# On Jacobi–Weierstrass mock modular forms

Claudia Alfes-Neumann  ,  Jens Funke  ,  Michael. H. Mertens  and  Eugenia
Rosu  Universität Bielefeld, Fakultät für Mathematik, Postfach 100 131, 33501
Bielefeld, Germany, E-Mail<EMAIL_ADDRESS>Durham University,
Department of Mathematical Sciences, Science Laboratories, South Rd, Durham
DH1 3LE, United Kingdom, E-Mail<EMAIL_ADDRESS>Universität zu Köln,
Department Mathematik/Informatik, Abteilung Mathematik, Weyertal 86–90, 50931
Köln, Germany, E-Mail<EMAIL_ADDRESS>Leiden University,
Mathematical Institute, Niels Bohrweg 1, 2333 CA Leiden Leiden, The
Netherlands E-Mail<EMAIL_ADDRESS>
###### Abstract.

We construct harmonic weak Maass forms that map to cusp forms of weight $k\geq
2$ with rational coefficients under the $\xi$-operator. This generalizes work
of the first author, Griffin, Ono, and Rolen, who constructed distinguished
preimages under this differential operator of weight $2$ newforms associated
to rational elliptic curves using the classical Weierstrass theory of elliptic
functions. We extend this theory and construct a vector-valued
Jacobi–Weierstrass $\zeta$-function which is a generalization of the classical
Weierstrass $\zeta$-function.

The first author is partially supported by the Daimler and Benz Foundation,
the Klaus Tschira Boost Fund and funded by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) – SFB-TRR 358/1 2023 – 491392403

The fourth author was partially supported by the Deutsche Forschungs-
Gemeinschaft (DFG) through the Collaborative Research Centre TRR 326 "Geometry
and Arithmetic of Uniformized Structures", project number 444845124.

##  1\. Introduction

Harmonic weak Maass forms are real-analytic generalizations of classical
modular forms with applications in combinatorics, number theory, as well as
representation theory and physics (see for instance [BFOR17] and the
references therein). They were first introduced by Bruinier and the second
author [BF04] in the context of theta lifts. A harmonic weak Maass form of
weight $k\in\frac{1}{2}\mathbb{Z}$ for a congruence subgroup $\Gamma$ of
$\operatorname{SL}_{2}(\mathbb{Z})$ is a smooth function
$F:\mathbb{H}\to\mathbb{C}$ that transforms like a classical modular form
under $\Gamma$, but which is harmonic rather than holomorphic (see Section 2.1
for a precise definition). If $F$ has poles in $\mathbb{H}$ it is called polar
harmonic weak Maass form.

Harmonic weak Maass forms are inherently connected to deep number-theoretic
questions. A beautiful example is Zwegers’ work [Zwe02] who showed that
Ramanujan’s mock theta functions are holomorphic parts of harmonic weak Maass
forms. Such functions are also called mock modular forms [Zag09]. Harmonic
weak Maass forms also appear prominently in the Kudla program and numerous
other places in mathematics.

The $\xi$-operator

$\xi_{k}=-2iv^{k}\overline{\frac{\partial}{\partial\overline{\tau}}},\ \
\tau=u+iv,$

plays a crucial role in relating the theory of classical modular forms to
harmonic weak Maass forms. It defines a surjective map from the space $H_{k}$
of harmonic weak Maass forms of weight $k$ to the space $S_{2-k}$ of cusp
forms of weight $2-k$. The image under the $\xi$-operator is called the shadow
of the corresponding harmonic weak Maass form resp. mock modular form. In
particular, there are infinitely many preimages in the space $H_{k}$ for any
given cusp form in $S_{2-k}$, yet they are surprisingly difficult to
construct. The quest for distinguished preimages under the $\xi$-operator is a
fundamental question, and as cusp forms are omnipresent in number theory, the
construction of distinguished preimages offers new routes to tackle related
problems.

There are various approaches to this problem: Bruinier [Bru02] and Bringmann
and Ono [BO07] showed that certain real-analytic Poincaré series, originally
introduced by Niebur [Nie74], map to the classical exponential type Poincaré
series of dual weight under the $\xi$-operator, provided that they converge,
e.g. when the weight is negative. The coefficients of cuspidal Poincaré series
are however not easy to handle, both practically and theoretically.

Another approach to the problem uses the _non-holomorphic Eichler integral_

$g^{*}(\tau)=\int_{-\overline{\tau}}^{\infty}\frac{\overline{f(-\overline{z})}}{(-i(z-\tau))^{k}}dz$

for a cusp form $g$ of weight $2-k$, as proposed and used first in work by
Andrews, Rhoades, and Zwegers [ARZ13] as well as Dabholkar, Murthy, and Zagier
[DMZ23]. Using a suitable auxiliary holomorphic cusp form $h$ and holomorphic
projection, they construct a harmonic weak Maass form $F(\tau)$ of weight $k$
such that $\xi_{k}F=cg$ for some constant $c$. This approach is particularly
well-suited when $g$ is a unary theta function, since then the Fourier
coefficients of the holomorphic projection can be evaluated explicitly (see
also [MOR21, MMR22]).

Recently, Ehlen, Li, and Schwagenscheidt [ELS22] gave an explicit procedure to
find a _good_ preimage of a CM modular form using a certain theta lift. It is
known from previous work by Bruinier, Ono, and Rhoades [BOR08] that such good
preimages always exist and in particular that their holomorphic parts have
algebraic Fourier coefficients at $\infty$. Using their explicit construction,
Ehlen, Li, and Schwagenscheidt could pinpoint the exact algebraic number field
containing these Fourier coefficients.

In this work we focus on the geometric approach and generalize the work of
Griffin, Ono, Rolen, and the first author [AGOR15]. They construct harmonic
weak Maass forms whose shadows are given by newforms of weight $2$ with
rational coefficients, generalizing previous work by Guerzhoy [Gue15].

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ and let $G_{E}$ be the
corresponding newform of weight $2$ for a congruence subgroup $\Gamma_{0}(N)$
of $\operatorname{SL}_{2}(\mathbb{Z})$ via the modularity theorem. Over
$\mathbb{C}$, $E$ is isomorphic to a torus $\mathbb{C}/\Lambda_{E}$, with the
isomorphism given by

$\wp^{\prime}(z)^{2}=4\wp(z)^{3}-g_{2}\wp(z)+g_{3},$

where $\wp$ is the Weierstrass $\wp$-function for the lattice $\Lambda_{E}$,
and $g_{2},g_{3}$ are the normalized Eisenstein series for the lattice
$\Lambda_{E}$.

The _Weierstrass $\zeta$-function_

(1.1)
$\zeta_{\Lambda_{E}}(z)=\frac{1}{z}+\sum_{\omega\in\Lambda_{E}\setminus\\{0\\}}\left(\frac{1}{z-\omega}+\frac{1}{\omega}+\frac{z}{\omega^{2}}\right),$

is not itself invariant under shifts by lattice points, but has a well-known
non-analytic completion $\zeta_{\Lambda_{E}}^{*}(z)$ which is indeed invariant
under $\Lambda_{E}$. This completion allows one to construct a canonical
preimage $Z_{E}$ of the modular form $G_{E}$ under the operator $\xi_{0}$ by
taking

$Z_{E}(\tau)=\zeta_{\Lambda_{E}}^{*}(\mathcal{E}_{E}(\tau)),$

where $\mathcal{E}_{E}(\tau)=\int_{\tau}^{\infty}G_{E}(t)dt$ is the
holomorphic Eichler integral. The error to modularity of the Eichler integral
is determined by the lattice, i.e.
$\mathcal{E}_{E}(\gamma\tau)=\mathcal{E}_{E}(\tau)+\omega$, for
$\gamma\in\Gamma_{0}(N)$ and for some $\omega\in\Lambda_{E}$, giving the
$\Gamma_{0}(N)$-invariance of $Z_{E}(\tau)$. We recall the details of this
construction in Section 3.

These preimages $Z_{E}$ can be computed very efficiently and are also of
theoretical interest. In [AGOR15], they were used to obtain a criterion for
the vanishing of critical $L$-derivatives of quadratic twists of $E$. Another
application in the context of vertex operator algebras can be found in [BM21].

In the current work we generalize the above construction to newforms $f$ with
rational Fourier coefficients of weight $k\geq 2$, for a congruence subgroup
$\Gamma\subset\operatorname{SL}_{2}(\mathbb{Z})$, with
$\Gamma_{1}(N)\subset\Gamma$ for some integer $N$. Our construction involves
vector-valued forms. For simplicity we restrict to the easiest case in the
introduction and focus on describing the central ideas of our work.

We start by noting that a modular form $f$ of weight $k\geq 2$ gives a vector-
valued modular form
$f:\mathbb{H}\rightarrow\operatorname{Sym}^{k-2}(\mathbb{C}^{2})$ of weight
$2$ by taking

(1.2) $G(\tau)=cf(\tau)(\tau e_{1}+e_{2})^{k-2},$

where $(e_{1},e_{2})$ is a fixed basis of $\mathbb{C}^{2}$ and
$\operatorname{Sym}^{k-2}(\mathbb{C}^{2})$ is the $(k-2)$-th symmetric power
of $\mathbb{C}^{2}$, which can be realized as the space of homogeneous
polynomials in $e_{1}$ and $e_{2}$ of degree $k-2$, and $c$ is a constant.
Note that this construction of vector-valued modular forms goes back to
classic work of Kuga and Shimura [KS60]. For further details, see Section 2.3.

The construction involves two central elements: a vector-valued generalization
of the Weierstrass $\zeta$-function $\widehat{\zeta}$ and the polynomial
Eichler integral $\mathcal{E}_{f}$.

In the simplest case (see Section 6), we define the multivariable function
$\zeta:\mathbb{H}\times\mathbb{C}^{k-1}\rightarrow\operatorname{Sym}^{k-2}(\mathbb{C}^{2})$
by

(1.3)
$\zeta(\tau,\boldsymbol{z})=\sum_{j=0}^{k-2}\zeta_{\Lambda_{\tau}}(z_{j})c_{j}e_{1}^{j}e_{2}^{k-2-j},\
\ c_{j}={k-2\choose j},\ \ \Lambda_{\tau}=\mathbb{Z}+\mathbb{Z}\tau.$

It has a natural completion

(1.4)
$\widehat{\zeta}(\tau,\boldsymbol{z})=\sum_{j=0}^{k-2}\zeta^{*}_{\Lambda_{\tau}}(z_{j})c_{j}e_{1}^{j}e_{2}^{k-2-j},$

where $\zeta_{\Lambda_{\tau}}^{*}$ is the completion of the classical
Weierstrass $\zeta$-function. Our first main result is the general
construction of Jacobi–Weierstrass $\zeta$-functions $\widehat{\zeta}$.

######  Theorem 1.1.

The function $\widehat{\zeta}$ is a (non-holomorphic) Jacobi form of weight
$1$ and index $0$, invariant under the lattice $\Lambda_{\tau}^{k-1}$.

To construct the higher degree generalization $\widehat{\zeta}$ of the
completed Weierstrass $\zeta$-function we follow Rolen [Rol16]. The key point
is to write the $\zeta$-function as the logarithmic derivative of the
Weierstrass $\sigma$-function which in turn is essentially given by a multiple
of Jacobi’s theta function. This definition of the $\sigma$-function can be
generalized to higher degree by giving an analogous construction using a
Jacobi theta function of lattice index. Replacing the logarithmic derivative
by a suitable weight raising operator we obtain the Jacobi–Weierstrass
$\zeta$-function.

We then use the Jacobi–Weierstrass $\zeta$-function to construct the natural
preimages of $f$ by evaluating it at the polynomial Eichler integral
$\mathcal{E}_{f}:\mathbb{H}\times\operatorname{SL}_{2}(\mathbb{Z})\to\operatorname{Sym}^{k-2}(\mathbb{C}^{2})$,

(1.5)
$\mathcal{E}_{f}(\tau,(X_{1},X_{2}))=\int_{\tau}^{\infty}f(t)(tX_{1}+X_{2})^{k-2}dt,$

for $\tau\in\mathbb{H}$, $M=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\operatorname{SL}_{2}(\mathbb{Z})$,
$(X_{1},X_{2})=(e_{1},e_{2})M$ is the change of basis of $\mathbb{C}^{2}$, and
$f$ is a cusp form of weight $k$.

To realize this, we first extend the Jacobi–Weierstrass $\zeta$-function to
include a possible change of basis for the space $\mathbb{C}^{2}$, given by
$M=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\operatorname{SL}_{2}(\mathbb{Z})$, as follows

(1.6)
$\widehat{\zeta}_{M}(\tau,\boldsymbol{z})=\sum_{j=0}^{k-2}\zeta^{*}_{\Lambda_{\tau}}(z_{j})X_{1}^{j}X_{2}^{k-2-j},\
\ (X_{1},X_{2})=(e_{1},e_{2})M.$

This gives the vector-valued Jacobi form
$\widehat{\zeta}:\mathbb{H}\times\mathbb{C}^{k-1}\times\operatorname{SL}_{2}(\mathbb{Z})\rightarrow\operatorname{Sym}^{k-2}(\mathbb{C}^{2})$
where we write
$\widehat{\zeta}(\tau,\boldsymbol{z},M)=\widehat{\zeta}_{M}(\tau,\boldsymbol{z}).$
Fixing further the standard basis of $\mathbb{C}^{2}$, we take the natural
isomorphism $\operatorname{Sym}^{k-2}(\mathbb{C}^{2})\simeq\mathbb{C}^{k-1}$,
which gives us the most general Jacobi–Weierstrass function
$\widehat{\zeta}:\mathbb{H}\times\operatorname{Sym}^{k-2}(\mathbb{C}^{2})\times\operatorname{SL}_{2}(\mathbb{Z})\rightarrow\operatorname{Sym}^{k-2}(\mathbb{C}^{2}).$
This form was carefully defined to account for the various actions of the
subgroup $\Gamma$ on the upper-half plane $\mathbb{H}$ and
$\operatorname{Sym}^{k-2}(\mathbb{C}^{2})$ (compare Section 5).

Finally, we define the vector-valued form
$F:\mathbb{H}\times\operatorname{SL}_{2}(\mathbb{Z})\rightarrow\operatorname{Sym}^{k-2}(\mathbb{C}^{2})$
by taking

(1.7)
$F(\tau,M)=\widehat{\zeta}_{M}(\Lambda_{f},\mathcal{E}(\tau,(e_{1},e_{2})M^{-1})),$

where $\Lambda_{f}$ is the lattice corresponding to $f$ (see Section 2.4, in
particular Remark 2.3).

We note that $F$ can also be written as a vector-valued form
$F:\mathbb{H}\rightarrow V,$ valued in the infinite-dimensional vector space
$V=\mathrm{Functions}\left\\{\operatorname{SL}_{2}(\mathbb{Z})\to\operatorname{Sym}^{k-2}(\mathbb{C}^{2})\right\\}$,
by sending $F(\tau)$ to the function $M\rightarrow F(\tau,M)$ in $V$.

The main result of the paper is the following:

######  Theorem 1.2.

The function $F:\mathbb{H}\rightarrow V$ is a vector-valued polar harmonic
weak Maass form of weight $0$ for $\Gamma$, whose image under $\xi_{0}$ is
given by

$G(\tau)=\frac{2\pi i}{\mathrm{Vol}(\Lambda_{f})}f(\tau)(\tau
e_{1}+e_{2})^{k-2}.$

Note that in the case of $k=2$ the functions we obtain through our
construction are in fact constant in the
$\operatorname{SL}_{2}(\mathbb{Z})$-variable, therefore we recover the results
from [AGOR15]. The poles of $F$ are explicitly computed in Proposition 6.3.

