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Generalised Kato classes on CM elliptic curves of rank 2

Francesc Castella

Abstract

Let $E/\mathbf{Q}$ be a CM elliptic curve and let $p\geq 5$ be a prime of good ordinary reduction for $E$. Suppose that $L(E,s)$ vanishes at $s=1$ and has sign $+1$ in its functional equation, so in particular ${\rm ord}_{s=1}L(E,s)\geq 2$. In this paper we slightly modify a construction of Darmon--Rotger to define a generalised Kato class $κ_p\in{\rm Sel}(\mathbf{Q},V_pE)$, and prove the following rank two analogue of Kolyvagin's result: \[ κ_p\neq 0\quad\Longrightarrow\quad{\rm dim}_{\mathbf{Q}_p}{\rm Sel}(\mathbf{Q},V_pE)=2. \] Conversely, when ${\rm dim}_{\mathbf{Q}_p}{\rm Sel}(\mathbf{Q},V_pE)=2$ we show that $κ_p\neq 0$ if and only if the restriction map \[ {\rm Sel}(\mathbf{Q},V_pE)\rightarrow E(\mathbf{Q}_p)\hat{\otimes}\mathbf{Q}_p \] is nonzero. The proof of these results, which extend and strenghten similar results of the author with Hsieh in the non-CM case, exploit a new link between the nonvanishing of generalised Kato classes and a main conjecture in anticyclotomic Iwasawa theory.

Generalised Kato classes on CM elliptic curves of rank 2

Abstract

Let be a CM elliptic curve and let be a prime of good ordinary reduction for . Suppose that vanishes at and has sign in its functional equation, so in particular . In this paper we slightly modify a construction of Darmon--Rotger to define a generalised Kato class , and prove the following rank two analogue of Kolyvagin's result: Conversely, when we show that if and only if the restriction map is nonzero. The proof of these results, which extend and strenghten similar results of the author with Hsieh in the non-CM case, exploit a new link between the nonvanishing of generalised Kato classes and a main conjecture in anticyclotomic Iwasawa theory.
Paper Structure (24 sections, 20 theorems, 159 equations)

This paper contains 24 sections, 20 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Diagonal cycle main conjecture
  3. Generalised Kato classes attached to $E$
  4. Application to rank two Selmer basis
  5. Acknowledgements
  6. $p$-adic $L$-functions
  7. Katz $p$-adic $L$-function
  8. Triple product $p$-adic $L$-function
  9. Congruence ideal
  10. Factorisation of $p$-adic triple product $L$-function
  11. Hida families with CM
  12. Congruence ideal of CM Hida families
  13. Proof of the factorisation
  14. Selmer group decompositions
  15. Selmer groups for Hecke characters
  16. ...and 9 more sections

Key Result

Theorem A

Let the triple be as above,and suppose that: Then $\kappa(\varphi,{\boldsymbol{g}},{\boldsymbol{h}})$ is not $\mathcal{R}_{\varphi{\boldsymbol{g}}{\boldsymbol{h}}}$-torsion, the modules ${\rm Sel}^{\rm bal}(\mathbf{Q},\mathbb{V}_{\varphi{\boldsymbol{g}}{\boldsymbol{h}}}^\dagger)$ and $X^{\rm bal}(\mathbf{Q},\mathbb{A}_{\varphi{\boldsymbol{g}}{ in $\mathcal{R}_{\varphi{\boldsymbol{g}}{\boldsymbol

Theorems & Definitions (49)

  • Conjecture A: Big diagonal class main conjecture
  • Theorem A
  • Theorem B
  • Theorem C
  • proof
  • Remark 1.3.1
  • Theorem 2.1.1
  • proof
  • Theorem 2.2.1
  • proof
  • ...and 39 more