Using the definition of ζ(s), we can show that ζ(s) has a simple pole ats=1 with a remainder of 1. 4) The Riemann function zeta has two rows of zeros. In this thesis we will review the inverse theorem of modular forms, following Hecke rather than Siegel.
Congruence subgroups
Fundamental domain and cusp
Modular form
We say that f ismeromorphic to the cuspi∞(resp. holomorphic to the cuspi∞) if there exists a negative integer −m such that an=0 for all n<−m (resp. an=0for all n<0). We say that f vanishes at the point a0=0 in the Fourier expansion of f|kγfor allγ∈SL(2,Z).
Modular forms with Dirichlet character
Hecke’s converse theorem
Using the Fourier expansion of the modular form f, we can define its Dirichlet series as follows. In the same way, ΛN(g,s) has a meromorphic continuation in the entire complex plane and satisfies.
Weil’s converse theorem
Let ψ be a primitive Dirichlet character of conductor m>1, and let Cψ be a constant that may depend on ψ. Let f and g be holomorphic functions on H that satisfy assumptions in Lemma 1, (1) in Theorem 5, and (1) in Theorem 1 for all primitive Dirichlet characterψ of conductor m>1 with the constant Cψ in Theorem 2.
More results for the converse theorem for modular forms
Finally, we conclude the proof to show that f and g are modular forms and their modularity relation as follows. In fact, there is a correspondence between modular forms with an Euler product and 'normalized eigenforms' (ie modular forms which are an eigenvector of all Hecke operators). From this result, he proved that Theorem 7 can be established if we reduce condition (2) (b) in Theorem 7 for all Dirichlet signs χ modulocNfor all such that 1≤c≤N.
Moreover, if it is prime, he also showed that the generators of Γ0(N) could be chosen more easily, and he proved that only characters moduloN. He proved that the primitive rotations ψ of modulus q Based on these terminologies, we can decompose an arbitrary Maass form f into even and odd form if f = 1. Moreover, when comparing coefficients in the Fourier-Whittaker expansion of f, we can easily obtain that if f is even, thenan=a−nfor a negative integer. Let f and g be a Maass form of type ν for Γ⊆SL(2,Z) with a Fourier-Whittaker expansion given in (25) with coefficient an and bn, respectively. Then the following applies. i) Ifε=0inν6=0, then Λν(f,s)andΛν(g,s) proceed to meromorphic functions on most simple poles in the set {±ν,1±ν}. ii) Ifε=0inν=0, then Λν(f,s)andΛν(g,s) proceed to meromorphic functions on at most double poles in the set {0,1}. iii) If = 1, Λν(f,s) and Λν(g,s) continue to the entire functions. We also have f(iy) = f(−1/iy) due to the automorphism of the Maass form, where the integrand is vanishingly small as y→0. Then we have an analogous calculation as in the case of ε=0, (iii) and the functional equation Λν(f,s). We will now derive that the above functions are null functions in the case of ε=0 and ε=1, respectively. To continue the proof, we introduce the following formula related to the hypergeometric function2F1. Finally, using Stirling's formula, both gamma factors are O(e−πt/4), so we have the following asymptotic behavior. In the proof of Lemma 3 (3) we used the fact that a holomorphic function opH which is invariant under an infinite order elliptic matrix is constant. Neurerer and Oliver solve this problem using the following 'two (hyperbolic) circles method'. They noted on their paper that this method was suggested by David Farmer.). In the same paper, Neurerer and Oliver proved a slightly different version of the converse theorem (Theorem 1.1 in [28]). Moreover, they applied this to show that the quotient of the symmetric square L-function of a Maass newform and the Riemann zeta function has infinitely many poles (Corollary 1.2 in [28]). These are similar in form to the Fourier expansion and L-function of the modular form. Finally, it completes the proof of the converse theorem in the same way as Maass's proof that we saw in Theorem 12, for every possible case of poles. In fact, we can see that his results are the same as the converse theorem of the non-cuspidal Maass form. Meanwhile, he wrote the following shortcomings of his work in his article: 'The ideas used in this article work only in the context of modular or Maass forms related to the full modular group, and indeed they cannot generalize to L-functions of automorphic forms associated with congruent subgroups of the modular group.' They were inspired by the fact that the transformation formula of the Jacobi theta function is derived from the Poisson summation formula in Hamburger's theorem. Their method requires more complicated information about the L-function, but has the advantage of extending the converse theorem to a more general Maass form (e.g., the weight of a half-integer). In this chapter, we introduce the basic theory of harmonic Maass forms, and then we intend to review the recently published inverse theorem for the harmonic Maass form of polynomial growth by Shankhadhar and Singh [34] and exponential growth by Diamantis, Lee, Raja, and Rolen [35]. If we substitute condition (iii) as below, then f is said to be a harmonic Maass form of manageable growth. The space of harmonic Maass forms of weight k, degree N and character χ is denoted by Hk(N,χ), and the space of harmonic Maass forms of controllable growth by Hk!