######  Remark 1.3.

The naive approach of fixing a basis $(e_{1},e_{2})$ of $\mathbb{C}^{2}$ and
plugging in the Eichler integral directly
$\widehat{\zeta}(\Lambda_{f},\mathcal{E}_{f}(\tau,(e_{1},e_{2})))$ loses the
$\Gamma$-invariance. In general we do not have a compatibility between the
action of $\operatorname{Sym}^{k-2}(\mathbb{C}^{2})$ on the outside and on the
inside

$\widehat{\zeta}\left(\Lambda_{f},\mathcal{E}_{f}\left(\tau,\left(e_{1},e_{2}\right)\right)\right)\neq\gamma\cdot\widehat{\zeta}\left(\Lambda_{f},\mathcal{E}_{f}\left(\gamma\tau,\left(e_{1},e_{2}\right)\right)\right).$

By controlling the change of basis with the extra variable
$M\in\operatorname{SL}_{2}(\mathbb{Z})$ we insure the $\Gamma$-invariance of
the vector-valued form $F$.

The paper is organized as follows. In Section 2 we introduce the notion of
harmonic weak Maass forms, give the necessary background on representation
theory, and introduce the vector-valued Eichler integral. In Section 3 we
review the construction of Weierstrass harmonic weak Maass forms in the case
of weight $2$. The generalization to higher degree of the completed
Weierstrass $\zeta$-function is given in Section 4. Section 5 contains our
main result, namely the construction of the vector-valued Jacobi–Weierstrass
harmonic weak Maass form $F:\mathbb{H}\rightarrow V$. In Section 6 we compute
the Laurent expansion of the Jacobi–Weierstrass $\zeta$-function and the poles
of the polar harmonic weak Maass form $F$. In Section 7 we provide two
examples of our construction.

##  2\. Preliminaries

Throughout this paper we let $\Gamma$ be a congruence subgroup of
$\operatorname{SL}_{2}(\mathbb{Z})$ such that $\Gamma_{1}(N)\subset\Gamma$ for
some positive integer $N$, and
$\Gamma_{1}(N)=\\{\gamma\in\operatorname{SL}_{2}(\mathbb{Z}):\gamma\equiv\left(\begin{smallmatrix}1&*\\\
0&1\end{smallmatrix}\right)\mod N\\}$. We call $f$ a newform for $\Gamma$ if
it is a newform for $\Gamma_{1}(N)$.

###  2.1. Harmonic weak Maass forms

By $\mathrm{Mp}_{2}(\mathbb{R})$ we denote the metaplectic group consisting of
pairs $(\gamma,\phi)$, where $\gamma=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\operatorname{SL}_{2}(\mathbb{R})$ and
$\phi:\mathbb{H}\to\mathbb{C}$ is a holomorphic function with
$\phi(\tau)^{2}=c\tau+d$. We let $\mathrm{Mp}_{2}(\mathbb{Z})$ be the inverse
image of $\operatorname{SL}_{2}(\mathbb{Z})$ under the covering map
$\mathrm{Mp}_{2}(\mathbb{R})\to\operatorname{SL}_{2}(\mathbb{R})$.

A twice continuously differentiable function $F:\mathbb{H}\to\mathbb{C}$ is
called a harmonic weak Maass form of weight $k\in\frac{1}{2}\mathbb{Z}$ for
$\mathrm{Mp}_{2}(\mathbb{Z})$ if it satisfies

  1. (1)

$\Delta_{k}F=0$, where $\Delta_{k}=-v^{2}\left(\frac{\partial^{2}}{\partial
u^{2}}+\frac{\partial^{2}}{\partial
v^{2}}\right)+ikv\left(\frac{\partial}{\partial u}+i\frac{\partial}{\partial
v}\right)$ is the hyperbolic weight $k$ Laplace operator and we write
$\tau=u+iv\in\mathbb{H}$.

  2. (2)

$F(\tau)|_{k}(\gamma,\phi):=\phi(\tau)^{-2k}F(\gamma\tau)=F(\tau)$ for
$(\gamma,\phi)\in\mathrm{Mp}_{2}(\mathbb{Z})$.

  3. (3)

There is a Fourier polynomial $P_{F}(\tau)=\sum_{n\leq 0}a^{+}(n)q^{n}$,
called the principal part of $F$, such that

$F(\tau)-P_{F}(\tau)=O(e^{-\varepsilon v})$

as $v\to\infty$, uniformly in $u$, for some $\varepsilon>0$. A similar
conditions holds at all cusps.

We define harmonic weak Maass forms of integral weight $k\in\mathbb{Z}$
transforming with respect to the group $\Gamma$ accordingly. If $F$ has poles
on $\mathbb{H}$ it is called polar harmonic weak Maass form.

###  2.2. The Jacobi theta function

Let $L\simeq\mathbb{Z}^{g}$ be an even lattice with positive definite inner
product defined through $\left(v,w\right):=v^{t}G_{L}w$, where $G_{L}$ is the
Gram matrix of $L$, a symmetric matrix with integer entries and even entries
on the diagonal, and let $V=L\otimes\mathbb{C}\simeq\mathbb{C}^{g}$ with
quadratic form $Q(v)=\left(v,v\right)/2$.

######  Definition 2.1.

Let $\tau\in\mathbb{H}$ and $\boldsymbol{z}\in\mathbb{C}^{g}$. We define the
Jacobi theta function by

$\theta(\tau,\boldsymbol{z})=\sum_{\ell\in L}e^{\pi i(\ell,\ell)\tau}e^{2\pi
i(\ell,\boldsymbol{z})}.$

The following two transformation properties of $\theta(\tau,\boldsymbol{z})$
are well-known

(2.1)
$\displaystyle\theta(\tau,\boldsymbol{z}+\boldsymbol{m}\tau+\boldsymbol{n})$
$\displaystyle=e^{-\pi i\tau(\boldsymbol{m},\boldsymbol{m})}e^{-2\pi
i(\boldsymbol{z},\boldsymbol{m})}\theta(\tau,\boldsymbol{z}),\ \
\boldsymbol{m},\boldsymbol{n}\in\mathbb{Z}^{g},$ (2.2)
$\displaystyle\theta\left(\tau+1,\boldsymbol{z}\right)$
$\displaystyle=\theta(\tau,\boldsymbol{z}).$

Under the additional assumption that $L$ is unimodular, see e.g. [Boy15], it
follows from Poisson summation that

(2.3)
$\displaystyle\theta\left(-\frac{1}{\tau},\frac{\boldsymbol{z}}{\tau}\right)$
$\displaystyle=(\tau/i)^{g/2}e^{2\pi
i\frac{Q(\boldsymbol{z})}{\tau}}\theta(\tau,\boldsymbol{z}).$

These functional equations imply that $\theta(\tau,\boldsymbol{z})$ is a
Jacobi form of weight $g/2$ and index $\frac{1}{2}G_{L}$ in the sense of
[BK93] for the full Jacobi group
$\operatorname{SL}_{2}(\mathbb{Z})\ltimes(\mathbb{Z}^{g})^{2}$. If $L$ is not
unimodular, $\theta(\tau,\boldsymbol{z})$ is a Jacobi form for some suitable
congruence subgroup $\mathcal{J}_{L}=\Gamma_{L}\ltimes(\mathbb{Z}^{g})^{2}$ of
the full Jacobi group (compare [Boy15, Corollary 3.34, 3.35]).

We follow Böcherer–Kohnen [BK93] and define the slash operator by

$\displaystyle(\phi|_{k,mL}A)(\tau,\boldsymbol{z})$
$\displaystyle:=\phi\left(\frac{a\tau+b}{c\tau+d},\frac{\boldsymbol{z}+\boldsymbol{m}\tau+\boldsymbol{n}}{c\tau+d}\right)(\sqrt{c\tau+d})^{-2k}$
$\displaystyle\quad\quad\quad e^{2\pi
im\left(-c\frac{2Q(\boldsymbol{z}+\boldsymbol{m}\tau+\boldsymbol{n})}{(c\tau+d)}+\left(\boldsymbol{m},\boldsymbol{m}\right)\tau+2\left(\boldsymbol{m},\boldsymbol{z}\right)\right)},$

where $A=\left[\left(\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right),\sqrt{c\tau+d}\right),(\boldsymbol{m},\boldsymbol{n})\right]\in\operatorname{Mp}_{2}(\mathbb{Z})\ltimes(\mathbb{Z}^{g})^{2}$
and $k,m\in\frac{1}{2}\mathbb{Z}$. Then the condition that $\theta$ is a
Jacobi form for the Jacobi subgroup $\mathcal{J}_{L}$ is equivalent to

(2.4) $\theta|_{g/2,1/2}A=\theta,\ \ \text{ for }A\in\mathcal{J}_{L}.$

As in [BRR15] we now define a weight raising operator for Jacobi forms by

(2.5) $Y_{+,z_{j}}^{g/2,1/2L}=\frac{\partial}{\partial z_{j}}+2\pi
i\frac{\operatorname{Im}((G_{L}\boldsymbol{z})_{j})}{\operatorname{Im}(\tau)},$

where $\boldsymbol{z}=(z_{1},\ldots,z_{g})\in\mathbb{C}^{g}$. As we will show
in Proposition 4.6, we have

(2.6)
$Y_{+,z_{j}}^{g/2,1/2L}\theta=(Y_{+,z_{j}}^{g/2,1/2L}\theta)|_{g/2+1,1/2}A$

for $A=\left[\left(\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right),\sqrt{c\tau+d}\right),(\boldsymbol{m},\boldsymbol{n})\right]\in\mathcal{J}_{L}$
and $k\in\frac{1}{2}\mathbb{Z}$.

###  2.3. A little bit of representation theory

We let $W(=\mathbb{C}^{2})$ be the standard complex representation of
$\operatorname{SL}_{2}(\mathbb{Z})$ and let $\langle\,,\,\rangle$ be the usual
symplectic form on $W$ with standard basis $e_{1},e_{2}$. This defines an
integral structure on $W$. We let $W_{m}=\operatorname{Sym}^{m}W$ be the
irreducible representation of dimension $m+1$ of highest weight $m$. Weight
vectors are (multiples of) $e_{1}^{j}e_{2}^{m-j}$ with corresponding weight
$j$ with respect to the standard rationally split torus
$S=\\{\left(\begin{smallmatrix}t&0\\\
0&t^{-1}\end{smallmatrix}\right)\,:\,t\in\mathbb{R}\\}$. Here we write $v^{k}$
for the symmetric tensor $v\bullet v\bullet\cdots\bullet v$.

Note that $W_{m}$ is self-dual, i.e. $W_{m}\simeq W_{m}^{*}$, where the
isomorphism is induced by the symplectic form on $W$. We will not distinguish
between $W_{m}$ and $W_{m}^{*}$.

The action of $\operatorname{SL}_{2}(\mathbb{Z})$ on
$W_{k-2}=\operatorname{Sym}^{k-2}(\mathbb{C}^{2})$ is given by acting on the
basis on $\mathbb{C}^{2}$, i.e.

$M\circ(e_{1},e_{2}):=\begin{pmatrix}e_{1}&e_{2}\end{pmatrix}M=\begin{pmatrix}ae_{1}+ce_{2}&be_{1}+de_{2}\end{pmatrix},\,\text{
for }M=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\operatorname{SL}_{2}(\mathbb{Z}).$

We recall the usual action of $\operatorname{SL}_{2}(\mathbb{Z})$ of the
upper-half plane $\mathbb{H}$ by fractional linear transformations
$M\tau=\frac{a\tau+b}{c\tau+d}$ for $M=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\operatorname{SL}_{2}(\mathbb{Z})$. Moreover, we
denote by $j(M,\tau)=c\tau+d$ the factor of automorphy.

We then have

(2.7) $M^{-1}\circ((M\tau)e_{1}+e_{2})^{k-2}=j(M,\tau)^{2-k}(\tau
e_{1}+e_{2})^{k-2}$

for $M\in\operatorname{SL}_{2}(\mathbb{Z})$.

For a modular form $f$ of weight $k$ for the congruence subgroup $\Gamma$,
this directly implies that the holomorphic $1$-form on $\mathbb{H}$ given by

$\eta_{f}:=f(\tau)d\tau\otimes(\tau e_{1}+e_{2})^{k-2}$

satisfies $\gamma^{-1}\circ\eta_{f}(\gamma\tau)=\eta_{f}(\tau)$ for
$\gamma\in\Gamma$.

###  2.4. Eichler integrals

Throughout this section let $f$ be a cusp form of weight $k$ for some finite
index subgroup $\Gamma$ of $\operatorname{SL}_{2}(\mathbb{Z})$ and let
$(X_{1},X_{2})$ be a basis for $W$. We define the _Eichler integral of $f$_ by

(2.8)
$\mathcal{E}_{f}(\tau,(X_{1},X_{2}))=\int_{\tau}^{\infty}f(t)(tX_{1}+X_{2})^{k-2}dt.$

It defines a function
$\mathcal{E}_{f}:\mathbb{H}\times\operatorname{SL}_{2}(\mathbb{Z})\rightarrow\operatorname{Sym}^{k-2}(\mathbb{C}^{2})$,
where the basis $(X_{1},X_{2})$ above corresponds to a matrix
$M\in\operatorname{SL}_{2}(\mathbb{Z})$ such that
$(X_{1},X_{2})=M\circ(e_{1},e_{2})$. Note that we recover the usual (scalar-
valued) Eichler integral by identifying $W_{m}$ with the space of homogeneous
polynomials of degree $m$ in two variables $X,Y$ and by replacing $X_{1}$ by
$1$ and $X_{2}$ by $\tau$.

For $\gamma\in\Gamma$ we consider

$\mathcal{E}_{f}(\gamma\tau,\gamma^{-1}\circ(X_{1},X_{2}))=\int_{\gamma\tau}^{\infty}f(t)\left[\gamma^{-1}\circ(tX_{1}+X_{2})^{k-2}\right]dt,\,\gamma\in\Gamma.$

######  Lemma 2.2.

Let $f$ be a cusp form of weight $k$ for $\Gamma$, we have

$\mathcal{E}_{f}(\gamma\tau,\gamma^{-1}\circ(X_{1},X_{2}))=\mathcal{E}_{f}(\tau,(X_{1},X_{2}))-\sum_{\ell=0}^{k-2}{k-2\choose\ell}X_{1}^{\ell}X_{2}^{k-2-\ell}\int_{\gamma^{-1}\infty}^{\infty}f(t)t^{\ell}dt.$

for $\gamma\in\Gamma$ and $(X_{1},X_{2})$ a basis of $\mathbb{C}^{2}$.

