Mellin transforms attached to certain automorphic integrals

Abstract

Let k be any real number with k < 2 . We will consider complex-valued smooth functions f , f ˜ on H of period 1, having exponential decay at infinity (i.e. they are ≪ e − c y for y = ℑ ( z ) → ∞ with c > 0 ) and such that f | k W N = C f ˜ + q g . Here | k is an appropriately defined Petersson slash operator in weight k, C ∈ C ⁎ is a constant and q g ( z ) : = ∫ 0 i ∞ g ( τ ) ( τ − z ¯ ) − k d τ ¯ ( z ∈ H ) is a period integral attached to a holomorphic function g : H → C such that both g and g | 2 − k W N have period 1, have only positive terms in their Fourier expansions and the Fourier coefficients are of polynomial growth. An arbitrary power of a non-zero complex number is defined by means of the principal branch of the complex logarithm. Under the assumption that k < 1 , we will show that the Mellin transform M ( f , s ) ( σ ≫ 1 ) naturally attached to f has meromorphic continuation to C and we will establish an explicit formula for it (Section 2, Theorem 1). There are possible simple poles at the points s = − n where n = 0 , 1 , 2 , … and the residue at s = − n essentially is equal to the “ n-th period” ∫ 0 ∞ g ( i t ) t n d t of g. Moreover, there again is a functional equation relating M ( f , s ) and M ( f ˜ , k − s ) .