CriticalGZ · Critical Slope Gross-Zagier formula and Perrin-Riou's Conjecture

Critical Slope Gross-Zagier formula and Perrin-Riou's Conjecture

One of the most fascinating features of L-functions is that their special values are expected to have an arithmetic interpretation, albeit being analytic objects by definition. A predecessor of this philosophy (linking the special values of L-functions to arithmetic data) is the celebrated BSD conjecture, one of the seven Clay Mathematics Institute’s Millennium problems, which predicts the rank of an elliptic curve E (and various other arithmetic invariants associated to E) in terms of the Hasse-Weil L-function of E. The main line of investigation the PI leads as part of the current project concerns also the leading coefficients of a specific family of (p-adic) L-functions, for which the PI and his collaborators utilize p-adic variational techniques. The themes that are covered within the scope of this project have received immense interest for the past 40 years and they still dominate a good portion of the research activity in Algebraic Number Theory (that goes under the title “Langlands’ Programme” and “Bloch-Kato Conjectures”). The principal goal of the proposed project is to prove a p-adic Gross-Zagier formula at critical-slope. Combined with the previous work of the PI, this formula allows us to deduce the strong form of Perrin-Riou’s conjectures. Furthermore, it also follows that at least one (of the two) p-adic height pairings are non-trivial, and consequently, this yields the full proof of the Birch and Swinnerton-Dyer Conjecture when the analytic rank equals 1, away from the support of the conductor of the elliptic curve. The research programme outlined above was carried out jointly with Antonio Lei (Laval University), Robert Pollack (Boston University and MPI-Bonn) and Shu Sasaki (Essen). We indicate why a proof of a Gross-Zagier formula at critical slope (the main objective of the current project) requires an approach different from its p-ordinary and p-supersingular counterparts. The method of Perrin-Riou and Kobayashi relies heavily on a Rankin-Selberg construction of the p-adic L-function they have studied (so as to allow them to express its derivative as a p-adic Petersson product of the modular attached to the elliptic curve and another modular form, whose Fourier coefficients are related to the values of height pairings involving Heegner points). This approach fails for the critical slope. Our strategy to deal with this technical obstacle is to resort to the theme of p-adic variation. In order to achieve the objectives recorded above, PI had proposed to carry out the following three tasks. A. Interpolation of Heegner Cycles in Coleman families B. p-adic Gross-Zagier formula for non-ordinary newforms of higher weight C. Construction of a 2-variable p-adic L-function for Coleman families over quadratic imaginary fields Besides the research oriented goals outlined above, this project aimed to reach out to a large group of scientific community, through presentations in conferences and workshops, contributions to training programs, regular interactions with the local mathematical community.

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