(N,χ). The space of harmonic Maass forms of weight k, plane N and characterχ is denoted by Hk(N,χ) and that of harmonic Maass forms of tractable growth by Hk!(N,χ). Case 2) Otherwise, let w=2πny, and rewrite the above equation as follows. Finally, our condition (iii) in Definition 16 and the asymptotic of incomplete gamma function Γ(s,x)∼xs−1e−x as|x| →∞ (50) to eliminate unnecessary terms in two series. In other words, harmonic Maass forms with trivial nonholomorphic parts are weakly holomorphic modular forms. Now we introduce two differential operators acting on the spaceHk!(N,χ), which are associated with Maass race and reduction operators and play an important role in studying the Fourier expansion of. For any cusp ρ of Γ0(N), the Fourier expansion of f at the cuspρ is given in Lemma 9. By a process similar to the proof of (1), we can calculate that the second series in (51) destroys, and thus meromorphy at other cusps follows directly. The shadow operator can be expressed by using the reduction operator and the hyperbolic Laplace operator axis. For each cuspρ ofΓ0(N), the Fourier expansion of f on the cuspρ is given in Lemma 9. Through a process similar to the proof of (1), we can calculate that the second sequence in(51) turns into the holomorphic part, and so meromorphicity in other cusps follows directly. Consequently, if we assume that f has at most polynomial growth, we have a Fourier expansion of the form. According to (4) and (5) in the above assertion, the only interesting case for our purpose is isk≤0 and k=2. From the Fourier expansion f∈Hk#(N,χ) given by (55), we define two Dirichlet series corresponding to holomorphic and non-holomorphic parts as follows. We can also predict that the L-function of L−(f,s) will be similar in shape to the L-function of Maass forms, but with an appropriate modification of the gamma factor. When proving the converse theorem of the Maass form (Theorem 13), the proof was completed by applying two equations derived from the even and odd Maass forms. Note that in the above definition, the function Wν is derived from the incomplete gamma function in the non-holomorphic parts of f (we will calculate this in the next section), and acts as the gamma factor of L−(f,s). From the Stirling approximation for the gamma function, for every µ>0 we have Γ(s)∼√. 3) By definition, we note that Wν(s) is the Mellin transform of Γ(ν,2x)ex. As in the proof of Lemma 2, using the Euler-Gauss formula for the gamma function, for n≥1 we have. The following lemma and the two specific functions serve to combine the two functions LN and ΩNwell in proving the converse theorem. By the same calculation, we see that Λ∗N(g,s) can be expressed by a similar integral representation as (59) and is thus complete and bounded on vertical bars. Similarly to the calculation of ΛN(f,s) by Lemma 16, we can obtain an integral representation of ΩN(f,s), and from Lemma 15, we can see that Ω∗N(f,s)andΩ∗ N (g,s) are complete and bounded on each vertical bar. In the proof of Lemma 15, we noted that the given Fourier expansion of f and g satisfies. Recall the set P given in definition 11: P=PN is a set of prime odd or 4 satisfying the following two conditions:. For any primitive Dirichlet character ψ with conductor mψ∈P, each of the L-functions ΛN(f,ψ,s),ΛN(g,ψ,s),ΩN(f,ψ,s),ΩN(g,ψ, s) can be analytically continued throughout the complex plane, bounded at any vertical bar, and satisfies the functional equation. Setting f = fψ, g=Cψgψ, and N=Nm2ψ in Theorem 15, we have the following proposition, which is an analogue of Proposition 1. Proposition 10 Let k be a negative integer and N a positive integer . These claims can be justified by the following proposition, which is an analogue of Lemma 3 (1) and (2). This can be induced in exactly the same way as the proof of Lemma 3 using (1). In the above calculations, we used the fact that the contour of the given complex integral is just a line from x0 to x0+t with fixed imaginary part y0 and the following formula. The converse theorem for harmonic Maass forms of general growth conditions is as follows. Martin and Osses [41] extended this result to the generalization of Weil's converse theorem for Jacobian cusp forms over subgroups Γ0(N)n Z2 of SL(2,Z)n Z2. This is a generalization of an inverse theorem for (holomorphic) Jacobi cusp forms thanks to Martin [40] to non-cuspid Jacobi forms. Singh, “An analogue of Weil's inverse theorem for harmonic Maass forms of polynomial growth,” Res. Osses, "On the analogue of Weil's inverse theorem for Jacobi forms and their lifting to half-integral weight modular forms," Ramanujan J., vol. Sato, “Inverse theorem for not necessarily cuspid Siegel modular forms of degree 2 and Saito-Kurokawa lifting,” Comment.Maass forms
Converse theorem for Maass form for level 1
Neurerer-Oliver’s converse theorem
More results for the converse theorem for Maass forms
Definition of harmonic Maass form
Fourier expansion of harmonic Maass forms
Differential operators related to harmonic Maass forms
Dirichlet series and L-functions
Hecke-type converse theorem
Shankhadhar and Singh’s converse theorem
More results for the converse theorem for harmonic Maass forms
