Fourier Coefficients Of Hilbert Eigenforms And Twisted Double Eisenstein Series

Modular forms are holomorphic functions on the upper half-plane that satisfy certain functional equation with respect to certain group action and certain moderate growth condition.The theory of modular forms therefore originates from complex analysis,but the main importance of the theory has traditionally been in its connections with number theory.Modular form theory is a special case of the more general theory of automorphic forms,and therefore can now be seen as just the most concrete part of a rich theory of discrete groups.Modular forms appear in many other areas,such as algebraic topology,sphere packing,and string theory,which makes the study of modular forms more significant.In recent years,the research on the theory of product identities and double Eisenstein series on modular form has developed steadily.This dissertation mainly discusses modular forms including Hecke eigenforms and twisted double Eisenstein series.On one hand,it proceeds to the study of the properties of the Fourier coefficients of Hilbert eigenforms and establishs the finiteness of the product identities for totally real number fields of any fixed degree.On the other hand,it studies the first Fourier coefficient of 1/2 twisted double Eisenstein series.Especially,by explicitly calculating and then analytically continuing the first Fourier coefficient of 1/2 twisted double Eisenstein series,we prove a formula of the Petersson inner product of Cohen’s kernel and one of its twists,and obtain a rationality result.Finally,we study the Petersson inner product of general Cohen’s kernel,and obtain its analytic continuation.The main contents of this dissertation are as follows:Firstly,we treat the Hecke eigenforms f and h and prove a conjecture of Joshi and Zhang on eigenform product identities for Hecke eigenforms of full level.The conjecture is as follows:For a fixed positive integer n,there are finitely many eigenform product identities over all totally real number fields of degree n and all Hecke eigenforms of weight2 or greater.We follow the strategy of Joshi and Zhang and separate the proof into two parts,according to whether the first Fourier coefficient of f · h vanishs or not.And then we compare the other Fourier coefficients of f ·h with Fourier coefficients of f and h and construct the related inequalities.we obtain the desired theorems finally.Secondly,we prove the rationality of the Petersson inner product of Cohen’s kernels.We follow the computation by Choie,Kohnen and Zhang and compute the Fourier expansion of the 1/2 twisted double Eisenstein series.By further study,we find that the first Fourier coefficient of the 1/2 twisted double Eisenstein series is equal to the Petersson inner product of Cohen kernels up to a scalar.And for the first Fourier coefficient,we carry out its analytic continuation by introducing Hurwitz zeta function and the Gaussian hypergeometric function.Finally,we simplify the resulting formula and obtain the rationality by computing the special values of the relevant functions.Finally,we obtain the analytic continuation of the first Fourier coefficient of generally twisted double Eisenstein series.By the same method as we compute the Fourier expansion of the 1/2 twisted double Eisenstein series,we obtain the first Fourier coefficient of the generally twisted double Eisenstein series.And a similar relation between Cohen kernels and its Fourier coefficients is established.Furthermore,we obtain the analytic continuation of the first Fourier coefficient by introducing finitely many Hurwitz zeta functions and Gaussian hypergeometric functions.