Abstract. -We prove a general formula for the p-adic heights of Heegner points on modular abelian varieties with potentially ordinary (good or semistable) reduction at the primes above p. The formula is in terms of the cyclotomic derivative of a Rankin-Selberg p-adic L-function, which we construct. It generalises previous work of Perrin-Riou, Howard, and the author, to the context of the work of Yuan-Zhang-Zhang on the archimedean Gross-Zagier formula and of Waldspurger on toric periods. We further construct analytic functions interpolating Heegner points in the anticyclotomic variables, and obtain a version of our formula for them. It is complemented, when the relevant root number is +1 rather than −1, by an anticyclotomic version of the Waldspurger formula.When combined with work of Fouquet, the anticyclotomic Gross-Zagier formula implies one divisibility in a p-adic Birch and Swinnerton-Dyer conjecture in anticyclotomic families. Other applications described in the text will appear separately.

Let f be a primitive Hilbert modular form of parallel weight 2 and level N for the totally real field F , and let p be a rational prime coprime to 2N . If f is ordinary at p and E is a CM extension of F of relative discriminant ∆ prime to N p, we give an explicit construction of the p-adic Rankin-Selberg L-function Lp(fE, ·). When the sign of its functional equation is −1, we show, under the assumption that all primes ℘|p are principal ideals of OF which split in OE, that its central derivative is given by the p-adic height of a Heegner point on the abelian variety A associated with f .This p-adic Gross-Zagier formula generalises the result obtained by Perrin-Riou when F = Q and (N, E) satisfies the so-called Heegner condition. We deduce applications to both the p-adic and the classical Birch and Swinnerton-Dyer conjectures for A.

We study the variation of the local Langlands correspondence for GLn in characteristic-zero families. We establish an existence and uniqueness theorem for a correspondence in families, as well as a recognition theorem for when a given pair of Galois-and reductive-group-representations can be identified as local Langlands correspondents. The results, which can be used to study local-global compatibility questions along eigenvarieties, are largely analogous to those of Emerton, Helm, and Moss on the local Langlands correspondence over local rings of mixed characteristic. We apply the theory to the interpolation of local zeta integrals and of L-and ε-factors.

We formulate a multi-variable p-adic Birch and Swinnerton-Dyer conjecture for pordinary elliptic curves A over number fields K. It generalises the one-variable conjecture of Mazur-Tate-Teitelbaum, who studied the case K = Q and the phenomenon of exceptional zeros. We discuss old and new theoretical evidence towards our conjecture and in particular we fully prove it, under mild conditions, in the following situation: K is imaginary quadratic, A = E K is the base-change to K of an elliptic curve over the rationals, and the rank of A is either 0 or 1.The proof is naturally divided into a few cases. Some of them are deduced from the purely cyclotomic case of elliptic curves over Q, which we obtain from a refinement of recent work of Venerucci alongside the results of Greenberg-Stevens, Perrin-Riou, and the author. The only genuinely multivariable case (rank 1, two exceptional zeros, three partial derivatives) is newly established here. Its proof generalises to show that the 'almost-anticyclotomic' case of our conjecture is a consequence of conjectures of Bertolini-Darmon on families of Heegner points, and of (partly conjectural) p-adic Gross-Zagier and Waldspurger formulas in families.

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