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Distribution of slopes for $\mathscr{L}$-invariants

Jiawei An

Abstract

Fix a prime $p\geq5$, an integer $N\geq1$ relatively prime to $p$, and an irreducible residual global Galois representation $\bar{r}: Gal_{\mathbb{Q}}\rightarrow GL_2(\mathbb{F}_p)$. In this paper, we utilize ghost series to study $p$-adic slopes of $\mathscr{L}$-invariants for $\bar{r}$-newforms. More precisely, under a locally reducible and strongly generic condition for $\bar{r}$: (1) we determine the slopes of $\mathscr{L}$-invariants associated to $\bar{r}$-newforms of weight $k$ and level $Γ_0(Np)$, with at most $O(log_pk)$ exceptions; (2) we establish the integrality of these slopes; (3) we prove an equidistribution property for these slopes as the weight $k$ tends to infinity, which confirms the equidistribution conjecture for $\mathscr{L}$-invariants proposed by Bergdall--Pollack recently.

Distribution of slopes for $\mathscr{L}$-invariants

Abstract

Fix a prime , an integer relatively prime to , and an irreducible residual global Galois representation . In this paper, we utilize ghost series to study -adic slopes of -invariants for -newforms. More precisely, under a locally reducible and strongly generic condition for : (1) we determine the slopes of -invariants associated to -newforms of weight and level , with at most exceptions; (2) we establish the integrality of these slopes; (3) we prove an equidistribution property for these slopes as the weight tends to infinity, which confirms the equidistribution conjecture for -invariants proposed by Bergdall--Pollack recently.

Paper Structure

This paper contains 31 sections, 48 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Equidistribution conjectures
  3. Local assumptions and main results
  4. Main input of our proof: The ghost conjecture
  5. Overview of the proof
  6. Recollection of the local ghost theorem
  7. Abstract set-up
  8. Local ghost series for $\bar{\rho}$
  9. Local ghost theorem
  10. Settings
  11. Equidistribution property for $k$-thresholds
  12. Breakpoints of ghost polygons
  13. Estimates of derivative slopes
  14. Good region
  15. Variations of $k$-newslopes
  16. ...and 16 more sections

Key Result

Theorem 1.6

Fix $p\geq11$. Assume that $\bar{r}$ is locally reducible and strongly generic. Then the following set of valuations coincides with the set of global $k$-thresholds corresponding to $\bar{r}$ (Definition global k-thresholds introduction), with at most $O(\log_p k)$ exceptions.

Theorems & Definitions (128)

  • Conjecture 1.2: Bergdall-Pollack
  • Conjecture 1.3: Gouvêa
  • Definition 1.5
  • Theorem 1.6
  • Corollary 1.7: Integrality
  • Remark 1.8
  • Theorem 1.9: Equidistribution property
  • Remark 1.11
  • Definition 1.13
  • Definition 1.14
  • ...and 118 more