The \(p\)-adic variation of the Gross-Kohnen-Zagier theorem (Q2317904)

Heegner points present, to this date, the main technique for constructing non-torsion points on rational elliptic curves of rank 1. The seminal paper of Gross, Kohnen, and Zagier [\textit{B. Gross} et al., Math. Ann. 278, 497--562 (1987; Zbl 0641.14013)] determines the relations between Heegner points associated with different discriminants, as their locations in the associated rank 1 group correspond to the coefficients of the Jacobi form arising as the theta lift of the newform associated with the elliptic curve. The norms of these points are also related (by the Gross-Zagier formula from [\textit{B. H. Gross} and \textit{D. B. Zagier}, Invent. Math. 84, 225--320 (1986; Zbl 0608.14019)]) to the values of the central derivatives of the various twists of the \(L\)-function associated with the elliptic curve. This result has been generalized in several directions, and the paper in question aims at such a generalization to the setting of \(p\)-adic families of modular forms. Indeed, modular forms of different weights, with integral coefficients, satisfy congruences modulo prime powers, allowing for the creation of \(p\)-adic modular forms, by \(p\)-adic interpolation. This paper investigates the variation of the associated Jacobi forms in the \(p\)-adic family, and relate them to some kind of \(p\)-adic \(L\)-functions, involving genus characters. These \(L\)-functions are also shown to be connected, via an appropriate element of the Selmer group of Greenberg associated with the big Galois representation attached to the \(p\)-adic family, to the cohomology classes constructed in Reference number 23. The requirement to work with newforms introduces additional Euler factors, arising from the \(p\)-stabilization involved. More precisely, the formal power series \(F_{\infty}\), with coefficients in the branch of the Hida family passing through our initial modular form \(f_{2k_{0}}\), has a natural image as a series with coefficients in rigid analytic functions, and the latter specialize to actual modular forms \(f_{2k}\) for positive even integers \(2k\) in the domain of convergence. On the other hand, there is a map on measure-theoretic modular symbols, sending a special elemnt to the function yielding the classical (orbit) modular symbol associated with \(f_{2k}\) for every such \(k\), up to an algebraic scalar. This is used for constructing the \(p\)-adic \(L\)-function, which is called a \emph{square-root \(L\)-function} because its square is related, again for such \(k\), to the algebraic \(L\)-function associated with \(f_{2k}\). Then, under certain assumptions (including the \(L\)-function being odd and having analytic rank 1 at some point), the cohomology classes extending the big Heegner points are non-torsion and the Selmer group of Greenberg has rank 1, showing that the former are essentially constant multiples of one another in the latter group. Then a Gross-Kohnen-Zagier type theorem is established in Reference number 37 in case the height pairing is positive definite (compared with Reference number 39, with a modified theta lift). The cohomology classes arising from the \(p\)-adic family and the étale Abel-Jacobi images of the classical Heegner cycles are related by a result from Reference number 9, again up to some explicit constants. Combining these results produces the relation between the cycle at the weight \(2k\) and the value of the (square-root) \(L\)-function at \(2k\), which motivates the conjecture of a GKZ theorem for the whole \(p\)-adic family. The paper is divided into 5 sections. Section 1 is the Introduction, with a crude form of the main conjecture and theorems. Section 2 skims briefly over binary quadratic forms and the (essentially Shimura) theta lift. Section 3 introduces the rings from Hida theory, the modular sybmols, and the \(p\)-adic \(L\)-functions. Section 4 defines the various types of Heegner cycles (or classes), explains the relations betwenn thema as well as the assumptions under which the main conjecture is posed. Finally, Section 5 contains the main result and the main conjecture.