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Newform Eisenstein Congruences of Local Origin

Dan Fretwell, Jenny Roberts

Abstract

We give a general conjecture concerning the existence of Eisenstein congruences between weight $k\geq 3$ newforms of square-free level $NM$ and weight $k$ new Eisenstein series of square-free level $N$. Our conjecture allows the forms to have arbitrary character $χ$ of conductor $N$. The special cases $M=1$ and $M=p$ prime are fully proved, with partial results given in general. We also consider the relation with the Bloch-Kato conjecture, and finish with computational examples demonstrating cases of our conjecture that have resisted proof.

Newform Eisenstein Congruences of Local Origin

Abstract

We give a general conjecture concerning the existence of Eisenstein congruences between weight newforms of square-free level and weight new Eisenstein series of square-free level . Our conjecture allows the forms to have arbitrary character of conductor . The special cases and prime are fully proved, with partial results given in general. We also consider the relation with the Bloch-Kato conjecture, and finish with computational examples demonstrating cases of our conjecture that have resisted proof.
Paper Structure (11 sections, 10 theorems, 84 equations)

This paper contains 11 sections, 10 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Background and notation
  3. The Setup
  4. A General Conjecture
  5. Initial results
  6. Necessary conditions for a newform Eisenstein congruence.
  7. Sufficient conditions for a newform Eisenstein congruence.
  8. Comparison with known results
  9. Low weight
  10. Relation with the Bloch-Kato Conjecture
  11. Examples

Key Result

Lemma 3.2

Each Eisenstein series $E_{\underline{\delta}}$ is a normalised eigenform in $\mathcal{E}_k(\Gamma_0(NM), \tilde{\chi})$. For each prime $p$ we have: We can also write where $\delta_m = \prod_{p \in \mathcal{P}_m} \delta_p$ for each $m\mid M$.

Theorems & Definitions (27)

  • Conjecture 1.1
  • Conjecture 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3: Spencer
  • Corollary 3.4
  • proof
  • Theorem 3.5
  • proof
  • Theorem 3.6
  • ...and 17 more