text:
We prove the weak analogue \cref{lem:notinpositroid} of Oh's theorem used in the proof of the unit necklace theorem. Then we prove the equivalence of the Coxeter-theoretic condition (2) and the weak-separation theoretic condition (3) from \cref{thm:main}.  Finally, we prove
\cref{thm:alignedIsQuasiEquiv} which says that aligned toggles on Grassmannlike necklaces induce quasi-equivalences of cluster structures.

\subsection{Proof of {\cref{lem:notinpositroid}}}\label{secn:rightweakproof}
Recall the definition of noncrossing and aligned chords and toggles from \cref{defn:noncrossing}.

By Remark~\ref{rmk:togglePluckerRel}, if we perform an aligned toggle at a necklace $\gnec$ satisfying $\gnec \subset \mcm_\pi$ then the new necklace $\gnec' \subset \mcm_\pi$ as well.

Since $\rho$ and $\iota$ determine $\gnec$, we will frequently omit the subsets $I_i$, writing the removal and insertion values in the following {\sl two-line notation}:
\begin{equation}\label{eq:chordsnotationii}
\gnec = \begin{matrix}\iota_1 \\ \rho_1\end{matrix}
\begin{matrix}\iota_2 \\ \rho_2\end{matrix} \cdots
\begin{matrix}\iota_{n-1} \\ \rho_{n-1}\end{matrix}
\begin{matrix}\iota_n \\ \rho_n\end{matrix}.
\end{equation}

Now we prove Lemma~\ref{lem:notinpositroid} which is needed in the proof of Theorem~\ref{thm:rightweak}.

\begin{proof}[Proof of Lemma~\ref{lem:notinpositroid}]

 Let $\rmci_{\pi}$ be a forward Grassmann necklace. Suppose $\gnec=(I_1, \dots, I_n)$ is a Grassmannlike necklace which can be obtained from $\rmci_\pi$ by a sequence of noncrossing toggles.

 We prove the following more specific claim which readily implies the desired statement. We abbreviate $L = I_{\rho^{-1}(a)}$ and $R = I_{\rho^{-1}(a)+1}$.

\textit{Claim: There exist sets
$\mcs,\mct \subset [n] \setminus \{\pi(a),a\}$,
with
\begin{align}\label{eq:LtRt}
L&= \rightI_{a} \setminus (\pi^{-1}\mct \cup \mcs)  \coprod \left(\mct  \cup \pi^{-1}\mcs \right) \\
R &= \rightI_{a+1} \setminus (\pi^{-1}\mct \cup \mcs)  \coprod \left(\mct  \cup \pi^{-1}\mcs \right)
\end{align}
such that the pair of chords $\pi^{-1}(s) \mapsto s$ and $a \mapsto \pi(a)$ are noncrossing for all $s \in \mcs$, and likewise the chords $\pi^{-1}(t) \mapsto t$ and $a \mapsto \pi(a)$ are noncrossing for all $t \in \mct$.
}

In \eqref{eq:LtRt}, let us clarify that the use of $\setminus$ implies that the second set is contained in the first. (We do not adopt that convention in most other parts of the paper.) However, it not important that the sets $\pi^{-1}(\mct), \mcs$ are disjoint, and it is also not important that the sets $\mcs$ and $\mct$ are disjoint (i.e., we allow for removing an element that is in $\mcs$ and then adding it back in if it is in $\mct$).

We will establish this claim by induction on $\ell(f)-\ell(i)$ and then explain why it implies the statement in the lemma.

The base case of \eqref{eq:LtRt} holds with $\mcs = \mct = \emptyset$. The subsets $L$ and $R$ only change meaning when we toggle at either $L$ or $R$. If we toggle at $R$, things look locally like
\begin{equation}
L \hspace{.3cm} \substack{\pi(a) \\ \rightleftarrows \\ a} \hspace{.3cm}  R \hspace{.3cm}  \substack{t \\ \rightleftarrows \\ \pi^{-1}(t)} \hspace{.3cm}  X.
\end{equation}

Let us denote by $L',R'$ the new versions of $L$ and $R$ after the toggle. Then $R' = X = R \setminus \pi^{-1}(t) \cup t$. The subset $L'$ is obtained by toggling at $R$; we clearly also have $L' = L\setminus \pi^{-1}(t)\cup t$. So $L',R'$ both evolve according to the formula \eqref{eq:LtRt} in this case. The claimed statement that these chords are noncrossing holds by assumption. Note also that $t \neq \pi(a)$. The argument in the case that we toggle at $L$ rather than at $R$ is similar, with the local picture looking like
\begin{equation}
X \hspace{.3cm} \substack{s \\ \rightleftarrows \\ \pi^{-1}(s)} \hspace{.3cm}  L \hspace{.3cm}  \substack{\pi(a) \\ \rightleftarrows \\ a} \hspace{.3cm}  R
\end{equation}
and the subsets evolving according to the formula $L' = L \setminus s \cup \pi^{-1}(s)$ and
$R' = R \setminus s \cup \pi^{-1}(s)$. Note again that $s \neq a,\pi(a)$. The claim holds by induction.

