Automorphic Forms on Feit’s Hermitian Lattices

Abstract

We consider the genus of 20 classes of unimodular Hermitian lattices of rank 12 over the Eisenstein integers. This set is the domain for a certain space of algebraic modular forms. We find a basis of Hecke eigenforms, and guess global Arthur parameters for the associated automorphic representations, which recover the computed Hecke eigenvalues. Congruences between Hecke eigenspaces, combined with the assumed parameters, recover known congruences for classical modular forms, and support new instances of conjectured Eisenstein congruences for U(2, 2) automorphic forms.

Automorphic forms for some even unimodular lattices

“…By Proposition 9.3(2), π i has global Arthur parameter ∆ 11 ⊕ [10] (where ∆ 11 is now the base change to GL 2 (A E ) of that appearing in §3). Hence λ i (T (2) ) = ((−24) 2 − 2 • 2 11 ) + 4 4 10 −1 4−1 + 2 12 −1 2+1 = 1395945, as in [DSc,Proposition 4.1], so i = 2. i = 4. The space S 11 (Γ 0 (3), χ −3 ) is 2-dimensional, spanned by a Hecke eigenform g = q + 12 √ −5q 2 +(−27 + 108 √ −5)q 3 + 304q 4 − 1272 √ −5q 5 + (−6480 − 324 √ −5)q 6 + 17324q 7 + .…”