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Heegner cycles and p-adic L-functions

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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 Nekovář’s work a proof of the parity conjecture in this setting.

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Notes

  1. See [17, Thm. 20.6] for the existence of the quadratic base change, and [18, §11] for the definition of L-functions on \(\mathrm{GL}(2)\).

  2. Here our convention is that p-adic cyclotomic character has Hodge–Tate weight \(+1\).

  3. As explained in [30, Example (5.3.4)(5)], this follows from properties [loc.cit.,(2)-(3)] for \(\mathcal T_\phi \), whose verification is immediate. Indeed, \((\mathcal T_\phi ,\mathcal T^+_{p,\phi })\) satisfies the Panchishkin condition of [30, Def. (3.3.1)] by construction, and \(\mathcal T_\phi \) is pure of weight 1 at all finite places, since Ramanujan’s conjecture holds for f; and anticyclotomic Hecke characters are pure of weight 0.

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Acknowledgements

Fundamental parts of this paper were written during the visits of the first-named author to the second-named author in Taipei during February 2014 and August 2014; it is a pleasure to thank NCTS and the National Taiwan University for their hospitality and financial support. We would also like to thank Ben Howard, Shinichi Kobayashi and David Loeffler for their comments and enlightening conversations related to this work.

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Authors and Affiliations

  1. Mathematics Department, Princeton University, Fine Hall, Washington Road, Princeton, NJ, 08544-1000, USA

    Francesc Castella

  2. Institute of Mathematics, Academia Sinica, Taipei, 10617, Taiwan

    Ming-Lun Hsieh

  3. National Center for Theoretic Sciences, Taipei, Taiwan

    Ming-Lun Hsieh

  4. Department of Mathematics, National Taiwan University, Taipei, Taiwan

    Ming-Lun Hsieh

Authors
  1. Francesc Castella
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  2. Ming-Lun Hsieh
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Correspondence to Francesc Castella.

Additional information

Communicated by Toby Gee.

During the preparation of this paper, F. Castella was partially supported by Grant MTM2012-34611 and by Prof. Hida’s NSF Grant DMS-0753991. M.-L. Hsieh was partially supported by a MOST Grant 103-2115-M-002-012-MY5.

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Castella, F., Hsieh, ML. Heegner cycles and p-adic L-functions. Math. Ann. 370, 567–628 (2018). https://doi.org/10.1007/s00208-017-1517-3

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