# Mass formulas and Eisenstein congruences in higher rank

## TL;DR

This paper develops a method using mass formulas to construct Eisenstein congruences for algebraic modular forms on higher rank groups, extending previous work on GL(2) and exploring their properties and implications.

## Contribution

It introduces a new construction of Eisenstein congruences in higher rank groups using mass formulas, and analyzes their cuspidality and endoscopic nature, extending prior results from GL(2).

## Key findings

- Constructs Eisenstein congruences with non-endoscopic cuspidal forms on unitary groups.
- Shows congruences often endoscopic when using unitary groups over fields.
- Provides conjectures relating to Eisenstein congruences for GL(2).

## 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).