## Periodic Reporting for period 2 - CriticalGZ (Critical Slope Gross-Zagier formula and Perrin-Riou's Conjecture)

Reporting period: 2018-08-01 to 2019-07-31

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.

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.

"All the three components (A, B and C above) of the project are complete. Our results are compiled within two preprints.

The first of these is entitled “p-adic Gross-Zagier formula at critical slope and a conjecture of Perrin-Riou”, we compete the tasks B and C above. This preprint is joint with Robert Pollack and Shu Sasaki. It is posted on ArXiv with identifier 1811.08216.

The goals of component A (construction of universal Heegner cycles) has been achieved in our preprint with Antonio Lei entitled ""Interpolation of Generalized Heegner Cycles in Coleman Families"". It is posted on ArXiv with identifier 1907.04086.

PI was also invited to address in important gatherings: “Iwasawa 2017” in Tokyo, “p-adic L-functions and algebraic cycles” in Taipei, “Special Cycles on Shimura Varieties and Iwasawa Theory” in Lausanne, “Stark's Conjectures, Iwasawa theory and related topics” in Exeter, ""Journees Arithmetiques"" in Istanbul and “Recent advances in the arithmetic of Galois representations” in Genoa. PI visited important mathematical centers; among others, he was invited to Bonn, Essen, London, Munich, Columbia U., Harvard U., MIT, Stanford U. and U. of Washington to speak at the local number theory seminars.

The dissemination activities the PI carried out within this action were not limited to scientific talks he delivered. He gave an instructional lecture series in Sofia University in Tokyo, he had regular mathematical exchange with Chi-Yun Hsu (a student of B. Mazur at Harvard) and he organized one of the problem sessions in the Arizona Winter School 2018.

In summary, the scientific portion of the action is complete. The PI worked hard to fulfill the dissemination goals of the action: He participated in 8 conferences/workshops (as an invited speaker in 7 of these), delivered 16 seminar talks in three continents and was invited to organize 2 instructional lecture series. We organized a satellite conference to Journees Arithmetiques XXXI in Istanbul, entitled ""p-adic modular forms""."

The first of these is entitled “p-adic Gross-Zagier formula at critical slope and a conjecture of Perrin-Riou”, we compete the tasks B and C above. This preprint is joint with Robert Pollack and Shu Sasaki. It is posted on ArXiv with identifier 1811.08216.

The goals of component A (construction of universal Heegner cycles) has been achieved in our preprint with Antonio Lei entitled ""Interpolation of Generalized Heegner Cycles in Coleman Families"". It is posted on ArXiv with identifier 1907.04086.

PI was also invited to address in important gatherings: “Iwasawa 2017” in Tokyo, “p-adic L-functions and algebraic cycles” in Taipei, “Special Cycles on Shimura Varieties and Iwasawa Theory” in Lausanne, “Stark's Conjectures, Iwasawa theory and related topics” in Exeter, ""Journees Arithmetiques"" in Istanbul and “Recent advances in the arithmetic of Galois representations” in Genoa. PI visited important mathematical centers; among others, he was invited to Bonn, Essen, London, Munich, Columbia U., Harvard U., MIT, Stanford U. and U. of Washington to speak at the local number theory seminars.

The dissemination activities the PI carried out within this action were not limited to scientific talks he delivered. He gave an instructional lecture series in Sofia University in Tokyo, he had regular mathematical exchange with Chi-Yun Hsu (a student of B. Mazur at Harvard) and he organized one of the problem sessions in the Arizona Winter School 2018.

In summary, the scientific portion of the action is complete. The PI worked hard to fulfill the dissemination goals of the action: He participated in 8 conferences/workshops (as an invited speaker in 7 of these), delivered 16 seminar talks in three continents and was invited to organize 2 instructional lecture series. We organized a satellite conference to Journees Arithmetiques XXXI in Istanbul, entitled ""p-adic modular forms""."

"As noted above, Antonio Lei (whom we hosted at UCD during the incoming phase) and the PI interpolated Heegner cycles in families. Our method is based on the theme of ""p-adic construction of rational points"" (which kicked off with Rubin's paper with epynomous title) and relies on BDP formulae for generalized Heegner cycles. Crucial ingredients are the big Perrin-Riou logarithm in this context and the two-variable BDP p-adic L-function, which we constructed along the way.

With Pollack and Sasaki, we have given an alternative construction of big Heegner cycles, by realizing the eigencurve within Emerton’s completed cohomology for GL2. One fundamental difficulty was to find a replacement for Hida's ordinary projector. At the top level of cohomology, Emerton has introduced what he calls the ""Jacquet module functor"", which cuts down the completed cohomology to its finite slope part. By establishing an explicit relation between Emerton's Jacquet module functor and Casselman's Jacquet functor, we were able to overcome this technical challenge.

All in all, both the outgoing and the incoming phase of this project have been fabulously productive: The PI benefited immensely from his regular discussions with Barry Mazur at Harvard, which were a great source of inspiration and motivation. In particular, the approach we pursued with A. Lei towards the construction of universal Heegner cycles owes greatly to these discussions. The secondment periods in Germany were extremely crucial for our timely progress in the project as well: If it was not for the extended periods of time the PI got to spend with his collaborators R. Pollack and S. Sasaki, we would be nowhere near the complete resolution of the proposed problem. Given the level competition in the field, it was very important to reach this stage in a timely manner. The PI learned a great deal of mathematics through his intense interactions with S. Sasaki: We have not only gained expertise in Emerton’s sweeping theory of completed cohomology in a relatively short period of time, we have managed to incorporate it with Iwasawa theoretic considerations. We believe that our joint work entitled “Big Heegner cycles with coefficients in Emerton's completed cohomology” will serve as a signpost for many other works in the field.

"

With Pollack and Sasaki, we have given an alternative construction of big Heegner cycles, by realizing the eigencurve within Emerton’s completed cohomology for GL2. One fundamental difficulty was to find a replacement for Hida's ordinary projector. At the top level of cohomology, Emerton has introduced what he calls the ""Jacquet module functor"", which cuts down the completed cohomology to its finite slope part. By establishing an explicit relation between Emerton's Jacquet module functor and Casselman's Jacquet functor, we were able to overcome this technical challenge.

All in all, both the outgoing and the incoming phase of this project have been fabulously productive: The PI benefited immensely from his regular discussions with Barry Mazur at Harvard, which were a great source of inspiration and motivation. In particular, the approach we pursued with A. Lei towards the construction of universal Heegner cycles owes greatly to these discussions. The secondment periods in Germany were extremely crucial for our timely progress in the project as well: If it was not for the extended periods of time the PI got to spend with his collaborators R. Pollack and S. Sasaki, we would be nowhere near the complete resolution of the proposed problem. Given the level competition in the field, it was very important to reach this stage in a timely manner. The PI learned a great deal of mathematics through his intense interactions with S. Sasaki: We have not only gained expertise in Emerton’s sweeping theory of completed cohomology in a relatively short period of time, we have managed to incorporate it with Iwasawa theoretic considerations. We believe that our joint work entitled “Big Heegner cycles with coefficients in Emerton's completed cohomology” will serve as a signpost for many other works in the field.

"