###### Proof.

Identity (2.7) implies

$\displaystyle\mathcal{E}_{f}(\gamma\tau,\gamma^{-1}\circ(X_{1},X_{2}))$
$\displaystyle=\int_{\gamma\tau}^{\infty}f(t)\cdot\left[\gamma^{-1}\circ(tX_{1}+X_{2})^{k-2}\right]dt$
$\displaystyle=\int_{\tau}^{\gamma^{-1}\infty}f(t)(tX_{1}+X_{2})^{k-2}dt.$

We split the integral

$\int_{\tau}^{\gamma^{-1}\infty}f(t)(tX_{1}+X_{2})^{k-2}dt=\int_{\tau}^{\infty}f(t)(tX_{1}+X_{2})^{k-2}dt-\int_{\gamma^{-1}\infty}^{\infty}f(t)(tX_{1}+X_{2})^{k-2}dt$

and use the binomial expansion to obtain the result. ∎

We now recall some facts about modular symbols. Further details and background
may be found for instance in Chapters $8$ and $10$ in [Ste07]. Let
$\mathbb{M}_{k}$ denote the module of _modular symbols_ , i.e. the
$\mathbb{Z}$-module generated by symbols $P\\{\alpha,\beta\\}$, where
$P\in\mathbb{Z}[X,Y]_{k-2}$ is a homogeneous polynomial of degree $k-2$ with
integer coefficients and $\alpha,\beta\in\mathbb{P}^{1}(\mathbb{Q})$ are
cusps, modulo the relations obtained from

$\\{\alpha,\beta\\}+\\{\beta,\gamma\\}+\\{\gamma,\alpha\\}=0$

and all torsion. The group $\Gamma$ acts from the left on modular symbols by
combining the usual action on $\mathbb{P}^{1}(\mathbb{Q})$ via Möbius
transformations and on homogeneous polynomials defined by

$(\left(\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right),P)\mapsto P(dX-
bY,-cX+aY).$

We denote by $\mathbb{M}_{k}(\Gamma)$ the module of modular symbols modulo all
torsion and all relations obtained from $x-\gamma.x$ where
$x\in\mathbb{M}_{k}$ and $\gamma\in\Gamma$. Every cusp form $f\in
S_{k}(\Gamma)$ defines a pairing

(2.9) $\displaystyle\Phi_{f}:\mathbb{M}_{k}(\Gamma)\to\mathbb{C},\ \
(f,P\\{\alpha,\beta\\})\mapsto\int_{\alpha}^{\beta}f(t)P(t,1)dt,$

which is consistent with the action of Hecke operators. If $f$ is a normalized
newform with integral coefficients, then the image
$\Phi_{f}(\mathbb{M}_{k}(\Gamma))$ is a lattice $\Lambda_{f}$ in $\mathbb{C}$.

######  Remark 2.3.

We write

(2.10)
$\mathcal{E}_{f}(\tau,(X_{1},X_{2}))=\sum_{\ell=0}^{k-2}\mathcal{E}_{\ell,f}(\tau)c_{\ell}X_{1}^{\ell}X_{2}^{k-2-\ell},$

with $c_{\ell}={k-2\choose\ell}$ and
$\mathcal{E}_{\ell,f}(\tau)=\int_{\tau}^{\infty}f(t)t^{\ell}dt$. If $f$ has
integral coefficients, the considerations above imply that

(2.11)
$\mathcal{E}_{\ell,f}(\gamma\tau)-\mathcal{E}_{\ell,f}(\tau)\in\Lambda_{f}.$

For $k=2$ and $\Gamma=\Gamma_{0}(N)$, the lattice $\Lambda_{f}$ coincides with
the period lattice of the rational elliptic curve associated to $f$.

We now choose the basis
$\\{c_{\ell}X_{1}^{\ell}X_{2}^{k-2-\ell}\\}_{0\leq\ell\leq k-2}$ of
$\operatorname{Sym}^{k-2}(\mathbb{C}^{2})$. In these coordinates the Eichler
integral is given as

(2.12)
$\mathcal{E}_{f}(\tau,(X_{1},X_{2}))=(\mathcal{E}_{0,f}(\tau),\dots,\mathcal{E}_{k-2,f}(\tau))_{(X_{1},X_{2})}.$

For later convenience, we reformulate Lemma 2.2 in a slightly different
fashion.

######  Lemma 2.4.

The Eichler integral $\mathcal{E}_{f}(\tau,(X_{1},X_{2}))$ is
$\Lambda_{f}^{k-1}$-invariant under the $\Gamma$-action, meaning that for
$\gamma\in\Gamma$ we have

$(\mathcal{E}_{0,f}(\gamma\tau),\dots,\mathcal{E}_{k-2,f}(\gamma\tau))_{\gamma^{-1}\circ(X_{1},X_{2})}=(\mathcal{E}_{0,f}(\tau),\dots,\mathcal{E}_{k-2,f}(\tau))_{(X_{1},X_{2})}+\omega,$

with $\omega=(\lambda_{0},\dots,\lambda_{k-1})_{(X_{1},X_{2})}$ and each
$\lambda_{i}\in\Lambda_{f}$.

A straightforward computation gives the Fourier expansion of the coefficients
$\mathcal{E}_{f,\ell}(\tau)$.

######  Lemma 2.5.

For a cusp form $f\in S_{k}(\Gamma)$ with a Fourier expansion
$f(\tau)=\sum\limits_{n\geq 1}a_{n}q^{n}$ we have

$\mathcal{E}_{f,\ell}(\tau)=\int_{\tau}^{\infty}f(t)t^{\ell}dt=\sum_{n\geq
1}\frac{a_{n}}{2\pi
n}\left(\sum_{j=0}^{\ell}\frac{\ell!}{(\ell-j)!}\left(\frac{i}{2\pi
n}\right)^{j}\tau^{\ell-j}\right)q^{n}.$

As is also apparent directly from the definition, $\mathcal{E}_{f,\ell}$ is
not 1-periodic except when $\ell=0$. In general the Fourier coefficients of
$\mathcal{E}_{f,\ell}$ are polynomials in $\tau$.

###  2.5. Change of basis

We note that each basis $(X_{1},X_{2})$ of $\mathbb{C}^{2}$ gives rise to a
basis

$\\{c_{\ell}X_{1}^{\ell}X_{2}^{k-2-\ell}\\}_{0\leq\ell\leq k-2}$ of
$\operatorname{Sym}^{k-2}(\mathbb{C}^{2})$. Let
$\boldsymbol{z}=(z_{0},\ldots_{,}z_{k-2})\in\mathbb{C}^{k-1}$. We write

(2.13)
$(\boldsymbol{z})^{\mathrm{t}}_{(X_{1},X_{2})}=(z_{0},\ldots,z_{k-1})^{\mathrm{t}}_{(X_{1},X_{2})}=\sum\limits_{\ell=0}^{k-2}z_{\ell}c_{\ell}X_{1}^{\ell}X_{2}^{k-2-\ell}$

for the corresponding element in $\operatorname{Sym}^{k-2}(\mathbb{C}^{2})$.
Then the action of $\operatorname{SL}_{2}(\mathbb{Z})$ is given by

$M\circ(\boldsymbol{z})^{\mathrm{t}}_{(e_{1},e_{2})}:=(\boldsymbol{z})^{\mathrm{t}}_{M\circ(e_{1},e_{2})}.$

This gives a change of basis from the standard basis
$\\{c_{\ell}e_{1}^{\ell}e_{2}^{k-2-\ell}\\}_{0\leq\ell\leq k-2}$ of
$\operatorname{Sym}^{k-2}(\mathbb{C}^{2})$ to the basis
$\\{c_{\ell}X_{1}^{\ell}X_{2}^{k-2-\ell}\\}_{0\leq\ell\leq k-2}$, where
$(X_{1},X_{2})=M\circ(e_{1},e_{2})$. We describe this change of basis
explicitly.

######  Lemma 2.6.

Let $M=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\operatorname{SL}_{2}(\mathbb{Z})$. Then

$(\boldsymbol{z})^{\mathrm{t}}_{M\circ(e_{1},e_{2})}=N(M)(\boldsymbol{z})^{\mathrm{t}}_{(e_{1},e_{2})}$

for a matrix $N(M)\in\operatorname{GL}_{k-1}(\mathbb{Z})$, whose entries
$(N_{\ell t})_{0\leq\ell,t\leq k-2}$ are given by

$N_{\ell t}=\sum\limits_{\begin{subarray}{c}0\leq i\leq t\\\ 0\leq\ell-i\leq
k-2-t\end{subarray}}{\ell\choose i}{k-2-\ell\choose
t-i}a^{i}c^{t-i}b^{\ell-i}d^{k-2-t-\ell+i}.$

###### Proof.

We note that

$(\boldsymbol{z})^{\mathrm{t}}_{M\circ(e_{1},e_{2})}=\sum\limits_{t=0}^{k-2}z_{t}c_{t}(ae_{1}+ce_{2})^{t}(be_{1}+de_{2})^{k-2-t}.$

By using the binomial theorem twice and rearranging the sums we obtain the
stated formula.

The claim that $N(M)\in\operatorname{GL}_{k-1}(\mathbb{Z})$ follows easily
from the properties mentioned in Remark 2.7 below: for each
$M\in\operatorname{SL}_{2}(\mathbb{Z})$, $N(M)$ has all integer entries and it
is a group homomorphism, which implies that $N(M)^{-1}=N(M^{-1})$ has integer
entries as well, proving the claim. ∎

We note that this formula is already contained in classical work by Kuga-
Shimura from the late 1950’s [KS60].

######  Remark 2.7.

For $M,M_{1},M_{2}\in\operatorname{SL}_{2}(\mathbb{Z})$ and the notation as in
Lemma 2.6, we note the following properties

$\displaystyle N(M_{1}M_{2})$ $\displaystyle=N(M_{2})N(M_{1}),$ $\displaystyle
N(M)^{\mathrm{t}}$ $\displaystyle=N(M^{\mathrm{t}}),$ $\displaystyle
N(M)^{-1}$ $\displaystyle=N(M^{-1}).$

Since $N(M)\in\operatorname{GL}_{k-1}(\mathbb{Z})$ by Lemma 2.6 we immediately
obtain the following observation.

######  Lemma 2.8.

Let $M\in\operatorname{SL}_{2}(\mathbb{Z})$ and the notation be as in Lemma
2.6. Then we have

$N(M)\Lambda_{f}^{k-1}=\Lambda_{f}^{k-1}.$

Here, we view an element of $\Lambda_{f}^{k-1}$ as a column vector.

##  3\. Weierstrass mock modular forms

We recall the classical construction of Weierstrass mock modular forms of
Guerzhoy [Gue15] and Alfes–Griffin–Ono–Rolen [AGOR15].

###  3.1. The completed Weierstrass $\zeta$-function

In this section we review Rolen’s [Rol16] construction of Eisenstein’s
completion of the Weierstrass $\zeta$-function.

We let $\Lambda_{\tau}=\mathbb{Z}+\mathbb{Z}\tau$ for $\tau\in\mathbb{H}$ and
define for $z\in\mathbb{C}$ the classical Weierstrass $\sigma$-function

$\sigma_{\Lambda_{\tau}}(z)=z\,\prod_{w\in\Lambda_{\tau}\setminus\\{0\\}}\left(1-\frac{z}{w}\right)\exp\left(\frac{z}{w}+\frac{z^{2}}{2w^{2}}\right).$

Its logarithmic derivative is the Weierstrass $\zeta$-function

$\zeta_{\Lambda_{\tau}}(z)=\frac{\frac{\partial}{\partial
z}\sigma_{\Lambda_{\tau}}(z)}{\sigma_{\Lambda_{\tau}}(z)}.$

Moreover, we need the following classical identity (cf. Theorem 3.9 of
[Pol03]) relating the Weierstrass $\sigma$-function to the standard Jacobi
theta function

(3.1)
$\vartheta(\tau,z)=-2\pi\eta(\tau)^{3}\exp\left(-\frac{\eta_{1}z^{2}}{2}\right)\sigma_{\Lambda_{\tau}}(z).$

Here, $\vartheta(\tau,z)$ is a Jacobi form of weight $1/2$ and index $1/2$
given by

$\vartheta(\tau,z)=\sum_{n\in\frac{1}{2}+\mathbb{Z}}e^{\pi in^{2}\tau+2\pi
in\left(z+\frac{1}{2}\right)},$

$\eta_{1}$ is the quasi-period defined by

$\eta_{1}=\eta_{1}(\tau):=\zeta_{\Lambda_{\tau}}(z+1)-\zeta_{\Lambda_{\tau}}(z),$

and $\eta(\tau)=q^{1/24}\prod_{n=1}^{\infty}(1-q^{n})$ is the Dedekind
$\eta$-function.

Then we find that

$\frac{\frac{\partial}{\partial
z}\vartheta(\tau,z)}{\vartheta(\tau,z)}=\zeta_{\Lambda_{\tau}}(z)-\eta_{1}z.$

As $\vartheta(\tau,z)$ is a holomorphic function in $z$, one may consider its
derivative. However, this is no longer a Jacobi form. We therefore replace the
derivative $\frac{\partial}{\partial z}$ by the canonical raising operator
$Y_{+}^{\frac{1}{2},\frac{1}{2}}$ on Jacobi forms and apply it to the Jacobi
theta function

$Y_{+}^{\frac{1}{2},\frac{1}{2}}(\vartheta(\tau,z))=\frac{\partial}{\partial
z}\vartheta(\tau,z)+2\pi
i\frac{\operatorname{Im}(z)}{\operatorname{Im}(\tau)}\vartheta(\tau,z)$

which then is a Jacobi form of weight $3/2$ and index $1/2$ (compare Section
2.2). Considering the corresponding analogue of the logarithmic derivative we
obtain

(3.2)
$\frac{Y_{+}^{\frac{1}{2},\frac{1}{2}}(\vartheta(\tau,z))}{\vartheta(\tau,z)}=\zeta_{\Lambda_{\tau}}(z)-\eta_{1}z+\frac{2\pi
i\operatorname{Im}(z)}{\operatorname{Im}(\tau)},$

which is a real-analytic Jacobi form of weight $1$ and index $0$. In
particular, it is an elliptic function in $z$. Using the relation
$G_{2}=\eta_{1}$ for the weight $2$ Eisenstein series, normalized to have
constant term $2\zeta(2)$, one can show that the function

$\zeta^{*}_{\Lambda_{\tau}}(z)=\zeta_{\Lambda_{\tau}}(z)-zG^{*}_{2}(\tau)-\pi\frac{\overline{z}}{\operatorname{Im}(\tau)}$

is a doubly-periodic function with respect to the lattice $\Lambda_{\tau}$.
Here, $G_{2}^{*}(\tau)=G_{2}(\tau)-\pi/\operatorname{Im}(\tau)$ is the non-
holomorphic completion of the weight $2$ Eisenstein series.

We note that

$\operatorname{Im}(\tau)=\mathrm{Vol}(\Lambda_{\tau}).$

###  3.2. Weierstrass mock modular forms

The completed Weierstrass $\zeta$-function can be used to produce harmonic
weak Maass forms of weight $0$. We let $f$ be a newform of weight $2$ for
$\Gamma_{0}(N)$ with rational Fourier coefficients. In the notation of Section
2.4, $\Lambda_{f}$ is the associated lattice and
$\mathcal{E}_{f}(\tau)=\sum_{n=1}^{\infty}\frac{a_{f}(n)}{n}q^{n}$ is the
Eichler integral of $f(\tau)=\sum_{n=1}^{\infty}a_{f}(n)q^{n}$. In particular,
we have

$\mathcal{E}_{f}(\gamma\tau)=\mathcal{E}_{f}(\tau)+\omega,\,\omega\in\Lambda_{f},$

for $\gamma\in\Gamma_{0}(N)$. We then define the Weierstrass form by

$Z_{f}(\tau)=\zeta^{*}_{\Lambda_{f}}(\mathcal{E}_{f}(\tau)).$

The following theorem was proven in [Gue15] and [AGOR15].

######  Theorem 3.1.

Assume the notation and hypotheses above. The following are true:

  1. (1)

The poles of the holomorphic part $Z_{f}^{+}(\tau)$ of $Z_{f}(\tau)$ are
precisely those points for which $\mathcal{E}_{f}(\tau)\in\Lambda_{f}$.

  2. (2)

If $Z_{f}^{+}(\tau)$ has poles in $\mathbb{H}$, then there is a canonical
modular function $M_{f}(\tau)$ with algebraic coefficients on $\Gamma_{0}(N)$
for which $Z_{f}^{+}(\tau)-M_{f}(\tau)$ is holomorphic on $\mathbb{H}$.

  3. (3)

The function $Z_{f}(\tau)-M_{f}(\tau)$ is a harmonic weak Maass form of weight
$0$ on $\Gamma_{0}(N)$.

