Exceptional zero formulae and a conjecture of Perr

Authors: Rodolfo Venerucci

Publish Date: 2015/07/15

Volume: 203, Issue: 3, Pages: 923-972

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Abstract

Let A/mathbfQ be an elliptic curve with split multiplicative reduction at a prime p We prove an analogue of a conjecture of PerrinRiou relating padic Beilinson–Kato elements to Heegner points in AmathbfQ and a large part of the rankone case of the Mazur–Tate–Teitelbaum exceptional zero conjecture for the cyclotomic padic Lfunction of A More generally let f be the weighttwo newform associated with A let f infty be the Hida family of f and let L pf infty ks be the Mazur–Kitagawa twovariable padic Lfunction attached to f infty We prove a padic Gross–Zagier formula expressing the quadratic term of the Taylor expansion of L pf infty ks at ks=21 as a nonzero rational multiple of the extended heightweight of a Heegner point in AmathbfQMuch of the work on this article was carried out during my PhD at the University of Milan It is a pleasure to express my sincere gratitude to my supervisor Prof Massimo Bertolini who constantly encouraged and motivated my work Every meeting with him has been a source of ideas and enthusiasm this paper surely originated from and grew up through these meetings I would like to thank Marco Seveso for a careful reading of the paper and for many interesting discussions related to this work I am also grateful to the anonymous referee the current version of the article is greatly inspired by his/her corrections and valuable comments which helped me to significantly clarify and improve the exposition

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