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Arithmetic of critical $p$-adic $L$-functions

Denis Benois, Kâzım Büyükboduk

Abstract

Our objective in the present work is to develop a fairly complete arithmetic theory of critical $p$-adic $L$-functions on the eigencurve. To this end, we carry out the following tasks: a) We give an "étale" construction of Bellaïche's $p$-adic $L$-functions at a $θ$-critical point on the cuspidal eigencurve. b) We introduce the algebraic counterparts of these objects (which arise as appropriately defined Selmer complexes) and develop Iwasawa theory in this context, including a definition of an Iwasawa theoretic $\mathscr L$-invariant $\mathscr{L}^{\rm cr}_{\rm Iw}$. c) We formulate the (punctual) critical main conjecture and study its relationship with its slope-zero counterparts. Along the way, we also develop descent theory (paralleling Perrin-Riou's work). d) We introduce what we call thick (Iwasawa theoretic) fundamental line and the thick Selmer complex to counter Bellaïche's secondary $p$-adic $L$-functions. This allows us to formulate an infinitesimal thickening of the Iwasawa main conjecture, and we observe that it implies both slope-zero and punctual critical main conjectures, but it seems stronger than both. e) We establish an $\mathcal{O}_{\mathcal{X}}$-adic leading term formula for the two-variable $p$-adic $L$-function over the affinoid neighbourhood $\mathcal{X}={\rm Spm}(\mathcal{O}_{\mathcal{X}})$ in the eigencurve about a $θ$-critical point. Using this formula we prove, when the Hecke $L$-function of $f$ vanishes to order one at the central critical point, that the derivative of the secondary $p$-adic $L$-function can be computed in terms of the second order derivative of an $\mathcal{O}_{\mathcal{X}}$-adic regulator (rather than a regulator itself).

Arithmetic of critical $p$-adic $L$-functions

Abstract

Our objective in the present work is to develop a fairly complete arithmetic theory of critical -adic -functions on the eigencurve. To this end, we carry out the following tasks: a) We give an "étale" construction of Bellaïche's -adic -functions at a -critical point on the cuspidal eigencurve. b) We introduce the algebraic counterparts of these objects (which arise as appropriately defined Selmer complexes) and develop Iwasawa theory in this context, including a definition of an Iwasawa theoretic -invariant . c) We formulate the (punctual) critical main conjecture and study its relationship with its slope-zero counterparts. Along the way, we also develop descent theory (paralleling Perrin-Riou's work). d) We introduce what we call thick (Iwasawa theoretic) fundamental line and the thick Selmer complex to counter Bellaïche's secondary -adic -functions. This allows us to formulate an infinitesimal thickening of the Iwasawa main conjecture, and we observe that it implies both slope-zero and punctual critical main conjectures, but it seems stronger than both. e) We establish an -adic leading term formula for the two-variable -adic -function over the affinoid neighbourhood in the eigencurve about a -critical point. Using this formula we prove, when the Hecke -function of vanishes to order one at the central critical point, that the derivative of the secondary -adic -function can be computed in terms of the second order derivative of an -adic regulator (rather than a regulator itself).
Paper Structure (124 sections, 102 theorems, 983 equations)

This paper contains 124 sections, 102 theorems, 983 equations.

Table of Contents

  1. Basic notation
  2. $p$-adic $L$-functions of modular forms
  3. The $\theta$-critical case
  4. The eigencurve at a $\theta$-critical point
  5. Overconvergent modular symbols
  6. Local $p$-adic representation
  7. Triangulation
  8. Two-variable $p$-adic $L$-function
  9. Overview of constructions and results
  10. Set-up
  11. $p$-adic $L$-functions
  12. Improved $p$-adic $L$-function and punctual Main Conjectures
  13. Critical $\mathscr L$-invariants
  14. Punctual main conjecture for the improved $p$-adic $L$-function and Iwasawa descent
  15. Main Conjectures for the infinitesimal deformation
  16. ...and 109 more sections

Key Result

Theorem B

Assume that $e=2$. At any classical point $x\in \mathcal{X} (E)\setminus\{x_0\}$, the specialization $L_{\mathrm{K},\eta}(\mathcal{X},\xi; x)$ of $L_{\mathrm{K},\eta}(\mathcal{X},\xi)$ to $x$ agrees with the Manin--Vishik $p$-adic $L$-function $L_{\mathrm{S},\alpha(x)}( f_x^\circ,\xi_x)$ in the foll where $\mathcal{E}_N(x)$ is the product of Euler-like factors at bad primes defined in eqn:interpol

Theorems & Definitions (247)

  • Remark A
  • Theorem B: Theorem \ref{['thm_interpolative_properties']}(i) and Equation \ref{['eqn:comparision with Manin-Vishik']}
  • Theorem C: Theorem \ref{['thm_interpolative_properties']}(ii) and Theorem \ref{["thm:comparision with Bellaiche's construction"]}
  • Conjecture D: see Conjecture \ref{['conjecture_GP']}
  • Proposition F: Proposition \ref{['prop: comparision p-adic L-functions for alpha and beta']}
  • Conjecture G
  • Conjecture H
  • Conjecture I
  • Theorem J
  • Conjecture K
  • ...and 237 more