  4. (4)

We have that

$\xi_{0}(Z_{f}(\tau))=-\frac{2\pi i}{\mathrm{vol}(\Lambda_{f})}f(\tau).$

##  4\. Vector-valued Jacobi–Weierstrass forms

We first construct the Jacobi–Weierstrass $\zeta$-function in Section 4.1,
which is a higher degree analogue of the Weierstrass $\zeta$-function. In
Section 4.2 we construct its completion based on Rolen’s approach and in
Section 4.3 we define a vector-valued analogue.

###  4.1. Elliptic functions of higher degree

Generalizing (3.1) we define a higher degree analogue of the Weierstrass
$\sigma$-function. We let $\Lambda_{\tau}=\mathbb{Z}+\mathbb{Z}\tau$ for
$\tau\in\mathbb{H}$.

######  Definition 4.1.

Let $\boldsymbol{z}\in\mathbb{C}^{g}$. We define the Jacobi–Weierstrass
$\sigma$-function by

(4.1)
$\sigma_{\Lambda_{\tau}}(\boldsymbol{z})=e^{-u(\tau)Q(\boldsymbol{z})}\theta(\tau,\boldsymbol{z}),$

where $u(\tau)$ is a function in $\tau$ defined in (4.7).

Note that for $g=1$ the above definition differs from the classical definition
of the Weierstrass $\sigma$-function by a (non-zero) constant factor depending
on the lattice. This is however of no further importance here.

Let $\boldsymbol{m},\boldsymbol{n}\in\mathbb{Z}^{g}$. Using the properties of
the Jacobi theta function from (2.1), we directly see that

(4.2)
$\sigma_{\Lambda_{\tau}}(\boldsymbol{z}+\boldsymbol{m}\tau+\boldsymbol{n})=e^{-u(\tau)(Q(\boldsymbol{z+\boldsymbol{m}\tau+\boldsymbol{n}})-Q(\boldsymbol{z}))}e^{-\pi
i\tau(\boldsymbol{m},\boldsymbol{m})}e^{-2\pi
i(\boldsymbol{z},\boldsymbol{m})}\sigma_{\Lambda_{\tau}}(\boldsymbol{z})$

for $\boldsymbol{m},\boldsymbol{n}\in\mathbb{Z}^{g}$. We now define higher
degree analogues of the $\wp$\- and $\zeta$-function.

######  Definition 4.2.

Let $\boldsymbol{z}\in\mathbb{C}^{g}$. We define the Jacobi–Weierstrass
$\zeta$-function by

(4.3) $\zeta_{\Lambda_{\tau},j}(\boldsymbol{z})=\frac{\partial}{\partial
z_{j}}\log\sigma_{\Lambda_{\tau},}(\boldsymbol{z}),\,1\leq j\leq g,$

and the Jacobi–Weierstrass $\wp$-function by

(4.4)
$\wp_{ji}(\boldsymbol{z})=\wp_{\Lambda_{\tau},ji}(\boldsymbol{z})=\frac{\partial}{\partial
z_{i}}\zeta_{\Lambda_{\tau},j}(\boldsymbol{z}),\,1\leq i,j\leq g.$

We first show that $\wp_{ji}$ is invariant under the lattice
$\Lambda_{\tau}^{g}$.

######  Lemma 4.3.

For $1\leq i,j\leq g$, the Jacobi–Weierstrass $\wp$-function $\wp_{ji}$ is
$\Lambda_{\tau}^{g}$-invariant, i.e. it holds that

(4.5)
$\wp_{ji}(\tau,\boldsymbol{z}+\boldsymbol{m}\tau+\boldsymbol{n})=\wp_{ji}(\tau,\boldsymbol{z}),$

for $\boldsymbol{m},\boldsymbol{n}\in\mathbb{Z}^{g}$.

###### Proof.

Let
$\boldsymbol{m}=(m_{1},\ldots,m_{g}),\,\boldsymbol{n}=(n_{1},\ldots,n_{g})\in\mathbb{Z}^{g}$.
Using the transformation properties of the higher degree $\sigma$-function
(compare (4.2)) we see that

$\frac{\left(\frac{\partial}{\partial
z_{j}}\sigma_{\Lambda_{\tau}}\right)(\boldsymbol{z}+\boldsymbol{m}\tau+\boldsymbol{n})}{\sigma_{\Lambda_{\tau}}(\boldsymbol{z}+\boldsymbol{m}\tau+\boldsymbol{n})}=\frac{\left(\frac{\partial}{\partial
z_{j}}\sigma_{\Lambda_{\tau}}\right)(\boldsymbol{z})}{\sigma_{\Lambda_{\tau}}(\boldsymbol{z})}+(-u(\tau)(G_{L}(\boldsymbol{m}\tau+\boldsymbol{n}))_{j}-2\pi
i(G_{L}\boldsymbol{m})_{j}),$

where, as earlier (see Section 2.2), $G_{L}$ denotes the Gram matrix of the
lattice $L$ used to define the theta series. This implies that

(4.6)
$\zeta_{\Lambda_{\tau},j}(\boldsymbol{z}+\boldsymbol{m}\tau+\boldsymbol{n})-\zeta_{\Lambda_{\tau},j}(\boldsymbol{z})=-u(\tau)(G_{L}(\boldsymbol{m}\tau+\boldsymbol{n}))_{j}-2\pi
i(G_{L}\boldsymbol{m})_{j}.$

The right hand side of the equation above is independent of $\boldsymbol{z}$
and depends only on the lattice, thus

$\frac{\partial}{\partial
z_{i}}\zeta_{\Lambda_{\tau},j}(\boldsymbol{z}+\boldsymbol{m}\tau+\boldsymbol{n})-\frac{\partial}{\partial
z_{i}}\zeta_{\Lambda_{\tau},j}(\boldsymbol{z})=0.$

This implies the desired invariance of $\wp_{ji}$. ∎

The independence of $\boldsymbol{z}$ of the difference in (4.6) justifies the
following definition.

######  Definition 4.4.

We define the quasi-periods to be

$\eta_{jl,1}(\tau)=\zeta_{\Lambda_{\tau},j}(\boldsymbol{z}+e_{l})-\zeta_{\Lambda_{\tau},j}(\boldsymbol{z})=0,\,\text{if
}j\neq l,$ (4.7)
$\eta_{jj,1}(\tau)=\zeta_{\Lambda_{\tau},j}(\boldsymbol{z}+e_{j})-\zeta_{\Lambda_{\tau},j}(\boldsymbol{z})=:u(\tau).$

Here, $e_{i}\in\mathbb{Z}^{g}$ denotes the $i$-th unit vector.

######  Remark 4.5.

This generalizes the definition of the quasi-periods in the classical setting,
where we have

$\eta_{1}(\tau)=\zeta_{\Lambda_{\tau}}(z+1)-\zeta_{\Lambda_{\tau}}(z).$

###  4.2. The completion of the Jacobi–Weierstrass $\zeta$-function

In this section we complete the Jacobi–Weierstrass $\zeta$-function such that
it is lattice invariant. This generalizes the construction in (3.2). Let
$\Lambda_{\tau}=\mathbb{Z}+\mathbb{Z}\tau$ for $\tau\in\mathbb{H}$. We define

(4.8)
$\zeta_{z_{j}}^{*}(\boldsymbol{z})=\zeta_{\Lambda_{\tau},j}^{*}(\boldsymbol{z})=\frac{(Y_{+,z_{j}}^{g/2,1/2L}\theta)(\tau,\boldsymbol{z})}{\theta(\tau,\boldsymbol{z})},\,\boldsymbol{z}\in\mathbb{C}^{g}.$

######  Proposition 4.6.

The function $\zeta^{*}_{z_{j}}(\boldsymbol{z})$ is a (non-holomorphic) Jacobi
form of weight $1$ and index $0$ for the Jacobi subgroup $\mathcal{J}_{L}$
(see Section 2.2). In particular, $\zeta^{*}_{z_{j}}(\boldsymbol{z})$ is
invariant under the lattice $\Lambda_{\tau}^{g}$.

###### Proof.

We first note that to show that $\zeta^{*}_{z_{j}}(\boldsymbol{z})$ is a
Jacobi form of weight $1$, it is enough to show the identity (2.6). To check
that $\zeta^{*}_{z_{j}}(\boldsymbol{z})$ is elliptic, we first compute the
partial derivative $\frac{\partial}{\partial z_{j}}$ of (2.1), namely

$\displaystyle\left(\frac{\partial}{\partial
z_{j}}\theta\right)(\tau,\boldsymbol{z}+\boldsymbol{m}\tau+\boldsymbol{n})$
$\displaystyle=e^{-\pi i\tau(\boldsymbol{m},\boldsymbol{m})}e^{-2\pi
i(\boldsymbol{z},\boldsymbol{m})}\left(\frac{\partial}{\partial
z_{j}}\theta\right)(\tau,\boldsymbol{z})$ $\displaystyle\quad-2\pi
i(G_{L}\boldsymbol{m})_{j}e^{-\pi
i\tau(\boldsymbol{m},\boldsymbol{m})}e^{-2\pi
i(\boldsymbol{z},\boldsymbol{m})}\theta(\tau,\boldsymbol{z}).$

Adding $2\pi
i\frac{(\operatorname{Im}(G_{L}(\boldsymbol{z}+\boldsymbol{m}\tau+\boldsymbol{n})))_{j}}{\operatorname{Im}(\tau)}e^{-\pi
i\tau(\boldsymbol{m},\boldsymbol{m})}e^{-2\pi
i(\boldsymbol{z},\boldsymbol{m})}\theta(\tau,\boldsymbol{z})$ and using that
$\frac{\operatorname{Im}(G_{L}(\boldsymbol{m}\tau+\boldsymbol{n}))}{\operatorname{Im}(\tau)}=G_{L}\boldsymbol{m},$
we get

(4.9)
$(Y_{+,z_{j}}^{g/2,1/2L}\theta)(\tau,\boldsymbol{z}+\boldsymbol{m}\tau+\boldsymbol{n})=e^{-\pi
i\tau(\boldsymbol{m},\boldsymbol{m})}e^{-2\pi
i(\boldsymbol{z},\boldsymbol{m})}(Y_{+,z_{j}}^{g/2,1/2L}\theta)(\tau,\boldsymbol{z}).$

Dividing by
$\theta(\tau,\boldsymbol{z}+\boldsymbol{m}\tau+\boldsymbol{n})=e^{-\pi
i\tau(\boldsymbol{m},\boldsymbol{m})}e^{-2\pi
i(\boldsymbol{z},\boldsymbol{m})}\theta(\tau,\boldsymbol{z})$, we obtain

$\frac{(Y_{+,z_{j}}^{g/2,1/2L}\theta)(\tau,\boldsymbol{z}+\boldsymbol{m}\tau+\boldsymbol{n})}{\theta(\tau,\boldsymbol{z}+\boldsymbol{m}\tau+\boldsymbol{n})}=\frac{(Y_{+,z_{j}}^{g/2,1/2L}\theta)(\tau,\boldsymbol{z})}{\theta(\tau,\boldsymbol{z})}$

which finishes the proof that $\zeta^{*}_{z_{j}}(\boldsymbol{z})$ is elliptic.

Now we compute the action under the matrix $B=[\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right),\sqrt{c\tau+d}]\in\mathcal{J}_{L}$. It is enough
to show identity (2.6),which is equivalent to showing

$(c\tau+d)^{-g/2-1}e^{-\pi
ic\frac{(\boldsymbol{z},\boldsymbol{z})}{c\tau+d}}(Y_{+,z_{j}}^{g/2,1/2L}\theta)\left(\frac{a\tau+b}{c\tau+d},\frac{\boldsymbol{z}}{c\tau+d}\right)=Y_{+,z_{j}}^{g/2,1/2L}\theta,$

We compute the two components of the action of
$Y_{+,z_{j}}^{g/2,1/2L}=\frac{\partial}{\partial z_{j}}+2\pi
i\frac{\operatorname{Im}(G_{L}\boldsymbol{z})_{j}}{\operatorname{Im}\tau}$.
The transformation property (2.4) implies that

$\theta\left(\frac{a\tau+b}{c\tau+d},\frac{\boldsymbol{z}}{c\tau+d}\right)=(c\tau+d)^{g/2}e^{\pi
ic\frac{(\boldsymbol{z},\boldsymbol{z})}{c\tau+d}}\theta(\tau,\boldsymbol{z}).$

We differentiate and obtain

$\displaystyle\left(\frac{\partial}{\partial
z_{j}}\theta\right)\left(\frac{a\tau+b}{c\tau+d},\frac{\boldsymbol{z}}{c\tau+d}\right)$
$\displaystyle=(c\tau+d)^{g/2}2\pi ic(G_{L}\boldsymbol{z})_{j}e^{\pi
ic\frac{(\boldsymbol{z},\boldsymbol{z})}{c\tau+d}}\theta(\tau,\boldsymbol{z})$
$\displaystyle\quad\quad+(c\tau+d)^{g/2+1}e^{\pi
ic\frac{(\boldsymbol{z},\boldsymbol{z})}{c\tau+d}}\left(\frac{\partial}{\partial
z_{j}}\theta\right)(\tau,\boldsymbol{z}).$

Multiplying by $(c\tau+d)^{-g/2-1}e^{-\pi
ic\frac{(\boldsymbol{z},\boldsymbol{z})}{c\tau+d}}$ yields

(4.10) $2\pi
ic(G_{L}\boldsymbol{z})_{j}(c\tau+d)^{-1}\theta(\tau,\boldsymbol{z})+\left(\frac{\partial}{\partial
z_{j}}\theta\right)(\tau,\boldsymbol{z}).$

For the second part, we have to multiply the term

$(c\tau+d)^{-g/2-1}e^{-\pi
ic\frac{(\boldsymbol{z},\boldsymbol{z})}{c\tau+d}}\theta\left(\frac{a\tau+b}{c\tau+d},\frac{\boldsymbol{z}}{c\tau+d}\right)=(c\tau+d)^{-1}\theta(\tau,\boldsymbol{z})$

by $2\pi
i(\operatorname{Im}\frac{(G_{L}\boldsymbol{z})_{j}}{c\tau+d})/(\operatorname{Im}\frac{a\tau+b}{c\tau+d})=(c\tau+d)\frac{\operatorname{Im}(G_{L}\boldsymbol{z})_{j}}{\operatorname{Im}\tau}-c(G_{L}\boldsymbol{z})_{j}$,
resulting in

(4.11) $2\pi
i\frac{\operatorname{Im}(G_{L}\boldsymbol{z})_{j}}{\operatorname{Im}\tau}\theta(\tau,\boldsymbol{z})-2\pi
ic(G_{L}\boldsymbol{z})_{j}(c\tau+d)^{-1}\theta(\tau,\boldsymbol{z}).$

Adding (4.10) and (4.11) we get $Y_{+,z_{j}}^{g/2,1/2L}\theta$, as desired.
This finishes the proof. ∎

We obtain a similar description of the completed Jacobi–Weierstrass
$\zeta$-function as in the degree $1$ case.

