ProKuPa

The framework of the project is the p-adic Kudla program which is a new research program and whoe potential amplitude has been unravalled by recent developpements only a few years ago: such as p-adic families of algebraic cycles or even being currently more fully developped (such as higher Hida theory). To better situate the general framework, I recall briefly what is the classical Kudla program. It aims to study special values of (derivatives of) Eisenstein series, or theta liftings of modular forms, in terms of so-called special cycles on Shimura varieties. The seminal works that inspired that program are: \item The Gross-Kohnen-Zagier theorem (1987) that shows that relative positions of Heegner points on a modular curve are Fourier coefficients of a Jacobi modular form. Notice that this result refines the Gross-Zagier formula that allowed to prove the Birch-Swinnerton-Dyer conjecture in ranks 0 and 1. \item The Hirzebruch-Zagier theorem (1979) that proves that intersection products of Hirzebruch--Zagier divisors (images of modular curves) on a Hilbert modular surface are Fourier coefficients of a classical modular form. \end{enumerate} One of the main driving ideas of the p-adic Kudla program is to use the flexibility of p-adic families of modular forms (and algebraic) cycles varying p-adically e.g., with respect to the weight. N.B. In the classical set-up, only Eisenstein series E(s,z) are explicitly equipped with a complex variable s whose variation can be studied, as modular cuspforms are devoid of such. My point of view is to use a framework encompassing all Shimura varieties associated to an arbitrary Shimura datum (G,X), where G is a reductive group. This way, functorial lifts in the sense of Langlands e.g., theta liftings especially, enter fully in this higher dimensional program. One of the first goals of the p-adic Kudla program with respect to p-adic families is to have a variety of general results covering all p-adic families of algebraic cycles associated to any group G, as well as p-adic variants of a large class functorial liftings including theta liftings.