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A refined Beilinson-Bloch conjecture for motives of modular forms


Authors: Matteo Longo and Stefano Vigni
Journal: Trans. Amer. Math. Soc. 369 (2017), 7301-7342
MSC (2010): Primary 14C25, 11F11
DOI: https://doi.org/10.1090/tran/6947
Published electronically: May 5, 2017
Previous version of record: Original version posted May 5, 2017
Corrected version of record: Current version corrects errors introduced by the publisher.
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Abstract: We propose a refined version of the Beilinson-Bloch conjecture for the motive associated with a modular form of even weight. This conjecture relates the dimension of the image of the relevant $ p$-adic Abel-Jacobi map to certain combinations of Heegner cycles on Kuga-Sato varieties. We prove theorems in the direction of the conjecture and, in doing so, obtain higher weight analogues of results for elliptic curves due to Darmon.


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Additional Information

Matteo Longo
Affiliation: Dipartimento di Matematica, Università di Padova, Via Trieste 63, 35121 Padova, Italy
Email: mlongo@math.unipd.it

Stefano Vigni
Affiliation: Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, 16146 Genova, Italy
Email: vigni@dima.unige.it

DOI: https://doi.org/10.1090/tran/6947
Keywords: Modular forms, Beilinson--Bloch conjecture, Heegner cycles
Received by editor(s): February 1, 2016
Published electronically: May 5, 2017
Additional Notes: The authors were partially supported by PRIN 2010–11 “Arithmetic Algebraic Geometry and Number Theory”. The first author was also partially supported by PRAT 2013 “Arithmetic of Varieties over Number Fields”. The second author was also partially supported by PRA 2013 “Geometria Algebrica e Teoria dei Numeri”.
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