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CM cycles on Kuga–Sato varieties over Shimura curves and Selmer groups

  • Yara Elias and Carlos de Vera-Piquero EMAIL logo
Published/Copyright: July 6, 2017
Forum Mathematicum
From the journal Forum Mathematicum Volume 30 Issue 2

Abstract

Given a modular form f of even weight larger than two and an imaginary quadratic field K satisfying a relaxed Heegner hypothesis, we construct a collection of CM cycles on a Kuga–Sato variety over a suitable Shimura curve which gives rise to a system of Galois cohomology classes attached to f enjoying the compatibility properties of an Euler system. Then we use Kolyvagin’s method [21], as adapted by Nekovář [28] to higher weight modular forms, to bound the size of the relevant Selmer group associated to f and K and prove the finiteness of the (primary part) of the Shafarevich–Tate group, provided that a suitable cohomology class does not vanish.

Keywords: Modular forms; Galois representations; Selmer groups; Heegner cycles
MSC 2010: 11G18; 11G40; 11R34; 11F80

Communicated by Jan Bruinier

Acknowledgements

We are very grateful to Henri Darmon and Victor Rotger for many useful discussions on the topic of this paper. We also thank Erick Knight for his helpful clarifications and suggestions on the proof of Lemma 3.2, and the anonymous referee for valuable comments that contributed to improve the exposition of the paper.

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Received: 2017-1-11
Revised: 2017-6-20
Published Online: 2017-7-6
Published in Print: 2018-3-1

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. The variance of divisor sums in arithmetic progressions
  3. Characterizing Lie groups by controlling their zero-dimensional subgroups
  4. CM cycles on Kuga–Sato varieties over Shimura curves and Selmer groups
  5. Towards a Goldberg–Shahidi pairing for classical groups
  6. On the non-existence of cyclic splitting fields for division algebras
  7. Sequential motion planning algorithms in real projective spaces: An approach to their immersion dimension
  8. Very ampleness of the bicanonical line bundle on compact complex 2-ball quotients
  9. Anharmonic solutions to the Riccati equation and elliptic modular functions
  10. A non-homogeneous local Tb theorem for Littlewood–Paley g*λ-function with Lp-testing condition
  11. Saturation rank for finite group schemes: Finite groups and infinitesimal group schemes
  12. On symplectic semifield spreads of PG(5,q2), q odd
  13. From Freudenthal’s spectral theorem to projectable hulls of unital Archimedean lattice-groups, through compactifications of minimal spectra
  14. Golodness and polyhedral products for two-dimensional simplicial complexes
https://doi.org/10.1515/forum-2017-0008
Keywords for this article
Modular forms; Galois representations; Selmer groups; Heegner cycles

Articles in the same Issue

  1. Frontmatter
  2. The variance of divisor sums in arithmetic progressions
  3. Characterizing Lie groups by controlling their zero-dimensional subgroups
  4. CM cycles on Kuga–Sato varieties over Shimura curves and Selmer groups
  5. Towards a Goldberg–Shahidi pairing for classical groups
  6. On the non-existence of cyclic splitting fields for division algebras
  7. Sequential motion planning algorithms in real projective spaces: An approach to their immersion dimension
  8. Very ampleness of the bicanonical line bundle on compact complex 2-ball quotients
  9. Anharmonic solutions to the Riccati equation and elliptic modular functions
  10. A non-homogeneous local Tb theorem for Littlewood–Paley g*λ-function with Lp-testing condition
  11. Saturation rank for finite group schemes: Finite groups and infinitesimal group schemes
  12. On symplectic semifield spreads of PG(5,q2), q odd
  13. From Freudenthal’s spectral theorem to projectable hulls of unital Archimedean lattice-groups, through compactifications of minimal spectra
  14. Golodness and polyhedral products for two-dimensional simplicial complexes
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