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Hilbert modular Eisenstein congruences of local origin

Dan Fretwell, Jenny Roberts

Abstract

Let $F$ be an arbitrary totally real field. Under weak conditions we prove the existence of certain Eisenstein congruences between parallel weight $k \geq 3$ Hilbert eigenforms of level $\mathfrak{mp}$ and Hilbert Eisenstein series of level $\mathfrak{m}$, for arbitrary ideal $\mathfrak{m}$ and prime ideal $\mathfrak{p}\nmid \mathfrak{m}$ of $\mathcal{O}_F$. Such congruences have their moduli coming from special values of Hecke $L$-functions and their Euler factors, and our results allow for the eigenforms to have non-trivial Hecke character. After this, we consider the question of when such congruences can be satisfied by newforms, proving a general result about this.

Hilbert modular Eisenstein congruences of local origin

Abstract

Let be an arbitrary totally real field. Under weak conditions we prove the existence of certain Eisenstein congruences between parallel weight Hilbert eigenforms of level and Hilbert Eisenstein series of level , for arbitrary ideal and prime ideal of . Such congruences have their moduli coming from special values of Hecke -functions and their Euler factors, and our results allow for the eigenforms to have non-trivial Hecke character. After this, we consider the question of when such congruences can be satisfied by newforms, proving a general result about this.

Paper Structure

This paper contains 19 sections, 17 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hilbert modular forms
  4. Hilbert Eisenstein series
  5. Constant term formulae
  6. Mod $l$ Hilbert modular forms
  7. Classical mod $l$ Hilbert modular forms
  8. Characters as weights
  9. Hilbert-Blumenthal abelian schemes
  10. The q-expansion principle
  11. Comparison to classical Hilbert modular forms
  12. Partial Hasse invariants
  13. $\theta$ map
  14. Adelic mod $l$ Hilbert modular forms
  15. Action on level structures
  16. ...and 4 more sections

Key Result

Theorem 1.1

Suppose that $F$ is a totally real field with ring of integers $\mathcal{O}$ and narrow class number $h^+ = 1$. Let $k>2$ be even and $\pi\in\mathcal{O}^+$ be a prime such that the level structure $U_1(\pi)$ is small enough (see Definition smallenough). If $l > k+1$ is a rational prime such that $l\ then there exists a normalised Hilbert eigenform $f \in S_{\textbf{k}}(\Gamma_{\mathcal{O}}(\pi))$

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Proposition 2.7
  • Proposition 3.1
  • ...and 20 more