Explicit calculations of automorphic forms for definite unitary groups

TL;DR

This paper presents an algorithm for explicitly computing automorphic forms for definite unitary groups over Q, demonstrating its application through examples, endoscopic liftings, and slope computations, advancing understanding of automorphic forms and Galois representations.

Contribution

It introduces a new algorithm for computing automorphic forms on definite unitary groups and applies it to specific cases, revealing new examples of liftings and slope behaviors.

Findings

Examples of endoscopic liftings from smaller automorphic forms.

An explicit non-endoscopic automorphic form linked to Galois representations.

Evidence for local constancy of 2-adic slopes in automorphic forms.

Abstract

I give an algorithm for computing the full space of automorphic forms for definite unitary groups over Q, and apply this to calculate the automorphic forms of level $G(Z-hat)$ and various small weights for an example of a rank 3 unitary group. This leads to some examples of various types of endoscopic lifting from automorphic forms for U_1 x U_1 x U_1 and U_1 x U_2, and to an example of a non-endoscopic form of weight (3,3) corresponding to a family of 3-dimensional irreducible l-adic Galois representations. I also compute the 2-adic slopes of some automorphic forms with level structure at 2, giving evidence for the local constancy of the slopes.