Number theory seminar in memory of F. Momose

A p-adic Gross-Zagier formula for diagonal cycles.


Victor Rotger Cerda


The conjectures of Birch and Swinnerton-Dyer, as generalized by Bloch, Kato and Beilinson, predict that the vanishing of the L-series of a motive at a critical point should be explained by the presence of rational algebraic null- homologous cycles. When the L-series has a simple zero, this expectation is made explicit through a (often conjectural) Gross-Zagier formula which relates the height of a canonical cycle to the first derivative at the critical point. These formulae tend to admit a p-adic avatar which is particularly useful for arithmetic applications: one expects that the image of the cycle under the p-adic Abel- Jacobi map can be recovered as the value of the p-adic L-function of the motive at a point outside the region of interpolation. We shall describe one instance of this philosophy, that we have worked out in detail in collaboration with Henri Darmon.


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