Asymptotics for cuspidal representations by functoriality from GL(2)

Abstract

Let π be a unitary automorphic cuspidal representation of GL2(Q A ) with Fourier coefficients λπ(n).Asymptotic expansions of certain sums of λπ(n) are proved using known functorial liftings from GL2, including symmetric powers, isobaric sums, exterior square from GL4 and base change. These asymptotic expansions are manifestation of the underlying functoriality and reflect value distribution of λπ(n) on integers, squares, cubes and fourth powers.

Some Density Results on Sets of Primes for Hecke Eigenvalues

“…In this section, we will briefly recall some fundamental facts about primitive automorphic L-functions and give the main tools and definitions. For more details to learn automorphic L-functions, refer [17,[20][21][22][23][24][25][26][27][28].…”

Resonance between the Representation Function and Exponential Functions over Arithemetic Progression

“…In addition, studying the behavior of the Fourier coefficients of automorphic forms has great significance in modern number theory. Analytic number theorists always estimate the mean value or the twisted sums such as S(X, α) mentioned above to obtain some information about the Fourier coefficients (for examples, see [3][4][5][6][7][8][9][10]).…”