Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Mass formulas and Eisenstein congruences in higher rank

Kimball Martin, Satoshi Wakatsuki

TL;DR

This work generalizes the Eichler–Mazur–Mazur-type Eisenstein-congruence framework to higher rank groups using mass formulas for algebraic modular forms on groups compact at infinity. The authors construct congruences on definite inner forms, then transfer them to quasi-split forms via endoscopic classification and functorial lifts, yielding both non-endoscopic and endoscopic congruences on unitary groups and related groups (GSp, SO, G2). They provide explicit local and global criteria, including mass divisibility and Bernoulli-number divisibilities, and develop lifting mechanisms (Kudla, symmetric powers, Ikeda) that preserve Eisenstein congruences. The results give new higher-rank Eisenstein-congruence phenomena and suggest conjectural links to weight-$k$ GL$(2)$ congruences, enriching our understanding of base change, endoscopy, and higher-dimensional modular forms. Overall, the paper offers a concrete, expandable roadmap linking mass formulas, endoscopic transfers, and higher-weight congruences across several classical groups, with substantial computational and conjectural implications for GL$(2)$ Eisenstein congruences.

Abstract

We use mass formulas to construct minimal parabolic Eisenstein congruences for algebraic modular forms on reductive groups compact at infinity, and study when these yield congruences between cusp forms and Eisenstein series on the quasi-split inner form. This extends recent work of the first author on weight 2 Eisenstein congruences for GL(2) to higher rank. Two issues in higher rank are that the transfer to the quasi-split form is not always cuspidal and sometimes the congruences come from lower rank (e.g., are "endoscopic"). We show our construction yields Eisenstein congruences with non-endoscopic cuspidal automorphic forms on quasi-split unitary groups by using certain unitary groups over division algebras. On the other hand, when using unitary groups over fields, or other groups of Lie type, these Eisenstein congruences typically appear to be endoscopic. This suggests a new way to see higher weight Eisenstein congruences for GL(2), and leads to various conjectures about GL(2) Eisenstein congruences. In supplementary sections, we also generalize previous weight 2 Eisenstein congruences for Hilbert modular forms, and prove some special congruence mod p results between cusp forms on U(p).

Mass formulas and Eisenstein congruences in higher rank

TL;DR

This work generalizes the Eichler–Mazur–Mazur-type Eisenstein-congruence framework to higher rank groups using mass formulas for algebraic modular forms on groups compact at infinity. The authors construct congruences on definite inner forms, then transfer them to quasi-split forms via endoscopic classification and functorial lifts, yielding both non-endoscopic and endoscopic congruences on unitary groups and related groups (GSp, SO, G2). They provide explicit local and global criteria, including mass divisibility and Bernoulli-number divisibilities, and develop lifting mechanisms (Kudla, symmetric powers, Ikeda) that preserve Eisenstein congruences. The results give new higher-rank Eisenstein-congruence phenomena and suggest conjectural links to weight- GL congruences, enriching our understanding of base change, endoscopy, and higher-dimensional modular forms. Overall, the paper offers a concrete, expandable roadmap linking mass formulas, endoscopic transfers, and higher-weight congruences across several classical groups, with substantial computational and conjectural implications for GL Eisenstein congruences.

Abstract

We use mass formulas to construct minimal parabolic Eisenstein congruences for algebraic modular forms on reductive groups compact at infinity, and study when these yield congruences between cusp forms and Eisenstein series on the quasi-split inner form. This extends recent work of the first author on weight 2 Eisenstein congruences for GL(2) to higher rank. Two issues in higher rank are that the transfer to the quasi-split form is not always cuspidal and sometimes the congruences come from lower rank (e.g., are "endoscopic"). We show our construction yields Eisenstein congruences with non-endoscopic cuspidal automorphic forms on quasi-split unitary groups by using certain unitary groups over division algebras. On the other hand, when using unitary groups over fields, or other groups of Lie type, these Eisenstein congruences typically appear to be endoscopic. This suggests a new way to see higher weight Eisenstein congruences for GL(2), and leads to various conjectures about GL(2) Eisenstein congruences. In supplementary sections, we also generalize previous weight 2 Eisenstein congruences for Hilbert modular forms, and prove some special congruence mod p results between cusp forms on U(p).

Paper Structure

This paper contains 42 sections, 36 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Congruences from mass formulas
  3. Algebraic modular forms
  4. Hecke operators
  5. Congruences
  6. Local congruences
  7. Algebraic groups, $L$-parameters and Satake parameters
  8. Some quasi-split reductive groups
  9. Discrete series of $\mathop{\mathrm{U}}\nolimits_{\mathbb C/\mathbb R}(n,\mathbb R)$
  10. Discrete series of $\mathop{\mathrm{GL}}\nolimits(2,\mathbb R)$
  11. Discrete series of $\mathop{\mathrm{GSp}}\nolimits(4,\mathbb R)$
  12. Discrete series of $G_2(\mathbb R)$
  13. Satake parameters
  14. Local congruence lifts
  15. Symmetric functions
  16. ...and 27 more sections

Key Result

Proposition 2.1

Suppose $p | m(K)$. Then there exists an eigenform $\phi \in \mathcal{A}_0(G, K)$ which is Hecke congruent to $\mathbbm{1} \bmod p$.

Theorems & Definitions (81)

  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Corollary 2.4
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • ...and 71 more