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$\mathcal L$-invariants and deep congruences between newforms

Andrea Conti, Peter Mathias Gräf

Abstract

We study congruences modulo powers of a prime $p$ between pairs of $p$-new modular Hecke eigenforms of level $Γ_0(p)$ and same weight $k$. Based on explicit computations, we conjecture that every such eigenform $f$ admits a twin to which it is congruent modulo a surprisingly high power of $p$, whose exponent is close to the opposite of the valuation of the $\mathcal L$-invariant of $f$, and whose Atkin--Lehner sign is opposite to that of $f$. This is a new phenomenon that is not explained by the known results on the $p$-adic variation of eigenforms. Inspired by the global picture, we formulate a local conjecture describing congruences between semistable representations of fixed weight, varying $\mathcal L$-invariant, and opposite Atkin--Lehner signs. We give some theoretical evidence towards our conjectures.

$\mathcal L$-invariants and deep congruences between newforms

Abstract

We study congruences modulo powers of a prime between pairs of -new modular Hecke eigenforms of level and same weight . Based on explicit computations, we conjecture that every such eigenform admits a twin to which it is congruent modulo a surprisingly high power of , whose exponent is close to the opposite of the valuation of the -invariant of , and whose Atkin--Lehner sign is opposite to that of . This is a new phenomenon that is not explained by the known results on the -adic variation of eigenforms. Inspired by the global picture, we formulate a local conjecture describing congruences between semistable representations of fixed weight, varying -invariant, and opposite Atkin--Lehner signs. We give some theoretical evidence towards our conjectures.
Paper Structure (17 sections, 9 theorems, 15 equations, 5 tables)

This paper contains 17 sections, 9 theorems, 15 equations, 5 tables.

Table of Contents

  1. Conjectural deep congruences between newforms
  2. Congruences of Hecke eigensystems and Galois representations
  3. Reminders on $\mathcal{L}$-invariants
  4. Doubling of $\mathcal{L}$-invariants
  5. A global conjecture
  6. Theoretical evidence
  7. Evidence for same $\mathcal{L}$-invariant valuation
  8. Evidence for opposite Atkin--Lehner signs
  9. Conjectural deep congruences between semistable representations
  10. A local conjecture
  11. The local conjecture implies a global congruence for $p\le 7$
  12. Same-sign congruences in small weight
  13. Computational evidence
  14. Computing $\mathcal{L}$-invariants
  15. Computing deep congruences
  16. ...and 2 more sections

Key Result

Theorem 1.7

Let $p\ge 11$, and let $f$ be an admissible newform of level $\Gamma_0(p)$ and weight $k$. Assume that $\overline\rho_f\vert_{I_{\mathbb Q_p}}$ is locally reducible and strongly generic in the sense of jiawei, i.e. isomorphic to a twist of with $2\le a \le p-5$. Then there exists a newform $g\ne f$ of level $\Gamma_0(p)$ and weight $k$ such that:

Theorems & Definitions (33)

  • Conjecture 1
  • Conjecture 2
  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Definition 1.4
  • Remark 1.5
  • Definition 1.6
  • Theorem 1.7: jiawei
  • Conjecture 1.9
  • ...and 23 more