######  Lemma 4.7.

The completed Jacobi–Weierstrass $\zeta$-function can be written as

$\zeta_{z_{j}}^{*}(\tau,\boldsymbol{z})=\zeta_{\Lambda_{\tau},j}(\boldsymbol{z})-u(\tau)z_{j}+2\pi
i\frac{\operatorname{Im}((G_{L}\boldsymbol{z})_{j})}{\mathrm{Vol}(\Lambda_{\tau})}.$

###### Proof.

Using the definition of the weight raising operator $Y_{+,z_{j}}^{g/2,1/2L}$
we find

$\frac{(Y_{+,z_{j}}^{g/2,1/2L}\theta)(\tau,\boldsymbol{z})}{\theta(\tau,\boldsymbol{z})}=\frac{(\frac{\partial}{\partial
z_{j}}\theta)(\tau,\boldsymbol{z})}{\theta(\tau,\boldsymbol{z})}+2\pi
i\frac{\operatorname{Im}((G_{L}\boldsymbol{z})_{j})}{\operatorname{Im}(\tau_{j})}.$

In order to compute the logarithmic derivative of the Jacobi theta-function,
we note that

$\frac{\partial}{\partial
z_{j}}\sigma_{\Lambda_{\tau}}(\boldsymbol{z})=c(\tau)e^{-u(\tau)Q(\boldsymbol{z})}\left(\frac{\partial}{\partial
z_{j}}\theta\right)(\tau,\boldsymbol{z})-z_{j}u(\tau)c(\tau)e^{-u(\tau)Q(\boldsymbol{z})}\theta(\tau,\boldsymbol{z}),$

which implies

$\frac{\frac{\partial}{\partial
z_{j}}\sigma_{\Lambda_{\tau}}(\boldsymbol{z})}{\sigma_{\Lambda_{\tau}}(\boldsymbol{z})}=e^{-u(\tau)Q(\boldsymbol{z})}\frac{(\frac{\partial}{\partial
z_{j}}\theta)(\tau,\boldsymbol{z})}{\sigma(\tau,\boldsymbol{z})}-z_{j}u(\tau)e^{-u(\tau)Q(\boldsymbol{z})}\frac{\theta(\tau,\boldsymbol{z})}{\sigma(\tau,\boldsymbol{z})}.$

Thus, using the definition (4.1) of the Jacobi-Weierstrass $\sigma$-function,
we find

(4.12) $\frac{\frac{\partial}{\partial
z_{j}}\theta(\tau,\boldsymbol{z})}{\theta(\tau,\boldsymbol{z})}=\zeta_{\Lambda_{\tau},j}(\boldsymbol{z})+z_{j}u(\tau).$

∎

###  4.3. Vector-valued Jacobi–Weierstrass forms

We now fix $g=k-2$ and introduce vector-valued Jacobi–Weierstrass forms. Let
$z\in\mathbb{C}^{k-1}$ and $M\in\operatorname{SL}_{2}(\mathbb{Z})$. Let
$(e_{1},e_{2})$ denote the standard basis of $\mathbb{C}^{2}$. We recall the
notation from Section 2.5 and write $(\boldsymbol{z})_{(e_{1},e_{2})}$ for the
element in $\operatorname{Sym}^{k-2}(\mathbb{C}^{2})$ corresponding to
$\boldsymbol{z}$.

######  Definition 4.8.

Let the notation be as above. We define the vector-valued Jacobi–Weierstrass
form by

$(\widehat{\zeta}_{\boldsymbol{z}}(\boldsymbol{z}))_{(X_{1},X_{2})}=\sum_{j=0}^{k-2}\zeta^{*}_{z_{j}}(\boldsymbol{z})c_{j}X_{1}^{j}X_{2}^{k-2-j}$

with $c_{j}=\binom{k-2}{j}$ as before, and
$(X_{1},X_{2})=M\circ(e_{1},e_{2})$.

Under the isomorphism
$\mathbb{C}^{k-1}\simeq\operatorname{Sym}^{k-1}(\mathbb{C}^{2})$ coming from
the standard basis, we can write $\widehat{\zeta}_{\boldsymbol{z}}$ as
$\widehat{\zeta}_{\boldsymbol{z}}((\boldsymbol{z})_{(e_{1},e_{2})})_{(X_{1},X_{2})}$.
It defines a function
$\widehat{\zeta}:\,\operatorname{Sym}^{k-2}(\mathbb{C}^{2})\times\operatorname{SL}_{2}(\mathbb{Z})\to\operatorname{Sym}^{k-2}(\mathbb{C}^{2})$,
where
$\widehat{\zeta}((\boldsymbol{z})_{(e_{1},e_{2})},M)=\widehat{\zeta}_{\boldsymbol{z}}((\boldsymbol{z})_{(e_{1},e_{2})})_{M\circ(e_{1},e_{2})}$.
In the definition we suppressed the dependence on the lattice
$\Lambda_{\tau}=\mathbb{Z}+\mathbb{Z}\tau$, $\tau\in\mathbb{H}$. Fixing a
basis $(X_{1},X_{2})$, we have the following result that is an immediate
consequence of Proposition 4.6

######  Theorem 4.9.

The function $(\widehat{\zeta}_{\boldsymbol{z}})_{(X_{1},X_{2})}$ is a (non-
holomorphic) Jacobi form of weight $1$ and index $0$, invariant under the
lattice $\Lambda_{\tau}^{k-1}$.

######  Remark 4.10.

Note that there are two sets of coordinates involved in the definition of the
Jacobi–Weierstrass form. First we have the element in
$\operatorname{Sym}^{k-2}(\mathbb{C}^{2})$ corresponding to
$\boldsymbol{z}\in\mathbb{C}^{k-2}$ in terms of the standard basis. Second,
the Jacobi–Weierstrass function itself is an element of
$\operatorname{Sym}^{k-2}(\mathbb{C}^{2})$ in terms of the basis
$(X_{1},X_{2})$.

We now consider directional derivatives of the Jacobi–Weierstrass form. Recall
that

$(\boldsymbol{z})^{\mathrm{t}}_{(e_{1},e_{2})}=\sum_{\ell=0}^{k-2}z_{\ell}c_{\ell}e_{1}^{\ell}e_{2}^{k-2-\ell}.$

We then define the directional derivative in the direction
$N(M)(\boldsymbol{z})^{\mathrm{t}}_{(e_{1},e_{2})}$ by

$\zeta_{(N(M)\boldsymbol{z})_{i}}(\boldsymbol{z})=\frac{\frac{\partial}{\partial(N(M)\boldsymbol{z})_{i}}\theta(\tau,\boldsymbol{z})}{\theta(\tau,\boldsymbol{z})}.$

Here, $N(M)$ is the matrix defined in Lemma 2.6 and we write
$(N(M)\boldsymbol{z})_{i}$ for the $i^{th}$ term in the vector
$N(M)\boldsymbol{z}$. We consider the global Jacobi–Weierstrass
$\zeta$-function

(4.13)
$\left(\zeta_{M^{-1}}(\boldsymbol{z})\right)_{(e_{1},e_{2})}=\sum_{\ell=0}^{k-2}\zeta_{(N(M)\boldsymbol{z})_{\ell}}(\boldsymbol{z})c_{\ell}e_{1}^{\ell}e_{2}^{k-2-\ell},$

as well as the global completed Jacobi–Weierstrass $\zeta$-function

(4.14)
$\left(\widehat{\zeta}_{M^{-1}}(\boldsymbol{z})\right)_{(e_{1},e_{2})}=\sum_{\ell=0}^{k-2}\zeta^{*}_{(N(M)\boldsymbol{z})_{\ell}}(\boldsymbol{z})c_{\ell}e_{1}^{\ell}e_{2}^{k-2-\ell}.$

We occasionally use the notation $\zeta_{N(M)\boldsymbol{z}}$ and
$\widehat{\zeta}_{N(M)\boldsymbol{z}}$ when we want to emphasize the indices.
To shorten notation we will also write
$\widehat{\zeta}_{M^{-1}}(\boldsymbol{z})$ for
$\left(\widehat{\zeta}_{M^{-1}}(\boldsymbol{z})\right)_{(e_{1},e_{2})}$ in
some places.

We note that in this notation
$\widehat{\zeta}_{\boldsymbol{z}}(\boldsymbol{z})=\widehat{\zeta}_{Id}(\boldsymbol{z})$
and we show that the global directional derivative corresponds to a change of
coordinates of $\widehat{\zeta}_{\boldsymbol{z}}$.

######  Lemma 4.11.

For $M\in\operatorname{SL}_{2}(\mathbb{Z})$,
$\boldsymbol{z}\in\mathbb{C}^{k-1}$ and $(X_{1},X_{2})$ a basis of
$\mathbb{C}^{2}$, we have

$\left(\widehat{\zeta}_{M}(\boldsymbol{z})\right)_{(X_{1},X_{2})}=\left(\widehat{\zeta}_{\boldsymbol{z}}(\boldsymbol{z})\right)_{M\circ(X_{1},X_{2})}.$

###### Proof.

We first consider the non-completed version of Definition 4.8, i.e.

$\left(\zeta_{M}(\boldsymbol{z})\right)_{(X_{1},X_{2})}=\sum_{\ell=0}^{k-2}\zeta_{(N(M^{-1})\boldsymbol{z})_{\ell}}(\boldsymbol{z})c_{\ell}X_{1}^{\ell}X_{2}^{k-2-\ell}.$

By definition this equals (in vector notation)

$\frac{1}{\theta(\tau,\boldsymbol{z})}\left(\frac{\partial}{\partial(N(M^{-1})\boldsymbol{z})_{0}}\theta(\tau,\boldsymbol{z}),\dots,\frac{\partial}{\partial(N(M^{-1})\boldsymbol{z})_{k-2}}\theta(\tau,\boldsymbol{z})\right)^{\mathrm{t}}_{(X_{1},X_{2})}.$

A short calculation using multivariable calculus yields

$\displaystyle\left(\frac{\partial}{\partial(N(M^{-1})\boldsymbol{z})_{0}}\theta(\tau,\boldsymbol{z}),\dots,\frac{\partial}{\partial(N(M^{-1})\boldsymbol{z})_{k-2}}\theta(\tau,\boldsymbol{z})\right)^{\mathrm{t}}_{(X_{1},X_{2})}$
$\displaystyle\quad=N(M)\left(\frac{\partial}{\partial
z_{0}}\theta(\tau,\boldsymbol{z}),\dots,\frac{\partial}{\partial
z_{k-2}}\theta(\tau,\boldsymbol{z})\right)^{\mathrm{t}}_{(X_{1},X_{2})}.$

Therefore, we have

$\left(\zeta_{M}(\boldsymbol{z})\right)_{(X_{1},X_{2})}=N(M)\left(\zeta_{\boldsymbol{z}}(\boldsymbol{z})\right)_{(X_{1},X_{2})}.$

Plugging this into the completion of the Jacobi–Weierstrass function yields
the desired result. ∎

##  5\. Vector-valued Jacobi–Weierstrass as polar harmonic weak Maass forms

In this section we define the higher weight analogue of Weierstrass harmonic
weak Maass forms. Throughout, we let $f$ be a newform of weight $k$ for
$\Gamma$ with rational Fourier coefficients, with $\Gamma_{1}(N)\subset\Gamma$
for some integer $N$. We will use the Eichler integral
$\mathcal{E}_{f}(\tau,(X_{1},X_{2}))$ and the lattice $\Lambda_{f}$ associated
to $f$ from Section 2.4.

By $V$ we denote the infinite-dimensional vector space

$V=\mathrm{Functions}\\{\operatorname{SL}_{2}(\mathbb{Z})\rightarrow\operatorname{Sym}^{k-2}(\mathbb{C}^{2})\\}.$

The action of $\Gamma$ on $V$ is given by

(5.1) $\rho(\gamma)F(M)=\gamma\circ F(\gamma^{-1}M),$

where $\gamma\in\Gamma$. Here $\Gamma$ acts on
$\operatorname{SL}_{2}(\mathbb{Z})$ by matrix multiplication and on
$\operatorname{Sym}^{k-2}(\mathbb{C}^{2})$ by acting on the basis as described
in Section 2.5.

######  Definition 5.1.

Let $\tau\in\mathbb{H}$ and $M\in\operatorname{SL}_{2}(\mathbb{Z})$. We define
the vector-valued Jacobi–Weierstrass form by

(5.2)
$F(\tau):=\left[M\mapsto\left(\widehat{\zeta}_{M}(\mathcal{E}_{f}(\tau,M^{-1}\circ(e_{1},e_{2})))_{(e_{1},e_{2})}\right)\right].$

We write

$F(\tau,M)=\widehat{\zeta}_{M}(\Lambda_{f},\mathcal{E}_{f}(\tau,M^{-1}\circ(e_{1},e_{2})))_{(e_{1},e_{2})},$

for the image of $F$ in $\operatorname{Sym}^{k-2}(\mathbb{C}^{2})$ and drop
the dependence on $\Lambda_{f}$ in the definition of $\widehat{\zeta}$.

######  Remark 5.2.

The function $F$ depends on the chosen basis of
$\operatorname{Sym}^{k-2}(\mathbb{C}^{2})$ in two places. We note two
important changes of coordinates. By Lemma 4.11 we see

(5.3)
$F(\tau,M)=\left(\widehat{\zeta}_{\boldsymbol{z}}\left(\mathcal{E}_{f}\left(\tau,M^{-1}\circ\left(e_{1},e_{2}\right)\right)\right)\right)_{M\circ(e_{1},e_{2})}.$

This can also be taken as the definition of the function $F(\tau,M)$.

Moreover, we have a second change of coordinates inside the function given by
changing the basis for the Eichler integral. As by definition we have
$\mathcal{E}_{f}\left(\tau,M^{-1}\circ\left(e_{1},e_{2}\right)\right)=N(M^{-1})\mathcal{E}_{f}(\tau,(e_{1},e_{2}))$
it follows that

(5.4)
$F(\tau,M)=\left(\widehat{\zeta}_{M}\left(N(M^{-1})\mathcal{E}_{f}\left(\tau,\left(e_{1},e_{2}\right)\right)\right)\right)_{(e_{1},e_{2})}.$

We now prove that $F$ defines a vector-valued polar harmonic weak Maass form
of weight $0$ that maps to $f$ with values in a coefficient system under
$\xi_{0}$.

######  Theorem 5.3.

The function $F(\tau)$ is a vector-valued polar harmonic weak Maass form of
weight $0$ for $\Gamma$ with respect to the representation $\rho$ and its
image under $\xi_{0}$ is given by

$G(\tau)=\frac{2\pi i}{\mathrm{Vol}(\Lambda_{f})}f(\tau)\ G_{L}\cdot(\tau
e_{1}+e_{2})^{k-2}.$

Here $G_{L}$ is the Gram matrix corresponding to the lattice $L$ which is
implicitly used in the construction of the Jacobi–Weierstrass $\zeta$-function
$\widehat{\zeta}$. For $\boldsymbol{z}=(z_{0},\dots,z_{k-2})^{t}$ written in
the basis $(X_{1},X_{2})$ of $\mathbb{C}^{2}$, the action of
$G_{L}=(g_{ij})_{0\leq i,j\leq k-2}$ is given by

(5.5)
$G_{L}\cdot(\boldsymbol{z})_{(X_{1},X_{2})}=\sum_{\ell=0}^{k-2}\left(\sum_{i=0}^{k-2}g_{\ell
i}z_{i}\right)c_{\ell}X_{1}^{\ell}X_{2}^{k-2-\ell}=\sum_{i=0}^{k-2}z_{i}\left(\sum_{\ell=0}^{k-2}g_{\ell
i}c_{\ell}X_{1}^{\ell}X_{2}^{k-2-\ell}\right).$

We note that for the lattice $\mathbb{Z}^{k-1}$ with the standard inner
product, the theorem simplifies to

$G(\tau)=\frac{2\pi i}{\mathrm{Vol}(\Lambda_{f})}f(\tau)(\tau
e_{1}+e_{2})^{k-2}.$

######  Remark 5.4.

  1. (1)

The function $G:\mathbb{H}\rightarrow\operatorname{Sym}^{k-2}(\mathbb{C}^{2})$
is a holomorphic vector-valued modular form of weight $2$.

