By definition of big-oh notation, we can rewrite this as
y→∞lim
c+f(n) +c−f (n)Γ(1−k,−4πny/t) yv−ke2π(n+κ)/t
<∞. (68)
By(50), we have
Γ(1−k,−4πny/t)∼(−4πny/t)−ke4πny/t as y→∞.
Then(68)can be write
y→∞lim
c+f(n) +c−f(n)(−4πny/t)−ke4πny/t yv−ke2π(n+κ)y/t
=lim
y→∞|c+f(n)y−(v−k)e−2π(n+κ)y/t
+c−f(n)(−4πn/t)−ky−ve(4πny−2π(n+κ)y)/t|<∞.
Now recall that k<0, v≥0, andκ∈[0,1). For n>0, the first term goes to0but the second term goes to∞as y→∞unless c−f(n) =0. Similarly, for n<0, the second term goes to0but the first term goes to∞as y→∞unless c+f(n) =0. Therefore we have
(f|kγ)(z) =
∞
∑
n=0
c+f(n)qn+κt +c−f(0)y1−kqκt +
∑
n<0
c−f(n)Γ(1−k,−4πny/t)qn+κt , and we conclude that f|kγhas a polynomial growth.
for eachs∈Cfor which the integral converges absolutely. Fors∈C, we defineϕs(x):=ϕ(x)xs−1. They defined the following:
Definition 23 Let f ∈Hk(N,χ)has a Fourier expansion as in Lemma 49.
f(z) =
∑
n−∞
c+f(n)qn+
∑
n<0
c−f(n)Γ(1−k,−4πny)qn. The generalized Dirichlet series of f twisted by Laplace transform is defined by
L(f,ϕ) =
∑
n−∞
c+f(n)(Lϕ)(2πn) +
∑
n<0
c−f(n)(−4πn)1−k Z ∞
0
(Lϕ2−k)(−2πn(2t+1)) (1+t)k dt.
In order to ensure the absolute and uniform convergence of the series, the following notion is required.
Let Ff be the space of piecewise smooth functionsϕ :R→C such that the above series converges absolutely. We will only consider L(f,ϕ)forϕ∈Ff.
In particular, for Re(s)>0, if we chooseϕ(x) = (2π)sxs−1 1
Γ(s), we obtain (Lϕ)(2πn) = 1
ns, (−4πn)1−k
Z ∞
0
(Lϕ2−k)(−2πn(2t+1))
(1+t)k dt= 1 (−n)sΓ(s)
Z ∞
0
Γ(1−k,−4πny)e−2πny(−2πny)sdy y . The second formula is derived from the following identity.
Γ(s,z) =zse−z Z ∞
0
e−zt
(1+t)1−sdt, (Re(z)>0).
Therefore we have
L(f,ϕ) =
∑
n−∞
c+f (n)n−s+W1−k(s) Γ(s)
∑
n<0
c−f(n)(−n)−s,
and so we can viewL(f,ϕ)as a generalized version ofΛN(f,s)defined in Definition 22.
The seriesL(f,ϕ)is represented by the following integral, similar to Mellin transform.
L(f,ϕ) = Z ∞
0
f(iy)ϕ(y)dy.
Next, analogously to Lemma 16, we define the following function δkf(z):=z∂f
∂x+k 2f(z).
Then we can compute another Dirichlet series related toδ as L(δkf,ϕ) =k
2L(f,ϕ)−2π
∑
n−∞
nc+f(n)(Lϕ2)(2πn)
−2π
∑
n<0
c−f (n)(−4πn)1−k Z ∞
0
(Lϕ3−k)(−2πn(2t+1)) (1+t)k dt
forϕ∈Fδkf. We also have
L(δkf,ϕ) = Z ∞
0
(δkf)(iy)ϕ(y)dy.
The second development is to twist the following generalized Gauss sum into a Dirichlet series instead of a primitive Dirichlet character.
τψ(n):=
∑
umodm
ψ(u)e2πinu/m
whereψ is a Dirichlet character modulom, andn∈Z. Definition 24 The twisted Fourier expansion of f as
fψ(z) =mk/2
∑
umodm
ψ(u) f|k√
mTu/m (z)
=mk/2
∑
umodm
ψ(u) f|k 1/√
m u/√ m
0 √
m
!!
