It is a projective limit of finite dimensional tori, so in particular is abelian.
If L* is the algebraic group with L*(A) the units of A⊗L, then L* is a torus with the same dimension as L, and its characters can be identified with integral functions on Gal(L/Q).
This module of rational characters can be identified with the integral functions λ on Gal(L/Q) such that for all σ in Gal(L/Q), where ι is complex conjugation.
The full Serre group S can be described similarly in terms of its module X*(S) of rational characters.
This module of rational characters can be identified with the locally constant integral functions λ on Gal(Q/Q) such that for all σ in Gal(Q/Q), where ι is complex conjugation.