Heegner points on Mumford-Tate curves

Hida families and rational points on elliptic curves

“…This L-function, introduced in Definition 3.5 and Remark 3.6 of Sect. 3 and denoted L p ( f ∞ /K ; k, s), interpolates the anticyclotomic p-adic L-functions attached to f k /K defined in [BD1] and in [BDIS]. Of special interest is its restriction to the central critical line…”

“…It would be worthwhile to understand Theorem 7 in the framework of the p-adic Birch and Swinnerton-Dyer conjectures for the anticyclotomic setting, in the same way that the conjectures of [BD1] inspired the main results of [BD2]. Section 3.3 uses work of Gross, Hatcher, Zhang and Hui Xue to describe a precise interpolation property relating L p ( f ∞ /K, k) to the algebraic parts of the corresponding classical central critical values L( f k /K, k/2) for even integers k ≥ 2.…”

“…Definition 2.1 differs from the one that is encountered in [BD1]- [BDIS], where an automorphic form on B × "of level Σ" is defined to be a functionφ :B × −→ A satisfying…”

The subconvexity problem for Rankin–Selberg L-functions and equidistribution of Heegner points. II

“…Indefinite quaternion algebras. For more background on Shimura curves and Heegner points, we refer to [BD96,Da04,Z02,Z04].…”

“…Here ν * q ,d g(z σ 0 ) = g(ν q ,d (z σ 0 )) = g((ν q ,d z 0 ) σ ); Indeed, one can check (e.g., by using the adelic formulation of ν q ,d and of the Heegner points given in [BD96]) that ν q ,d z 0 is a Heegner point on X 0 (q − , q ) and then, since ν q ,d is defined over Q, it commutes with the Galois action. This reduces us to the case where g is an L 2 -normalized Maass newform of level q + .…”