Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Congruences with Eisenstein series and mu-invariants

Joël Bellaïche, Robert Pollack

Abstract

We study the variation of mu-invariants in Hida families with residually reducible Galois representations. We prove a lower bound for these invariants which is often expressible in terms of the p-adic zeta function. This lower bound forces these mu-invariants to be unbounded along the family, and moreover, we conjecture that this lower bound is an equality. When U_p-1 generates the cuspidal Eisenstein ideal, we establish this conjecture and further prove that the p-adic L-function is simply a power of p up to a unit (i.e. lambda=0). On the algebraic side, we prove analogous statements for the associated Selmer groups which, in particular, establishes the main conjecture for such forms.

Congruences with Eisenstein series and mu-invariants

Abstract

We study the variation of mu-invariants in Hida families with residually reducible Galois representations. We prove a lower bound for these invariants which is often expressible in terms of the p-adic zeta function. This lower bound forces these mu-invariants to be unbounded along the family, and moreover, we conjecture that this lower bound is an equality. When U_p-1 generates the cuspidal Eisenstein ideal, we establish this conjecture and further prove that the p-adic L-function is simply a power of p up to a unit (i.e. lambda=0). On the algebraic side, we prove analogous statements for the associated Selmer groups which, in particular, establishes the main conjecture for such forms.

Paper Structure

This paper contains 36 sections, 61 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. Setting the stage
  3. $p$-adic $L$-functions and mod $p$ multiplicity one
  4. A more general picture
  5. Selmer groups
  6. Overconvergent modular symbols and $p$-adic $L$-functions
  7. Modular symbols
  8. Measures and $p$-adic $L$-functions
  9. Overconvergent modular symbols and $p$-adic $L$-functions
  10. $p$-adic $L$-functions of ordinary Eisenstein series
  11. Analytic results
  12. Notations and the like
  13. Gorenstein conditions and the principality of the Eisenstein ideal
  14. Freeness of spaces of modular symbols
  15. Lower bounds
  16. ...and 21 more sections

Key Result

Theorem 1.1

Fix an irregular pair $(p,j)$ and assume that r1 holds for this pair. Then in ${\mathbb Z}_p[[\Gamma_{w,j} \times {\mathbb Z}_p^\times]]$. In particular, for each even $a$ with $0 \leq a \leq p-1$ and for classical $k \equiv j \pmod{p-1}$.

Theorems & Definitions (129)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Conjecture 1.9
  • Theorem 1.10
  • ...and 119 more