(z)
=
∑
n≥−n0
τψ(n)c+f(n)qn/m+
∑
n<0
τψ(n)c−f(n)Γ(1−k,−4πny/m)qn/m.
(Compare this with Proposition 11 (1).) For each ϕ ∈Ffψ, the twisted generalized Dirichlet series associated to fψ is defined as
L(fψ,ϕ) =
∑
n−∞
τψ(n)c+f(n)(Lϕ)(2πn/m) +
∑
n<0
τψ(n)c−f(n)(−4πn/m)1−k Z ∞
0
(Lϕ2−k)(−2πn(2t+1)/m) (1+t)k dt, and for eachϕ∈Fδkfψ, we also have
L(δkfψ,ϕ) =k
2L(fψ,ϕ)−2π
m
∑
n−∞
nc+f(n)(Lϕ2)(2πn/m)
−2π
m
∑
n<0
c−f(n)(−4πn/m)1−k Z ∞
0
(Lϕ3−k)(−2πn(2t+1)/m) (1+t)k dt.
Finally, denoteWN=√
N−1ωN. The converse theorem for harmonic Maass forms of general growth condition as in the following.
Theorem 17 (Theorem 4.5 and 5.1 in [35]) Fix two integers k and N with N>0, and letχbe a Dirich- let character modulo N. Let f and g be two functions defined onH given by the formal Fourier series
f(z) =
∑
n≥−n0
c+f(n)qn+
∑
n<0
c−f(n)Γ(1−k,−4πny)qn. g(z) =
∑
n≥−n0
c+g(n)qn+
∑
n<0
c−g(n)Γ(1−k,−4πny)qn, with c±f ,c±g =O(ec
√
N)as|n| →∞for some constant c>0. Then the followings are equivalent.
(1) The functions f and g belong to Hk(N,χ)and Hk(N,χ), respectively, and g= f|kWN.
(2) Letψ be a Dirichlet character modulo m∈ {1,2,· · ·,N2−1}with(m,N) =1. Let Sc(R+)be a set of complex-valued, compactly supported and piecewise smooth functions onR+which satisfy the following condition: for any y∈R+, there existsϕ∈Sc(R+)such thatϕ(y) =0. For anyψ, andϕ∈Sc(R+), f and g satisfies the following functional equations
L(fψ,ϕ) =ikχ(m)ψ(−N)
Nk/2−1 L(gψ,ϕ|2−kWN), L(δkfψ,ϕ) =−ikχ(m)ψ(−N)
Nk/2−1 L(δkgψ,ϕ|2−kWN).
The proof follows the methods of Theorem 15 and Theorem 16 with using twistedL-series instead.
We will skip the detailed proof and pay attention to some of the following changes.
Remark 7 (1) In the assumption, the growth condition for the Fourier coefficients c±f,c±g =O(ec
√ N) follows from Bruiner and Funke’s result (Lemma 3.4 in [36]). In fact, this asymptotic holds for all elements in Hk(N,χ).
(2) The set Sc(R+) simplifies the problem of simultaneous convergence of several Dirichlet series, especially the series in the above functional equations.
Supposeϕ∈Sc(R+) withsupp(ϕ) ={x∈R+|ϕ(x)6=0} ⊂(c1,c2) (c1,c2>0). Then for all x>0
(Lϕ)(x) = Z c2
c1
ϕ(x)e−xydy=Oc1,c2,ϕ(e−c1x).
Together with c±f,c±g =O(ec
√
N)andτ(ψ)(n) =Om(1), we deduce thatFfψ contains Sc(R+)for arbitrary characterψ. Moreover, Sc(R+)is closed under the action of WN.
(3) We can see that the meromorphic continuation did not appear in the theorem because the element of the set Sc(R+)is compactly support. However, in general, the functional equations in above theorem are vaild ifϕ contains the following set.
\
ψmodm
{ϕ|ϕ∈Ffψ∩Fδkfψ,ϕ|2−kWN∈Fgψ∩Fδkgψ},
which contains Sc(R+). Therefore, if we choose a subset of the above set instead of Sc(R+), we may have the problem of meromorphic continuation. For a instance, the authors proved the following theorem.
