A Hecke Correspondence for Automorphic Integrals With in_Nitely Many Poles

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Addis Ababa University


In 1936 Hecke proved a correspondence theorem between Dirichlet series with functional equations and automorphic forms with certain growth conditions. In this correspondence, the Dirichlet series has at most simple poles at 0 and k: Since Hecke's original paper, many authors have generalized the correspondence theorem in di_erent directions. Among the generalizations, we shall be interested in is the one by Salomon Bochner in 1951. In Bochner's version, the correspondence is between automorphic integrals with _nite log{polynomial sum and Dirichlet series with a functional equation. The most important feature of this generalization is the presence of the log{polynomial sum. Here a logpolynomial sum is a sum of the form q(z) = Pn l=1 z_l Pm(l) j=0 _(l; j)(log z)j ; the coe_cients, _(l; j) and _l are complex numbers, n; l; m(l); and j are non negative integers. In Bochner's version, the Dirichlet series has a pole of order m(l) + 1 at _l: Austin Daughton in 2012, extended Bochner's result to the Dirichlet series with in_nitely many poles for the theta group and for weight k _ 0: This thesis will extend the Bochner correspondence theorem between automorphic integrals with in_nite log-polynomial period function and of _ weight arbitrary real k on Hecke group which is generated by S_ = : The theta group correspond to _ = 2: We will deal with the cases when _ > 2 and _ = 2 cos _=p; p 2 Z and p _ 3:



A Hecke Correspondence for Automorphic Integrals