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CriticalGZ SIGNED

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

Total Cost €

EC-Contrib. €

Partnership

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Outcomes and
results

CriticalGZ project word cloud

Explore the words cloud of the CriticalGZ project. It provides you a very rough idea of what is the project "CriticalGZ" about.

antonio theme forms independent beilinson weights family critical base body heavily fellowship cyclotomic variation note coleman kobayashi first collaborator right perrin context rely intend specialized interpolate affinoid denis weight full ordinary quadratic prove things goals height benjamin formula zagier believe proof bellaiche carry functions families readily desired riou lei rationals constructed attached cycles function precise francesc brief complemented modular imaginary kato pairings deform noting conjecture slope final yield thanks say gives gross joel generalisation adic derivative castella befitting construction individual howard members formulae suitable consist points benois heegner

Project "CriticalGZ" data sheet

The following table provides information about the project.

Coordinator	UNIVERSITY COLLEGE DUBLIN, NATIONAL UNIVERSITY OF IRELAND, DUBLIN Organization address address: BELFIELD city: DUBLIN postcode: 4 website: www.ucd.ie contact info title: n.a. name: n.a. surname: n.a. function: n.a. email: n.a. telephone: n.a. fax: n.a.
Coordinator Country	Ireland [IE]
Project website	https://maths.ucd.ie/
Total cost	152˙988 €
EC max contribution	152˙988 € (100%)
Programme	1. H2020-EU.1.3.2. (Nurturing excellence by means of cross-border and cross-sector mobility)
Code Call	H2020-MSCA-IF-2016
Funding Scheme	MSCA-IF-GF
Starting year	2017
Duration (year-month-day)	from 2017-08-01 to 2019-07-31

Partnership

Take a look of project's partnership.

#	participants	country	role	EC contrib. [€]
1	UNIVERSITY COLLEGE DUBLIN, NATIONAL UNIVERSITY OF IRELAND, DUBLIN UNIVERSITY COLLEGE DUBLIN, NATIONAL UNIVERSITY OF IRELAND, DUBLIN Organization address address: BELFIELD city: DUBLIN postcode: 4 website: www.ucd.ie contact info title: n.a. name: n.a. surname: n.a. function: n.a. email: n.a. telephone: n.a. fax: n.a.	IE (DUBLIN)	coordinator	152˙988.00
2	KOC UNIVERSITY KOC UNIVERSITY Organization address address: RUMELI FENERI YOLU SARIYER city: ISTANBUL postcode: 34450 website: www.ku.edu.tr contact info title: n.a. name: n.a. surname: n.a. function: n.a. email: n.a. telephone: n.a. fax: n.a.	TR (ISTANBUL)	participant	0.00
3	PRESIDENT AND FELLOWS OF HARVARD COLLEGE PRESIDENT AND FELLOWS OF HARVARD COLLEGE Organization address address: MASSACHUSETTS AVENUE 1350 city: CAMBRIDGE postcode: 2138 website: www.bidmc.harvard.edu contact info title: n.a. name: n.a. surname: n.a. function: n.a. email: n.a. telephone: n.a. fax: n.a.	US (CAMBRIDGE)	partner	0.00

Map

Project objective

The main objective of this project is to prove a p-adic Gross-Zagier formula for the critical slope p-adic L-functions attached to p-ordinary modular forms. As we will explain in the main body of this proposal, such a formula will lead, among other things, to a full proof of a conjecture of Perrin-Riou (that gives a precise comparison between p-adic Beilinson-Kato elements and Heegner points).

Our approach will rely heavily on the theme of p-adic variation and will consist of three major steps (which, we believe, are independent on their own right):

As the first step, we would like to interpolate the Heegner cycles associated to modular forms along Coleman families. This has been carried out for p-ordinary forms by Benjamin Howard (and complemented by the work of Francesc Castella, befitting our goals).

The second step is to carry out a construction of the two-variable p-adic L-function for the base change of a Coleman family (over an affinoid A, say) to the suitable imaginary quadratic field. We note here that such a p-adic L-function over the field of rationals has been constructed by Joel Bellaiche.

The third and final step is to prove p-adic Gross-Zagier formulae for individual (p-non-ordinary) members of the family. This has been carried out by S. Kobayashi for weight 2 forms; we aim to provide a generalisation of his work to higher weights.

Noting that p-adic height pairings readily deform well in families (thanks to the work of Denis Benois, in this context), we aim to prove a A-adic Gross-Zagier formula for the cyclotomic derivative of the base change p-adic L-function. This formula, when specialized to weight 2, will yield the desired formula.

In the duration of this fellowship, we also intend to carry out several projects with our long-term collaborator Antonio Lei. We shall provide a brief account for these in the main body of our proposal.

Publications

List of publications.
year	authors and title	journal	last update
2019	KÃ¢zÄ±m BÃ¼yÃ¼kboduk, Antonio Lei Rank-two Euler systems for symmetric squares published pages: 1, ISSN: 0002-9947, DOI: 10.1090/tran/7860	Transactions of the American Mathematical Society	2019-08-29

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The information about "CRITICALGZ" are provided by the European Opendata Portal: CORDIS opendata.