  2. (2)

For the special case of weight $k=2$ it is possible to construct a modular
function $M$ with algebraic Fourier coefficients which eliminates all poles of
$F$ on the upper half-plane (see [AGOR15] resp. Theorem 3.1). Then, $F+M$ is a
harmonic weak Maass forms whose shadow is the newform $f$. The proof given in
loc. cit. uses the proof for the well-known fact that $j(\tau)$ and $j(N\tau)$
generate the field of modular functions for the group $\Gamma_{0}(N)$ (see e
g. [Cox13, Theorem 11.9]). To the authors’ knowledge, there is, however, no
systematic theory of modular functions with respect to the (infinite-
dimensional) representation $\rho$ from (5.1) which would allow to prove an
analogous result in general.

We prove Theorem 5.3 in several steps. We first show the $\Gamma$-invariance
of each one of the terms
$\widehat{\zeta}_{M}\left(\mathcal{E}_{f}\left(\tau,M^{-1}\circ\left(e_{1},e_{2}\right)\right)\right)$.

######  Lemma 5.5.

Let the notation be as above. For $\gamma\in\Gamma$, we have

$\widehat{\zeta}_{M}\left(\mathcal{E}_{f}\left(\tau,(\gamma^{-1}M)^{-1}\circ\left(e_{1},e_{2}\right)\right)\right)=\widehat{\zeta}_{M}\left(\mathcal{E}_{f}\left(\gamma\tau,M^{-1}\circ\left(e_{1},e_{2}\right)\right)\right).$

###### Proof.

We write $(X_{1},X_{2})=M^{-1}\circ(e_{1},e_{2})$ and note that

$(\gamma^{-1}M)^{-1}\circ\left(e_{1},e_{2}\right)=\left(e_{1},e_{2}\right)M^{-1}\gamma=\left(X_{1},X_{2}\right)\gamma.$

By definition of the change of basis matrix $N(\cdot)$, for the Eichler
integral we have

$\mathcal{E}_{f}\left(\tau,\gamma\circ\left(X_{1},X_{2}\right)\right)=N(\gamma)\mathcal{E}_{f}(\tau,(X_{1},X_{2}))\quad\text{and}\quad\mathcal{E}_{f}(\tau,(X_{1},X_{2}))=N(M^{-1})\mathcal{E}_{f}(\tau,(e_{1},e_{2})).$

Moreover, from Lemma 2.4 we have

(5.6)
$\mathcal{E}_{f}(\gamma\tau,(X_{1},X_{2}))=\mathcal{E}_{f}(\tau,\gamma\circ(X_{1},X_{2}))-\omega,$

where $\omega\in\Lambda_{f}^{k-1}$ with $\Lambda_{f}^{k-1}$ written in the
basis $\gamma\circ(X_{1},X_{2})=(e_{1},e_{2})M^{-1}\gamma$, thus it is an
element of $N(M^{-1}\gamma)\Lambda_{f}^{k-1}$. By Lemma 2.8 we know that
$N(M^{-1}\gamma)\Lambda_{f}^{k-1}=\Lambda_{f}^{k-1}$. Therefore we obtain

$\mathcal{E}_{f}(\gamma\tau,M^{-1}\circ(e_{1},e_{2}))-\mathcal{E}_{f}(\tau,(\gamma^{-1}M)^{-1}\circ(e_{1},e_{2}))\in\Lambda_{f}^{k-1}.$

Since $\widehat{\zeta}_{M}$ is $\Lambda_{f}^{k-1}$-invariant from Proposition
4.6, this gives the result. ∎

###### Proof of Theorem 5.3.

We first prove that $F$ is invariant under the action of $\Gamma$. The
harmonicity (away from possible poles) will follow from the fact that
$\xi_{0}(F)$ is a holomorphic cusp form (with values in a coefficient system),
since the Laplacian equals $\Delta_{k}=-\xi_{2-k}\circ\xi_{k}$.

Let $\gamma\in\Gamma$. For $M\in\operatorname{SL}_{2}(\mathbb{Z})$ and
$\tau\in\mathbb{H}$, we need to show that

$F(\gamma\tau,M)=\rho(\gamma)F(\tau,M),$

where the action $\rho:\Gamma\rightarrow\operatorname{End}(V)$ was defined in
(5.1), thus

$\rho(\gamma)F(\tau,M)=\gamma\circ\left(\widehat{\zeta}_{(\gamma^{-1}M)}\left(\mathcal{E}_{f}\left(\tau,\left((\gamma^{-1}M)^{-1}\right)\circ\left(e_{1},e_{2}\right)\right)\right)\right)_{(e_{1},e_{2})}.$

By Lemma 4.11 this equals

$\gamma\circ\widehat{\zeta}_{\boldsymbol{z}}\left(\mathcal{E}_{f}(\tau,\left(\gamma^{-1}M\right)^{-1}\circ\left(e_{1},e_{2}\right))\right)_{(e_{1},e_{2})\gamma^{-1}M}$

and this expression simplifies to

$\widehat{\zeta}_{\boldsymbol{z}}\left(\mathcal{E}_{f}(\tau,\left(\gamma^{-1}M\right)^{-1}\circ\left(e_{1},e_{2}\right))\right)_{(e_{1},e_{2})\gamma\gamma^{-1}M}=\widehat{\zeta}_{M}\left(\mathcal{E}_{f}(\tau,\left(\gamma^{-1}M\right)^{-1}\circ\left(e_{1},e_{2}\right))\right)_{(e_{1},e_{2})}.$

Finally, we use Lemma 5.5 and obtain that

$\rho(\gamma)F(\tau,M)=\widehat{\zeta}_{M}\left(\mathcal{E}_{f}\left(\gamma\tau,M^{-1}\circ\left(e_{1},e_{2}\right)\right)\right)_{(e_{1},e_{2})}=F(\gamma\tau,M).$

Now we will compute the image of $F(\tau,M)$ under
$\xi_{0}=-2i\overline{\frac{\partial}{\partial\overline{\tau}}}$. This gives a
function

$g(\tau,M)=\xi_{0}F(\tau,M),$

with $g:\mathbb{H}\rightarrow V$. We will compute $g$ explicitly and show that
it is indeed constant on $\operatorname{SL}_{2}(\mathbb{Z})$. We write

(5.7)
$F(\tau,M)=\left(\widehat{\zeta}_{\boldsymbol{z}}\left(\mathcal{E}_{f}(\tau,M^{-1}\circ\left(e_{1},e_{2}\right)\right)\right)_{(Y_{1},Y_{2})},$

with $(Y_{1},Y_{2})=M\circ(e_{1},e_{2})$ and consider each of the coordinates
$\zeta_{z_{\ell}}^{*}$ in the basis $(Y_{1},Y_{2})$. By Lemma 4.7 we obtain

$\zeta_{z_{\ell}}^{*}(\boldsymbol{z})=\zeta_{z_{\ell}}(\boldsymbol{z})-u(\tau_{f})z_{\ell}+2\pi
i\frac{\operatorname{Im}((G_{L}\boldsymbol{z})_{\ell})}{\operatorname{Im}(\tau_{f})}.$

Writing $\operatorname{Im}(z)=(z-\overline{z})/2i$ and plugging in
$\boldsymbol{z}=\mathcal{E}_{f}(\tau,M^{-1}\circ(e_{1},e_{2})))$ we see that

$\overline{\frac{\partial}{\partial\overline{\tau}}\zeta_{z_{\ell}}^{*}\left(\mathcal{E}_{f}\left(\tau,M^{-1}\circ\left(e_{1},e_{2}\right)\right)\right)}=-\frac{\pi}{\operatorname{Im}(\tau_{f})}\overline{\left(G_{L}\frac{\partial}{\partial\overline{\tau}}\overline{\mathcal{E}_{f}\left(\tau,M^{-1}\circ\left(e_{1},e_{2}\right)\right)}\right)}_{\ell}.$

As $G_{L}$ accounts for changing the basis from (5.5)

$G_{L}\cdot(\boldsymbol{z})_{(Y_{1},Y_{2})}=\sum_{i=0}^{k-2}z_{i}\left(\sum_{\ell=0}^{k-2}g_{\ell
i}c_{\ell}Y_{1}^{\ell}Y_{2}^{k-2-\ell}\right),$

it is enough to compute each component
$\overline{\frac{\partial}{\partial\overline{\tau}}\left(\overline{\mathcal{E}_{f}\left(\tau,M^{-1}\circ\left(e_{1},e_{2}\right)\right)}\right)}_{\ell}$,
where

$\displaystyle\left(\mathcal{E}_{f}\left(\tau,M^{-1}\circ\left(e_{1},e_{2}\right)\right)\right)_{\ell}=\int_{\tau}^{\infty}f(t)(dt-b)^{\ell}(-ct+a)^{k-2-\ell}dt,$

with $M=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\operatorname{SL}_{2}(\mathbb{Z})$. This gives
us

$\overline{\frac{\partial}{\overline{\partial\tau}}\overline{\left(\mathcal{E}_{f}\left(\tau,M^{-1}\circ\left(e_{1},e_{2}\right)\right)\right)_{\ell}}}=f(\tau)(d\tau-b)^{\ell}(-c\tau+a)^{k-2-\ell}$

and with the action of $G_{L}$ we get

$\displaystyle\xi_{0}F(\tau,M)$ $\displaystyle=\frac{2\pi
i}{\operatorname{Im}(\tau_{f})}G_{L}\cdot\sum_{\ell=0}^{k-2}f(\tau)(d\tau-b)^{\ell}(-c\tau+a)^{k-2-\ell}c_{\ell}Y_{1}^{\ell}Y_{2}^{k-2-\ell}$
$\displaystyle=\frac{2\pi i}{\operatorname{Im}(\tau_{f})}f(\tau)\
G_{L}\cdot\left((dY_{1}-cY_{2})\tau+(-bY_{1}+aY_{2})^{k-2}\right)$
$\displaystyle=\frac{2\pi i}{\operatorname{Im}(\tau_{f})}f(\tau)\
G_{L}\cdot\left(M^{-1}\circ(Y_{1}\tau+Y_{2})^{k-2}\right)$
$\displaystyle=\frac{2\pi i}{\operatorname{Im}(\tau_{f})}f(\tau)\
\left(G_{L}\cdot(e_{1}\tau+e_{2})^{k-2}\right).$

The last expression is independent of $M$. The holomorphicity is obvious and
the modularity follows from (2.7). Moreover, we see that $F(\tau)$ is
harmonic.

It remains to show that $F$ grows at most linear exponentially when $\tau$
approaches a cusp of $\Gamma$. For this let
$\sigma\in\operatorname{SL}_{2}(\mathbb{Z})$ be arbitrary. The same
computation as in the proof of Lemma 2.2 shows that

$\mathcal{E}_{f}(\sigma\tau,M^{-1}\circ(e_{1},e_{2}))=\mathcal{E}_{f|\sigma}(\tau,M^{-1}\sigma\circ(e_{1},e_{2}))+C$

for a certain constant vector $C$. Since $f$ is a cusp form and $f|\sigma$
yields the Fourier expansion of $f$ at the cusp $\sigma\infty$ by definition,
it vanishes exponentially as $\tau\to\infty$. Since $\theta$ is holomorphic in
all elliptic variables, it follows that the partial logarithmic derivatives of
$\theta$ all have at most a finite order pole at the divisor of $\theta$. But
this implies that, as $\tau\to\infty$, the analytic part of $F(\sigma.\tau,M)$
grows at most linear exponentially if $C$ lies in the divisor of $\theta$ and
is bounded otherwise. This shows the claim, as the non-analytic part is
clearly bounded. ∎

##  6\. Laurent expansion and poles

In the definition of the Jacobi theta function we allowed an arbitrary
positive definite integral bilinear form on $\mathbb{Z}^{g}$. Especially when
computing examples (see Section 7), it is most convenient to take the standard
bilinear form, but allow for a non-trivial characteristic. In what follows we
define

(6.1)
$\displaystyle\theta(\tau,\boldsymbol{z})=\sum_{n\in\left(\frac{1}{2}+\mathbb{Z}\right)}e^{\pi
i\left(\sum_{i}n_{i}^{2}\right)\tau+2\pi
i\sum_{i}n_{i}z_{i}}=\prod_{i=1}^{g}\vartheta(\tau,z_{i}).$

In this setting we immediately find from (4.8) that

$\zeta_{z_{j}}^{*}(\tau,\boldsymbol{z})=\frac{Y_{+,z_{j}}^{g/2,1/2}\theta(\tau,\boldsymbol{z})}{\theta(\tau,\boldsymbol{z})}=\frac{Y_{+,z_{j}}^{1/2,1/2}\vartheta(\tau,z_{j})}{\vartheta(\tau,z_{j})}=\zeta_{\Lambda_{\tau}}^{*}(z_{j}),$

where $\zeta^{*}$ here denotes the completion of the usual $g=1$ Weierstrass
$\zeta$-function.

Similarly, the directional derivatives occurring in the definition of the
function $F$ in (5.2) can be expressed explicitly as

$\widehat{\zeta}_{(N(M^{-1})\boldsymbol{z})_{j}}(\tau,\boldsymbol{z})=\sum_{i}N(M)_{ij}\zeta_{\Lambda_{\tau}}^{*}(z_{i}),$

which follows directly from the definition of the directional derivative,
where $N(M)=\\{N(M)_{ij}\\}_{0\leq i,j\leq k-2}$.

We recall the Laurent expansion of the completed Weierstrass $\zeta$-function

$\zeta_{\Lambda_{\tau}}^{*}(z)=\frac{1}{z}-\sum_{n\geq
1}G_{2n+2}(\Lambda_{\tau})z^{2n+1}-S(\Lambda_{\tau})z-\frac{\pi}{\operatorname{Im}(\tau)}\overline{z},$

where $G_{2n}(\Lambda)=\sum_{\omega\in\Lambda\setminus\\{0\\}}\omega^{-2n}$ is
the classical Eisenstein series of weight $2n$ and
$S(\Lambda_{\tau})=\lim\limits_{s\rightarrow
0^{+}}\sum_{\omega\in\Lambda_{\tau}\setminus\\{0\\}}\frac{1}{\omega^{2}|\omega|^{2}}$.

For the Laurent expansion of the Jacobi–Weierstrass $\zeta$-function

$\widehat{\zeta}_{\boldsymbol{z}}(\tau,\boldsymbol{z})=\sum\limits_{\ell=0}^{k-2}\zeta^{*}_{\Lambda_{\tau}}(z_{\ell})e_{1}^{\ell}e_{2}^{k-2-\ell}c_{\ell}$

we then obtain the following result

######  Lemma 6.1.

With the standard choice of lattice, the Laurent expansion of the
Jacobi–Weierstrass $\zeta$-function $\zeta_{\boldsymbol{z}}$ is given by

$\zeta_{\boldsymbol{z}}(\tau,\boldsymbol{z})=\sum_{\ell=0}^{k-2}\frac{1}{z_{\ell}}e_{1}^{\ell}e_{2}^{k-2-\ell}c_{\ell}+\sum_{n\geq
1}G_{2n+2}(\Lambda_{\tau})\sum_{\ell=0}^{k-2}z_{\ell}^{2n+1}e_{1}^{\ell}e_{2}^{k-2-\ell}c_{\ell}.$

Moreover, the Laurent expansion of its completion is given by

$\widehat{\zeta}_{\boldsymbol{z}}(\tau,\boldsymbol{z})=\zeta_{\boldsymbol{z}}(\tau,\boldsymbol{z})-S(\Lambda_{\tau})\sum_{\ell=0}^{k-2}z_{\ell}e_{1}^{\ell}e_{2}^{k-2-\ell}c_{\ell}-\frac{\pi}{\operatorname{Im}(\tau)}\overline{z_{\ell}}e_{1}^{\ell}e_{2}^{k-2-\ell}c_{\ell}.$

We note that when plugging in $\mathcal{E}_{f}(\tau,(e_{1},e_{2}))$, we get
the poles of the vector-valued function
$F(\tau,Id):\mathbb{H}\rightarrow\operatorname{Sym}^{k-2}(\mathbb{C}^{2})$,
for $\tau\in\mathbb{H}$ such that

$\mathcal{E}_{\ell}(\tau,(e_{1},e_{2}))\in\Lambda_{f}.$

The theory easily extends to general $M\in\operatorname{SL}_{2}(\mathbb{Z})$
for

$\zeta_{M^{-1}}(\Lambda_{\tau},N(M^{-1})\boldsymbol{z})=\sum\limits_{\ell=0}^{k-2}\zeta_{\Lambda_{\tau}}((N(M^{-1})\boldsymbol{z})_{\ell})X_{1}^{\ell}X_{2}^{k-2-\ell}c_{\ell},$

where $(X_{1},X_{2})=M\circ(e_{1},e_{2})$, and its completed version.

