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On the Iwasawa Main Conjecture for generalized Heegner classes in a quaternionic setting

Maria Rosaria Pati

Abstract

We prove one divisibility relation of the anticyclotomic Iwasawa Main Conjecture for a higher weight ordinary modular form $f$ and an imaginary quadratic field satisfying a "relaxed" Heegner hypothesis. Let $Λ$ be the anticyclotomic Iwasawa algebra. Following the approach of Howard and Longo--Vigni, we construct the $Λ$-adic Kolyvagin system of generalized Heegner classes coming from Heegner points on a suitable Shimura curve. As its application, we also prove one divisibility relation in the Iwasawa-Greenberg main conjecture for the $p$-adic $L$-function defined by Magrone.

On the Iwasawa Main Conjecture for generalized Heegner classes in a quaternionic setting

Abstract

We prove one divisibility relation of the anticyclotomic Iwasawa Main Conjecture for a higher weight ordinary modular form and an imaginary quadratic field satisfying a "relaxed" Heegner hypothesis. Let be the anticyclotomic Iwasawa algebra. Following the approach of Howard and Longo--Vigni, we construct the -adic Kolyvagin system of generalized Heegner classes coming from Heegner points on a suitable Shimura curve. As its application, we also prove one divisibility relation in the Iwasawa-Greenberg main conjecture for the -adic -function defined by Magrone.
Paper Structure (19 sections, 14 theorems, 111 equations)

This paper contains 19 sections, 14 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Galois representations
  3. Galois representation of a modular form
  4. Galois representation of a modular form in a quaternionic setting
  5. Generalized Kuga-Sato varieties fibered over a Shimura curve
  6. $p$-adic Abel-Jacobi maps and Heegner classes
  7. Selmer groups and the Kolyvagin method
  8. Selmer structures
  9. Bloch-Kato Selmer groups of a modular form
  10. The Bloch-Kato Selmer group of $f$ over the anticyclotomic $\mathds Z_p$-extension of $K$
  11. Greenberg's Selmer groups of a modular form
  12. Greenberg's Selmer group of $\Lambda$-adic representations
  13. Comparison of Selmer groups
  14. Bounding Selmer groups
  15. A $\Lambda$-adic Kolyvagin system of generalized Heegner cycles in a quaternionic setting
  16. ...and 4 more sections

Key Result

Theorem 1.1

If $\kappa_\infty$ is not torsion, then there exists a finitely generated torsion $\Lambda$-module $M_\infty$ such that where $\sim$ means pseudo-isomorphism and $\iota\colon \Lambda\rightarrow \Lambda$ is the involution induced by the inversion on $\mathop{\mathrm{Gal}}\nolimits(K_\infty/K)$.

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.2
  • Definition 3.2
  • Theorem 3.4
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 19 more