Heegner cycles and p-adic L-functions

2015/05/29 by Francesc Castella, Castella, Francesc, Ming-Lun Hsieh +1 · 3 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1505.08165

openalex publication_date 2015/05/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we deduce the vanishing of Selmer groups for the Rankin-Selberg convolution of a cusp form with a theta series of higher weight from the nonvanishing of the associated L-value, thus establishing the rank 0 case of the Bloch-Kato conjecture in these cases. Our methods are based on the connection between Heegner cycles and p-adic L-functions, building upon recent work of Bertolini, Darmon and Prasanna, and on an extension of Kolyvagin's method of Euler systems to the anticyclotomic setting. In the course of the proof, we also obtain a higher weight analogue of Mazur's conjecture (as proven in weight 2 by Cornut-Vatsal), and as a consequence of our results, we deduce from Nekovar's work a proof of the parity conjecture in this setting.

Cited by

• On the Tamagawa number conjecture for modular forms twisted by anticyclotomic Hecke characters
• On quaternionic ordinary families of modular forms and p-adic L-functions
• Iwasawa Main Conjecture for Heegner Points: Supersingular Case

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• Heegner Point Kolyvagin System and Iwasawa Main Conjecture