######  Lemma 6.2.

The Laurent expansion of the Jacobi–Weierstrass $\zeta$-function
$\zeta_{M^{-1}}(\Lambda,(\boldsymbol{z})_{M^{-1}})$ is given by

$\zeta_{M^{-1}}(\Lambda,(\boldsymbol{z})_{M^{-1}})=\sum_{\ell=0}^{k-2}\frac{1}{(\boldsymbol{z})_{M^{-1},\ell}}X_{1}^{\ell}X_{2}^{k-2-\ell}c_{\ell}+\sum_{k\geq
0}G_{k-2}(\Lambda)\sum_{\ell=0}^{k-2}(\boldsymbol{z})_{M^{-1},\ell}^{k-1}X_{1}^{\ell}X_{2}^{k-2-\ell}c_{\ell},$

where $(X_{1},X_{2})=M\circ(e_{1},e_{2})$, and
$(\boldsymbol{z})_{M^{-1},\ell}=(N(M^{-1})(\boldsymbol{z}))_{l}$ is the
$\ell^{th}$ component of the vector $N(M^{-1})(\boldsymbol{z})$. We get a
similar expansion for the completion

$\widehat{\zeta}_{M^{-1}}(\Lambda,(\boldsymbol{z})_{M^{-1}})=\zeta_{M^{-1}}(\Lambda,(\boldsymbol{z})_{M^{-1}})+S(\Lambda_{\tau})\sum_{\ell=0}^{k-2}z_{\ell}X_{1}^{\ell}X_{2}^{k-2-\ell}c_{\ell}-\frac{\pi}{\operatorname{Im}(\tau)}\overline{z_{\ell}}X_{1}^{\ell}X_{2}^{k-2-\ell}c_{\ell}.$

We recall that
$F(\tau,M)=\widehat{\zeta}_{M^{-1}}\left(\Lambda_{f},N(M^{-1})\mathcal{E}_{f}\left(\tau,\left(e_{1},e_{2}\right)\right)\right)$.
Therefore, we can compute the poles of the Jacobi–Weierstrass $\zeta$-function
for the standard choice of symmetric form on $\mathbb{Z}^{k-1}$ explicitly.

######  Proposition 6.3.

Under the conditions above, the polar harmonic weak Maass form
$F:\mathbb{H}\rightarrow V$ has poles of order $1$ at the values
$\tau\in\mathbb{H}$ for which there exists
$M\in\operatorname{SL}_{2}(\mathbb{Z})$ and $0\leq\ell\leq k-1$ such that

$(N(M^{-1})\mathcal{E}_{f}(\tau,(e_{1},e_{2})))_{\ell}\in\Lambda_{f}.$

##  7\. Examples

We consider the setting of the previous section. In particular we fix the
choice of the lattice $L$ in the definition of the Jacobi–Weierstrass
$\zeta$-function to be the standard lattice $\mathbb{Z}^{k-1}$ with the
standard bilinear form. We also just evaluate the image of the function $F$,
plugging in the identity matrix.

###  7.1. A Weierstrass form for $\Delta$

Consider the Ramanujan $\Delta$-function,

$\Delta(\tau):=q\prod_{n=1}^{\infty}(1-q^{n})^{24}=\sum_{n\geq
1}\tau(n)q^{n}.$

Using the built-in functions for modular symbols in Pari/Gp [BBB+], we can
compute the period lattice of $\Delta$. Since
$\operatorname{SL}_{2}(\mathbb{Z})$ is generated by
$T=\left(\begin{smallmatrix}1&1\\\ 0&1\end{smallmatrix}\right)$ and
$S=\left(\begin{smallmatrix}0&-1\\\ 1&0\end{smallmatrix}\right)$, the period
lattice is generated by the coefficients of the period polynomial
$\int_{0}^{\infty}\Delta(t)(e_{1}+te_{2})^{10}dt$, which equals

$\displaystyle\alpha\left(e_{1}^{10}-\frac{691}{1620}\binom{10}{8}e_{1}^{8}e_{2}^{2}+\frac{691}{2520}\binom{10}{6}e_{1}^{6}e_{2}^{4}-\frac{691}{2520}\binom{10}{4}e_{1}^{4}e_{2}^{6}+\frac{691}{1620}\binom{10}{2}e_{1}^{2}e_{2}^{8}-e_{2}^{10}\right)$
$\displaystyle\qquad\qquad\qquad+\beta\left(\binom{10}{9}e_{1}^{9}e_{2}-\frac{25}{48}\binom{10}{7}e_{1}^{7}e_{2}^{3}+\frac{5}{12}\binom{10}{5}e_{1}^{5}e_{2}^{5}-\frac{25}{48}\binom{10}{3}e_{1}^{3}e_{2}^{7}+\binom{10}{1}e_{1}e_{2}^{9}\right).$

with

$\alpha=0.00595896\ldots i\quad\text{and}\quad\beta=0.00370771\ldots.$

Therefore we see that the period lattice for $\Delta$ is given by

$\Lambda_{\Delta}=\omega_{1}\mathbb{Z}\oplus\omega_{2}\mathbb{Z}$

with $\omega_{1}=\frac{1}{48}\beta=7.7243968...\cdot 10^{-7}$ and
$\omega_{2}=2.6274096...\cdot 10^{-7}i$.

With this data we can compute the Laurent expansion of the completed function
$\widehat{\zeta}_{\Lambda_{\Delta}}(z)$ to be approximately

$\displaystyle\widehat{\zeta}_{\Lambda_{\Delta}}(z)$
$\displaystyle=z^{-1}+0.0016910\ldots
z-454230029641788589613076734.309657\ldots z^{3}+O(z^{5})$
$\displaystyle\qquad\qquad\qquad-154795208574.9957812\ldots\overline{z}.$

For illustration purposes, we only consider the Eichler integral

$\mathcal{E}_{\Delta,0}(\tau)=\frac{1}{2\pi}(q-12q^{2}+84q^{3}-368q^{4}+966q^{5}-1008q^{6})+O(q^{7})$

since it has a proper Fourier expansion. Plugging this into the holomorphic
part of the Laurent expansion above yields the $0$-th component of the vector-
valued harmonic weak Maass form $F$ for the $\Delta$-function (or rather its
holomorphic part),

$\displaystyle\pi\left(q^{-1}+12+60.0000428\ldots q+79.999485\ldots
q^{2}\right.$
$\displaystyle\qquad\qquad\qquad\left.-291444838990458826940712.635494\ldots
q^{3}+O(q^{4})\right).$

We note that the Fourier coefficients grow very quickly. For instance, the
coefficient of $q^{10}$ is approximately $1.33163\cdot 10^{61}.$

Since the Eichler integrals $\mathcal{E}_{\Delta,\ell}(\tau)$ can be evaluated
quite efficiently, we can illustrate the modularity of the function $F$ we
constructed by giving numerical evaluations of it and acting with elements of
$\operatorname{SL}_{2}(\mathbb{Z})$ on them.

For example let $\tau=2i$ and $\gamma=\left(\begin{smallmatrix}2&5\\\
1&3\end{smallmatrix}\right)$. In the standard basis
$\\{\binom{10}{\ell}e_{1}^{\ell}e_{2}^{10-\ell}\\}$ the coefficients of
$\mathcal{E}_{\Delta}(\tau,(e_{1},e_{2}))$ are given by

$v=\left(\begin{smallmatrix}-17511.494570...i\\\ 7431.817430...\\\
3204.517440...i\\\ -1400.899032...\\\ -619.775633...i\\\ 277.055319...\\\
124.975219...i\\\ -56.821709...\\\ -26.014701...i\\\ 11.983426...\\\
5.550045...i\end{smallmatrix}\right)\cdot 10^{-7}.$

On the other hand we compute that $\mathcal{E}_{\Delta}(\gamma\tau)$ in the
standard basis yields

$\left(\begin{smallmatrix}-68.879683...+42.252473...i\\\
36.852496...-27.426477...i\\\ -19.386784...+17.474359...i\\\
9.977402...-10.932230...i\\\ -4.995272...+6.715450...i\\\
2.416272...-4.051512...i\\\ -1.118039...+2.402667...i\\\
0.4863669...-1.402420...i\\\ -0.191789...+0.806993...i\\\
0.062025...-0.458557...i\\\
-0.009620...+0.257699...i\end{smallmatrix}\right)=N(\gamma^{-1})v+\left(\begin{smallmatrix}-23814000\omega_{1}\\\
11895660\omega_{1}+12960\omega_{2}\\\ -5251302\omega_{1}-12912\omega_{2}\\\
1943634\omega_{1}+9159\omega_{2}\\\ -503319\omega_{1}-5456\omega_{2}\\\
-14030\omega_{1}+2860\omega_{2}\\\ 136923\omega_{1}-1336\omega_{2}\\\
-123396\omega_{1}+551\omega_{2}\\\ 81046\omega_{1}-192\omega_{2}\\\
-45360\omega_{1}+48\omega_{2}\\\ 22680\omega_{1}\end{smallmatrix}\right).$

Indeed, applying $\zeta_{\Lambda_{\Delta}}^{*}(z)$ to each component of these
vectors yields approximately

$\left(\begin{smallmatrix}-11432504.181072...-6.201719...\cdot 10^{-12}i\\\
6701461.733071...\\\ -6966824.048050\\\ 2360012.1697371...\\\
6631363.825398\\\ 10786753.122386...+8.786101...\cdot 10^{-33}i\\\
-8634302.260257...\\\ -92484.930082...\\\ 9744076.055919...+5.980872...\cdot
10^{-68}i\\\ -11495943.166401-2.586110...\cdot 10^{-9}i\\\
1267421.546264...\end{smallmatrix}\right).$

The difference between the results for the vectors after applying $\zeta^{*}$
is less than $10^{-96}$, where the computations were carried out to 115
significant digits.

###  7.2. A Weierstrass form for a CM form

We consider the newform

$f(\tau)=\eta(3\tau)^{8}=q-8q^{4}+20q^{7}-70q^{13}+64q^{16}+56q^{19}+O(q^{21})\in
S_{4}(\Gamma_{0}(9)).$

Note that this form has complex multiplication by the field
$\mathbb{Q}(\sqrt{-3})$. Note that there is a rigid Calabi–Yau threefold $X$
over $\mathbb{Q}$ such that $L(X,s)=L(f,s)$ (compare [WvG90]).

The group $\Gamma_{0}(9)$ is generated by the parabolic elements $T$ and
$-I_{2}$, as well as the matrices $\sigma_{1}=\left(\begin{smallmatrix}4&-1\\\
9&-2\end{smallmatrix}\right)$ and $\sigma_{2}=\left(\begin{smallmatrix}7&-4\\\
9&-5\end{smallmatrix}\right)$. Therefore the period lattice of $f$ is
generated by the coefficients in the polynomials

$\displaystyle p_{1}$
$\displaystyle=\int_{\sigma_{1}.\infty}^{\infty}f(t)(te_{1}+e_{2})^{2}dt$
$\displaystyle=-0.693005\binom{2}{2}e_{1}^{2}+(0.288752+0.033342i)\binom{2}{1}e_{1}e_{2}$
$\displaystyle\quad+(-0.115500-0.022228i)\binom{2}{0}e_{2}^{2}$

and

$\displaystyle p_{2}$
$\displaystyle=\int_{\sigma_{2}.\infty}^{\infty}f(t)(te_{1}+e_{2})^{2}dt$
$\displaystyle=(0.346502-0.600160i)\binom{2}{2}e_{1}^{2}+(-0.288752+0.433449i)\binom{2}{1}e_{1}e_{2}$
$\displaystyle\quad+(0.231001-0.311194i)\binom{2}{0}e_{2}^{2}.$

A basis for the lattice in $\mathbb{C}$ generated by these coefficients is
given by $\Lambda_{f}=\omega_{1}\mathbb{Z}\oplus\omega_{2}\mathbb{Z}$ with

$\omega_{1}=0.057750\quad\text{and}\quad\omega_{2}=0.011114i.$

With this we can compute the Laurent expansion of
$\widehat{\zeta}_{\Lambda_{f}}(z)$ as

$z^{-1}+21739.040942z-141870582.946988z^{3}+1079581634085.963275z^{5}+O(z^{7})\\\
+4894.639140\overline{z}.$

Since the coefficient $\mathcal{E}_{f,0}(\tau)$ has a Fourier expansion, we
can plug it into this expansion, yielding the following expression for the
$0$-th component of the holomorphic part of our harmonic weak Maass form,

$\displaystyle\pi\left(q^{-1}+21739.040942q+8q^{2}-141870582.946988q^{3}-173912.327537q^{4}++O(q^{5})\right).$

We now pick some point in the upper half-plane, say
$\tau=\frac{1+i\sqrt{7}}{2}$. In the standard basis
$\\{e_{1}^{2},2e_{1}e_{2},e_{2}^{2}\\}$ we obtain the value

$\mathcal{E}_{f}(\tau,(e_{1},e_{2}))=\left(\begin{smallmatrix}5.792643+7.706733i\\\
-5.792643+1.954292i\\\ -3.908585i\end{smallmatrix}\right)\cdot 10^{-5}.$

On the other hand we compute

$\mathcal{E}_{f}(\sigma_{1}\tau,(e_{1},e_{2}))=\begin{pmatrix}-0.115037-0.020877i\\\
0.287651+0.030313i\\\
-0.690398+0.006789i\end{pmatrix}=v_{1}+\left(\begin{smallmatrix}-2\\\ 5\\\
-12\end{smallmatrix}\right)\omega_{1}+\left(\begin{smallmatrix}-2\\\ 3\\\
0\end{smallmatrix}\right)\omega_{2}$

and

$\mathcal{E}_{f}(\sigma_{2}\tau,(e_{1},e_{2}))=\left(\begin{smallmatrix}0.230596-0.306949i\\\
-0.288288+0.427988i\\\
0.345981-0.593136i\end{smallmatrix}\right)=v_{2}+\left(\begin{smallmatrix}4\\\
-5\\\ 6\end{smallmatrix}\right)\omega_{1}+\left(\begin{smallmatrix}-28\\\
39\\\ -54\end{smallmatrix}\right)\omega_{2}.$

Plugging either one of the vectors above into
$\widehat{\zeta}_{\Lambda_{f}}(z)$ yields

$\left(\begin{smallmatrix}235.526014-626.338523i\\\ -129.035666+208.728000i\\\
165.268142+59.208840i\end{smallmatrix}\right)\quad\text{resp.}\quad\left(\begin{smallmatrix}-33.090396-108.245540i\\\
32.306903+7.539680i\\\ -44.064442+121.275441i\end{smallmatrix}\right).$

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Multidimensional Quantum Fourier Transformation
Philipp Pfeffer
Institute of Thermodynamics and Fluid Mechanics, Technische Universität Ilmenau, Postfach 100565, D-98684 Ilmenau, Germany

The Quantum Fourier Transformation (QFT) is a well-known subroutine for algorithms on qubit-based universal quantum computers. In this work,  the known QFT circuit is used to derive an efficient circuit for the  multidimensional QFT. The complexity of the algorithm is $\mathcal{O}(  \log^2(M)/d )$ for an array with $M=(2^n)^d$ elements $(n \in \mathbb{N})$ equally separated along $d$ dimensions. Relevant properties for application are discussed. An example on current hardware is depicted by a 6 qubit 2D-QFT with an IBM quantum computer.

