Asymptotics for cuspidal representations by functoriality from GL(2)

• Asymptotics were proved for the following functorial liftings from GL(2)GL(2):
• The symmetric square, cube, and fourth power representations of π.
• The isobaric product π1⊠π2π1⊠π2 of unitary automorphic cuspidal representations π1π1 and π2π2 of GL2(QA)GL2(QA).
• The isobaric product π1⊠Sym2(π2)π1⊠Sym2(π2).
• The exterior square representation ∧2(π1⊠π2)∧2(π1⊠π2).
• The base change from π   to a Galois number field of prime degree over QQ.

Let π   be a unitary automorphic cuspidal representation of GL2(QA)GL2(QA) with Fourier coefficients λπ(n)λπ(n). Asymptotic expansions of certain sums of λπ(n)λπ(n) are proved using known functorial liftings from GL2GL2, including symmetric powers, isobaric products, exterior squares, and base change. These asymptotic expansions are manifestation of the underlying functoriality and reflect value distribution of λπ(n)λπ(n) on integers, squares, cubes and fourth powers.