Galois action on them is given by a Shimura reciprocity law, and their heights relate to the derivative of the L-function of E over K at 1 (see, e.g., [Zha01] or [Dar04, §7.5]). We assume therefore from now on that K has at least one real place. The goal of the theory of Darmon points (earlier called Stark-Heegner points) is to extend to such K the construction of Heegner points. In [Dar01], Darmon started the theory in the case F = Q and K a totally real quadratic field. In [Gre09], Greenberg generalizes this work to give a conjectural construction of points in the case where F is arbitrary totally real quadratic field of narrow class number one, E is semistable, N 6= (1), the sign of the functional equation of E over K is −1, there is a prime dividing the conductor of E that is inert in K, and the discriminant of K is coprime to N . The techniques used are p-adic in nature. In [Gar11], Gartner generalizes the work of [Dar04, Chap 8] and [DL03] to give a construction of what he calls Darmon points in certain situations using archimedian techniques. The conditions under which Gartner’s construction works are a bit too technical to describe here, but we shall describe them in Section 2.1. All the constructions share a basic outline: one computes an archimedean or p-adic integral of the modular form associated to E/F , and plugs the resulting value into a Weierstrass of Tate parametrization of E to produce the Darmon point. Let O ⊆ K be an OF -order such that Disc(O/OF ) is coprime to N . In this article, we show that if the sign of the functional equation of E over K is −1, the discriminant of K is coprime to N , and the part of N divisible by primes that are inert in K is square-free, then one can apply either the construction of Gartner or the construction of Greenberg (after removing the assumption that F has narrow class number one, which we show how to do) to conjecturally associate to O a point that we call a Darmon point (actually there are choices in the construction, and so one gets several Darmon points). This point is intially defined over a transcendental extension of K, but we conjecture that the point is algebraic, defined over the narrow ring class field extension of K associated to the order O. This point comes with an action of the narrow class group of O, and we state a conjectural Shimura receprocity law for this action. In Section 2 we recall and slightly modify the constructions of Greenberg and Gartner. In Section 3 we show how one of the two constructions can be carried out under our hypotheses. We assume throughout this article that the reader is familiar with [Gre09] and [Gar11].
Hyperbolic tessellations, modular symbols, and elliptic curves over complex quadratic fields
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The rationality of Stark-Heegner points over genus fields of real quadratic fields
We study the algebraicity of Stark-Heegner points on a modular elliptic curve E. These objects are /?-adic points on E given by the values of certain /?-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.