The Quantum Fourier Transformation [1] (QFT) is a key subroutine in quantum information processing, most prominently used within the quantum phase estimation [2] and the factoring algorithm of Shor [3]. Comparing to its classical counterpart, the fast fourier transformation [4] (FFT) solves the same problem with effort $\mathcal{O}(N\log(N))$ while the QFT needs  $\mathcal{O}(\log^2(N))$ operations [5] for a vector with $N=2^n$ elements ($n \in \mathbb{N}$). Though this speed-up does not lead to the broad replacement of the FFT, the aspect of quantum parallelism is important for the construction of further quantum algorithms. Single operations acting on a large quantum state, such that the whole state is affected, enable the speed-up of the QFT and can be utilized even better for a multidimensional QFT.

The idea of this dimensional extension is not new, as the 2-dimensional QFT (2D-QFT) is of crucial use in the field of quantum image processing [6, 7, 8]. There, it can be used for edge detection [9], watermarking [10] and for the implementation of a discrete cosine transform [11], which is useful for interpolation [12]. The motivation here is to extend the concept to more dimensions and give an easily understandable summary of the constructed circuit and its complexity. Furthermore, this work aims at readers new to the field in order to help them understand the concept of quantum parallelism as well as the consequences hindering its success.

The exact task is to construct a quantum circuit which calculates the d-dimensional discrete Fourier transformation of a d-dimensional array A where each dimension $i$ spreads over $N_i=2^{n_i}$ elements ($n_i \in \mathbb{N}$). The transformation is given by the formula

\begin{equation}
\tilde{a}_{\delta_1,..,\delta_d} = \sum\limits_{k_d=0}^{N_d-1}  \omega_{N_d}^{k_d \delta_d} \sum\limits_{k_{d-1}=0}^{N_{d-1}-1}  \cdots \left( \sum\limits_{k_1=0}^{N_1-1} \omega_{N_1}^{k_1\delta_1} a_{k_1,..,k_d}  \right)
\label{Eq:MDFT}
\end{equation}

with $\omega_{N_i}=e^{j2\pi/N_i}$ and $a_{k_1,...,k_d}$ as the indexed elements of A. For simplification, indices are abbreviated by

$$ N_1\cdot...\cdot N_{i-1}=:N_{i_\downarrow } \ \ , \ \  N_{ i^\uparrow}:= N_{i+1}\cdot... \cdot N_{d}$$

and the number of all array elements is $M=N_1\cdot ... \cdot N_d$.

The 1-dimensional FFT has a matrix representation given by the Vandermonde-matrix

\begin{equation*}
V_{N_i} = \begin{pmatrix}
\omega_{N_i}^0 & \omega_{N_i}^0 & \omega_{N_i}^0 & \cdots & \omega_{N_i}^0 \\
\omega_{N_i}^0 & \omega_{N_i} & \omega_{N_i}^2 & \cdots & \omega_{N_i}^{N_i-1} \\
\omega_{N_i}^0 & \omega_{N_i}^2 & \omega_{N_i}^4 & \cdots & \omega_{N_i}^{2(N_i-1)} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
\omega_{N_i}^0 & \omega_{N_i}^{N_i-1} & \omega_{N_i}^{2(N_i-1)} & \cdots & \omega_{N_i}^{\left( (N_i-1)^2\right)}
\end{pmatrix}.
\end{equation*}

The QFT in terms of a matrix is the rescaled version of this matrix, namely

\begin{equation*}
QFT_{N_i} = \frac{1}{\sqrt{N_i}} V_{N_i}.
\end{equation*}

Thereby the matrix is unitary. The circuit implementing this matrix (see [5]) will be called $QFT_{n_i}$. Note that the prefactor of $QFT_{N_i}$ matters for comparing the results with the classical algorithm, as done in Fig. <ref>.

In classical computing, multidmensional FFT algorithms utilize the FFT recursively [13]. For the solution, one can start by applying the FFT to the last sum of Eq. (<ref>) to get an interim array where every entry is transformed along the first dimension. This array is then the input for the FFT along the next dimension, which can be repeated until every sum is evaluated. The $d=2$ case gives this procedure the name row-column algorithm, since there one transforms the array first along its rows or columns in order to transform along the other next. This idea is essential for the following algorithm.

The quantum circuit for the multidimensional QFT is shown in Fig. <ref> and works as follows. The elements of $A$ can be aligned into a vector

$$\vec{v} = (a_{1,1,..,1}\ ,\ ...\ , \ a_{N_1,1,..,1}\ ,\ a_{1,2,..,1}\ , \ ... \ , \ a_{N_1,N_2..,N_d} )^T.  $$

This vector has to be reformulated into a quantum state $\ket{v}$, which requires to scale the vector by

\begin{equation}
\ket{v} = \frac{\vec{v}}{\sqrt{\sum\limits_{k=1}^{N_1 \cdot ... \cdot N_d} |v_k|^2}}.
\label{Eq:Scale}
\end{equation}

With that $\ket{v}$ can be initialized on a quantum computer by a state loading procedure[14, 15]. As pointed out later on, it is more favourable if the input construction already is an efficient algorithm for a specific problem.

The formalism for circuit-to-matrix conversion is that the upper qubit is the least significant bit, or in other words the last bit of the bit string. Thereby, if the action of $QFT_{n_1}$ in Fig. <ref> is of interest, the corresponding matrix acting on the input state is by definition [5]

\begin{equation}
U_1 = \mathcal{I}_{N_{1^\uparrow}} \otimes QFT_{N_1} = \begin{pmatrix} QFT_{N_1} &   & \mathbb{O} \\
\  & \ddots &  \\
\mathbb{O} &   & QFT_{N_1} \\
\end{pmatrix}
\label{Eq:U1}
\end{equation}

where $\mathcal{I}_N$ denotes the N$\times$N identity matrix, $\mathbb{O}$ represents all zeros besides the block diagonal and $\otimes$ is the tensor product. The tensor product is the mathematical representation of quantum parallelism since it duplicates the operation to the size of the quantum state it acts on. As a result, this circuit gives

$$ U_1\ket{v} = \begin{pmatrix}
QFT_{N_1}\    (\ v_1\ ,\ ...\ ,\ v_{N_1}\ )^T    \\
QFT_{N_1}\ (v_{N_1+1},...,v_{2N_1})^T \\
\vdots \\
QFT_{_1}\ (v_{M-N_1},...,v_{M})^T
\end{pmatrix} $$

which creates the first interim matrix comparable to the classical approach. In other words, a single QFT finishes the first dimension globally. In general for any dimension i, the large unitary matrix corresponding to the quantum circuit and its position in Fig. <ref> is

\begin{equation}
U_i = \mathcal{I}_{N_{i^\uparrow}} \otimes \underbrace{QFT_{N_i} \otimes \mathcal{I}_{N_{i_\downarrow}}}_{\tilde{U}_i}.
\label{Eq:Ui}
\end{equation}

For edge cases, $\mathcal{I}_{1_\downarrow} =\mathcal{I}_{d^\uparrow} = 1$ is the empty product convention. The arrangement of A as a vector in combination with Eq. (<ref>) yields the transformation along dimension i. To further clarify this, writing out $\Tilde{U}_i$ of Eq. (<ref>) by the tensor product definition gives

\begin{equation}
\Tilde{U}_i = \frac{1}{\sqrt{N_i}} \begin{pmatrix}
\omega^0_{N_i} \ \mathcal{I}_{N_{i_\downarrow}} & \omega^0_{N_i} \mathcal{I}_{N_{i_\downarrow}}  & ... & \omega^0_{N_i} \mathcal{I}_{N_{i_\downarrow}} \\
\omega^0_{N_i} \mathcal{I}_{N_{i_\downarrow}}  & \omega^1_{N_i}\mathcal{I}_{N_{i_\downarrow}} & ... & \omega^{N_i-1}_{N_i} \mathcal{I}_{N_{i_\downarrow}} \\
\vdots & \vdots & \ddots & \vdots \\
\omega^0_{N_i} \mathcal{I}_{N_{i_\downarrow}} & \omega^{N_i-1}_{N_i} \mathcal{I}_{N_{i_\downarrow}} & ... & \omega_{N_i}^{(N_i-1)^2} \mathcal{I}_{N_{i_\downarrow}}
\end{pmatrix}.
\label{Eq:Ui_Ti}
\end{equation}

The overall working principle of the algorithm can be explained by observing Eq. (<ref>) with the help of Eq. (<ref>) and Eq. (<ref>). The position of $QFT_{n_i}$ in the circuit structure of Fig <ref> forces the tensor product of Eq. (<ref>).  The evaluation of the latter tensor product, namely for $\Tilde{U}_i$, creates a matrix as shown in Eq. (<ref>). Here, the elements of the 1-dimensional QFT are separated such that, combined with the vector form of the original array, they act only along the dimension i. The first tensor product, similar to Eq. (<ref>), then duplicates this correctly spaced matrix to the necessary size in a block diagonal structure, which assures the action on all array elements. All QFTs act on separate qubits, which means that all $U_i$ of Eq. (<ref>) commute with each other [5]. Thereby, one can choose that they act in ascending order, i.e. they evaluate the sums as done in Eq. (<ref>), dimension by dimension, similar to the classical algorithm. To summarize, each QFT transforms the array along the corresponding dimension with a single QFT while all QFTs are executed simultaneously.

The difference of computational complexity is thereby significant. For simplicity, let A be a d-dimensional array equally sized along every dimension i such that every dimension has $N_i=N=2^n$ elements and $M=N^d$. It is knwon for the classical multidimensional FFT that it has computational effort $\mathcal{O}{(M\log(M))}$ [13], similar to the 1-dimensional case. For the quantum algorithm, a single QFT has a known complexity of $\mathcal{O}(n^2)$, and the algorithm needs d QFTs in total, so $\mathcal{O}(d n^2)$ operations. In terms of $M$, one can deduce that $n = \log_2(\sqrt[\leftroot{1} \uproot{2} d ]{M})$ which means for a direct comparison that

$$\text{FFT:} \  \mathcal{O}{(M \log(M))} \leftrightarrow \text{QFT:} \ \mathcal{O}( \log^2(M)/d ).$$

The applicability of this speed up remains to be shown since it is already known for the $d=1$ case that this algorithm will not replace all applications of the FFT [5]. The computational effort of rescaling arrays according to Eq. (<ref>), then initializing them on a quantum computer [14, 15] in order to approximate the spectrum with measurements will almost always scale at least linear in terms of array elements $M$, which makes the computational advantage either shallow or non-existing. Additionally, the approximation of the final spectrum by measurements will return $|\hat{v}_i|^2$, which will erase both the sign as well as the phase information of the complex Fourier coefficient $\hat{v}_i$. These points have to be addressed in competition with the classical multidimensional FFT, which is efficiently executable with parallel computing approaches [13, 16].

In favor of the proposed algorithm speak the existing applications of QFTs, as is already the case in the field of quantum image processing [9, 12, 10, 11]. Another application could use the multidimensional QFT to solve Laplace operators by variational quantum algorithms (see [17]) with more dimensions. Furthermore, it is worth noting that the algorithm itself does not scale with the amount of parallel initialized arrays. In detail, if one manages to initialize not only a single array $M$ as $\ket{v}$ done by Eq. (<ref>) but multiple ones aligned one after another, then the circuit of Fig. <ref> can be appended where only the initialization spreads over more qubits. In the herein notation, these qubits are added below the illustrated circuit, which then creates a structure similar to Eq. (<ref>) whereas the 1-dimensional QFT is replaced by the d-dimensional QFT.

Concluding, an optimal use-case for this algorithm would be that multiple input arrays need to be calculated first, and it exists an efficient quantum algorithm to do so. The result state after the QFT should then either be strongly localized or further processible on a quantum computer, for instance if the difference of multiple Fourier transformed arrays is of interest.

As a final example, a real quantum device of IBM ('ibm_lagos') is used to calculate the Fourier coefficients of an 8$\times$8 picture ($d=2$ and $n_x=n_y=3$), where two  simplifications are used. First, the final swap operations in the QFTs (see [5]) are avoided by measuring the qubits in a swapped order. Further circuit optimization has been inhibited rigorously. Second, the input picture is generated by

\begin{equation}
f(x,y) = sin(\pi x/2) cos(\pi y/2 )
\label{Eq:fx}
\end{equation}

where $x,y \in \{0,1,...7\}$. This is easy to initialize, as Fig. <ref>.a shows. The image itself is illustrated in Fig. <ref>.b. For this, four distinct peaks are expected in the Fourier spectrum, as depicted in Fig. <ref>.c. The result of the real quantum computer, shown in Fig. <ref>.d , has its largest peaks at the correct locations. Here, the expectable influence of noisy quantum computers corrupts the result such that the peaks are not equally sized and additional peaks rise. The lowest correct and highest incorrect peak still show  significant difference.

[] (-1.5,-4.7) node[anchor=south west];
[] (0,0) – (1.6,0);
i̊n 1,...,5
(0,*̊) – (1.6,*̊);

[fill=white] (0.25,-0.225) node[anchor=south west]H rectangle ++(0.475,0.45);
[fill=white] (0.25,0.325) node[anchor=south west]X rectangle ++(0.475,0.45);
[fill=white] (0.85,0.325) node[anchor=south west]H rectangle ++(0.475,0.45);
[fill=white] (0.25,1.425) node[anchor=south west]H rectangle ++(0.475,0.45);
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[dashed] (1.5,-0.25) – ++(0,3.25);

[decoration=calligraphic brace,amplitude=5pt, decorate, line width=1.25pt] (-0.055,1.45) – ++(0,1.5) ++(0,-0.75) node[anchor =  east] $n_x$   ;
[decoration=calligraphic brace,amplitude=5pt, decorate, line width=1.25pt] (-0.055,-0.25) – ++(0,1.5) ++(0,-0.75) node[anchor =  east] $n_y$   ;
[] (-1,2.8) node[](a);
[] (-1,-1.6) node[](c);
[] (2.5,2.8) node[](b);
[] (2.5,-1.6) node[](d);

Example of a 2D-QFT on the IBM quantum computer 'ibm_lagos'. (a) represents the quantum circuit that initializes $f(x,y)$ of Eq. (<ref>). (b) shows the corresponding initialized image. (c) illustrates the absolute values of the classical 2D-FFT. (d) shows the result extracted from 'ibm_lagos' by $2^{14}$ samples.

Closing with acknowledgments, I like to thank Theo Käufer for bringing up the fundamental motivation for this work. I thank Darius Becher and Clara Stolzenberg for helpful comments. During this work, I was supported by the project no. P2018-02-001 "DeepTurb – Deep Learning in and of Turbulence" of the Carl Zeiss Foundation and the Deutsche Forschungsgemeinschaft under grant no. DFG-SPP 1881.

I acknowledge the use of IBM Quantum services for this work. The views expressed are those of the author, and do not reflect the official policy or position of IBM or the IBM Quantum team. In this paper 'ibm_lagos' was used, which is an IBM Quantum Falcon Processor.

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