Theorem 18 (Theorem 4.6 in [35]) Let k∈Z, f ∈Hk(N,χ), and let g= f|kWN. Suppose the followings.
(1) f(z),g(z) =O(e2πn0y)as y→∞for some n0.
(2) A non-zero piecewise smooth functionϕ:R→Csatisfies
ϕ(y)e2π(n0+ε)y,ϕ(1/y)e2π(n0+ε)y→0 as y→∞ for someε>0.
(3) L(f,ϕ)converges absolutely (i.e.,ϕ∈Ff).
Then the series L(f,ϕs)converges absolutely forRe(s)>k−1
2, and has an analytic continuation to all s∈C, and satisfies the functional equation
L(f,ϕs) =N−s−k2−1ikL(g,(ϕ|1−kWN)1−s).
This can be seen as an analogue of Hecke’s theorem (Theorem 5), and can be proved in a similar way.
V Converse theorem for various modular forms
In addition to the theorems discussed above sections, various generalizations of Hecke and Weil’s inverse theorem to the other automorphic forms are known. We list some of these results.
(1) Doi and Naganuma [39] stated the converse theorem for certain Hilbert modular forms. They did not provide a detailed proof, but they mentioned that this can be prove using Hecke or Weil’s method.
(2) Martin [40] proved a generalized Hecke’s converse theorem for Jacobi cusp forms (of degree 1) over the full Jacobi group SL(2,Z)n Z2.
Martin and Osses [41] extended this result to the generalization of Weil’s converse theorem for Jacobi cusp forms over subgroupsΓ0(N)n Z2of SL(2,Z)n Z2.
Recently, Kohnen, Martin, and Shankhadhar [42] improved this result to a Jacobi cusp forms of degree 2, which is a generalization of Jacobi forms to the Siegel upper-half plane of degree 2 which is invariant under the generalized Jacobi group Sp(2,Z)n(Z2×Z2). Here, the Siegel upper-half plane of degree 2 is defined as
H2={M∈M2×2(C)|tM=M,Im(M)positive definite}, and Sp(2,R)is the symplectic group. (Note that Sp(2,Z) =SL(2,Z).)
(3) Imai [43] proved the converse theorem for degree 2 Siegel modular forms. This result was ex- tended to Siegel cusp forms of any degree by Weissauer [44], and revisited for non-cuspidal Siegel forms of degree 2 by Arakawa, Makino and Sato [45].
(4) Bringmann and Hayashida [46] established the converse theorem for Hilbert-Jacobi cusp forms over the Hilbert-Jacobi group SL(2,OK)n(OK×OK). HereK is a totally real number field of degreeg= [K:Q]with discriminantDKand narrow class number 1, andOKis its ring of integers.
Note that ifg=1, this is just the case of Jacobi forms overQwe mentioned above.
(5) Herrero-Miranda [47] established the converse theorem for Jacobi-Maass forms. This is a gen- eralization of a converse theorem for (holomorphic) Jacobi cusp forms due to Martin [40] to non-cuspidal Jacobi forms. He also provided a converse theorem for skew-holomorphic Jacobi forms.
(6) The most generalized result is the converse theorem for automorphic representations of GL(n,AF).
HereF is a global field, andAF is the adele ring ofF. This is proved by several authors: forn=2 by Jacquet and Langlands [48], forn=3 by Jacquet, Piatetski-Shapiro and Shalika [49], [50] and for generalnby Cogdell and Piatetski-Shapiro [51]. In particular, the Jacquet-Langlands converse theorem applies to holomorphic modular and Maass forms of levelN.
The proof of these does not deviate much from the original method of Hecke and Weil, but requires some ideas according to the characteristics of each modular form. For instance, Jacobi form is a function fromH ×CintoCwith some conditions. Then we cannot apply the ordinary Mellin transform directly, so we need a proper modification of it to obtainL-function associated to Jacobi forms.
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Acknowledgements
First and foremost, I would like to express my gratitude to my supervisor, Hae-Sang Sun, for his gener- ous support and guidance. I would also like to express my gratitude to professors Chol Park and Jaehyun Cho, who examined my thesis, and all members of the UNIST number theory group who helped me a lot during my UNIST life.
I would also like to thank my family who have made me where I am today.
I would finally like to thank the members of ‘Room 604’ who are my alumni, colleagues, and best friends.