A Generalization Of A Twisting Equation

This thesis will generalize the integral equation of twisting of automorphic L-function on any totally real field or function field F without any coprime condition.Firstly we will decompose the global twisting L-function of any cuspidal admissible automorphic representation into a product of a global zeta integral with some local twisting L-functions and the inverse of local zeta integral.Then,in some cases,we will use the test vector theory to choose some special local Whittaker functions according to the ramifiedness of local multiple character in order to insure those local zeta integral are all non-zero.Summarize these two steps,we can get the integral equation of the global twisting L-function.The first step is based on the Jacquet-Langlands' automorphic representations and forms theory on GL(2),and the second step is based on the Casselman's newform theory,Gross-Prasad test vector theory and Cai-Shu-Tian's generalization.In the modu-lar form theory,we will also transform this integral equation into the integral equation of twisting L-function of cuspidal modular newforms,which is the non-coprime generaliza-tion of the integral equation from the Hecke twisting.For the ambiguous choice of test vectors,we also give an equation as a corollary of our main theorem.This thesis will sum-marize the Tate's theory and its application on the class number formula,which has the similar method of proof with our main theorem.Next we will summarize the automorphic form theory on GL(2)in order to get the first step of the proof of our main theorem.Then we will introduce the Casselman's newform theory.This is the non-twisting case,i.e.the test vector theory in the case that the local multiple character is unramified.And then we will introduce the twisting theory,which is the Gross-Prasad test vector theory and Cai-Shu-Tian's generalization on this theory.Meanwhile,we also summarize the test vector theory on archimedean field,and other papers over this field recently.Finally,we will comprehend all above,and use a slightly different test vector from the Cai-Shu-Tian's result to prove the main theorem and the corollary in the modular form theory and the case of ambiguous test vectors.And we will retrospect our main theorem to the classical theorems in modular form theory on both number field and function field.