Stark-Heegner points over real quadratic fields

Motivated by the conjectures of “Mazur-Tate-Teitelbaum type” formulated in [BD1] and by the main result of [BD3], we describe a conjectural construction of a global point PK ∈ E(K), where E is a (modular) elliptic curve over Q of prime conductor p, and K is a real quadratic field satisfying suitable conditions. The point PK is constructed by applying the Tate p-adic uniformization of E to an explicit expression involving geodesic cycles on the modular curve X0(p). These geodesic cycles are a natural generalization of the modular symbols of Birch and Manin, and interpolate the special values of the Hasse-Weil L-function of E/K twisted by certain abelian characters of K. In the analogy between Heegner points and circular units, the point PK is analogous to a Stark unit, since it has a purely conjectural definition in terms of special values of L-functions, but no natural “independent” construction of it seems to be known. We call the conjectural point PK a “Stark-Heegner point” to emphasize this analogy. The conjectures of section 4 are inspired by the main result of [BD3], in which the real quadratic field is replaced by an imaginary quadratic field. The methods of [BD3], which rely crucially on the theory of complex multiplication and on the Cerednik-Drinfeld theory of p-adic uniformization of Shimura curves, do not seem to extend to the real quadratic situation. One must therefore content oneself with numerical evidence for the conjectures. This evidence is summarized in the last section.