Since the chords $\pi^{-1}(s) \mapsto s$ and $a \mapsto \pi(a)$ are noncrossing, we either have that  $\{s,\pi^{-1}(s)\} \subset (a,\pi(a))$ or that
$\{s,\pi^{-1}(s)\} \subset (\pi(a),a)$ (with both of these considered as cyclic subintervals of $[n]$). And similarly for $\{\pi^{-1}(t),t\}$. From \eqref{eq:LtRt}, it follows that
$$\#\left(R \cap (a,\pi(a))\right) = \#\rightI_{a+1} \cap (a,\pi(a)).$$
If $J = L \setminus a \cup y= R \setminus \pi(a) \cup y$ where $y<_a \pi(a)$, then
$$\#\left(J \cap (a,\pi(a))\right) = \#\left(R \cap (a,\pi(a))\right)+1 >\#(\rightI_{a+1} \cap (a,\pi(a)))$$
so that $J \notin \mcm_\pi$ using Oh's Theorem.

Likewise, let $J = L \setminus y \cup \pi(a)$ in the situation $\pi(a)<_a y$. Then
$$\#\rightI_{a} \cap [a,\pi(a)] = \#\left(L \cap [a,\pi(a)]\right) = \#(J \cap [a,\pi(a)])-1 ,$$
so that $J \notin \mcm_\pi$ using Oh's Theorem.

\end{proof}

\subsection{Proof of \cref{thm:main}: (1) $\iff$ (2) }\label{secn:conjugationproof}
By \cref{rmk:togglePluckerRel}, aligned toggles correspond to those in which the Pl\"ucker relation has signs $I_iI_i'=I_{i-1}I_{i+1}+S_1S_2$. Such a relation ``looks like''
the three-term Pl\"ucker relation encoding a square move on weakly separated collections. However, such a Pl\"ucker relation does not necessarily correspond to a square move on weakly separated collections; that is, performing an aligned toggle does not preserve weak separation. So not all of the necklaces $\gnec_{\bullet, \iota, \pi}$ with $\iota \ler \pi$ are weakly separated.

Recall our general setup: we have $i \leq_R f$ which are lifts of permutations $\iota,\pi$ of type $(k,n)$. We define $\mu = \iota^{-1}\pi\iota$ which is a permutation of type $(k,n)$ with lift $m \in {\rm Bound}(k,n)$. By \cref{lem:conjugationLift}, $m=i^{-1}fi$. We always have $\ell(m) \leq \ell(f)$ and we want to characterize when $\ell(m) = \ell(f)$.

To simplify statements, let $r=f^{-1}i$, so $m=r^{-1}i$.

\begin{lem}\label{lem:conjugation}
We have $\ell(m) < \ell(f)$ if and only if there exists a transposition $t \in T$ satisfying both $\ell(tr) < \ell(r)$ and
$\ell(ti) < \ell(i)$.
\end{lem}

In other words, if $t = t_{ij}$, then the values $i$ and $j$ are ``out of order'' in both $\rho$ and $\iota$ (when both permutations are appropriately upgraded to affine permutations).

\begin{proof}
Since $i \leq_R f$, the factorization $f = ir^{-1}$ is length-additive: $ \ell(f) = \ell(i)+
\ell(r^{-1})$. So $\ell(m) = \ell(f)$ if and only if the factorization $m = r^{-1}i$ is length-additive:
$\ell(r^{-1}i) = \ell(r^{-1})+\ell(i)$.

By \cref{lem:lengthAddAndReflections}, $\ell(r^{-1}i) = \ell(r^{-1})+\ell(i)$ if and only if $T_R(r^{-1}) \cap T_L(i)= \emptyset$. Also $T_R(r^{-1})=T_L(r)$, so we are done.

\end{proof}

We now prove (1) implies (2) in Theorem~\ref{thm:main}, namely that $\ell(m) = \ell(f)$ is sufficient to guarantee that $\gnec_{\bullet,\iota,\pi}$ is weakly separated.
\begin{proof}[Proof of sufficiency]
We will suppose that $\gnec$ is not weakly separated and show that there exists a transposition $t \in T$ as in the statement of Lemma~\ref{lem:conjugation}.

First we rephrase weak separation of $\gnec$ as a condition on the removal and insertion permutations $\rho$ and $\iota$. A 4-tuple of circularly ordered numbers $a < b < c < d$ are a witness for nonseparation of $\gnec$ if and only if there are values $x,y \in [n]$, such that
\begin{align}
\{a,c\} \subset \iota([y,x)) &\text{ and } \{b,d\} \subset \rho([y,x))  \label{eq:wxyz}\\
\{a,c\} \subset \rho([x,y)) &\text{ and } \{b,d\} \subset \iota([x,y)). \label{eq:wxyzb}
\end{align}

Visually, we can ``chop'' $\gnec$ in the positions $I_x$ and $I_y$, breaking $[n] = \bb_1 \coprod \bb_2$ in two cyclic intervals $\bb_i$. In $\bb_1$ we see
$\{a,c\}$ in the insertion permutation and $\{b,d\}$ in the removal permutation while in $\bb_2$ we see the opposite.

Now we switch from thinking about permutations $\rho,\iota$ to thinking about affine permutations.  We consider the two-line notation \eqref{eq:chordsnotationii} (extended bi-infinitely and $n$-periodically in both directions) whose numerator is $i$ and whose denominator is $r=f^{-1}i$. Reducing values modulo $n$ yields the permutations $\iota,\rho$ respectively.

As in the proof of Lemma~\ref{lem:conjugationLift}, we can reach this two-line notation by starting with the two-line notation whose numerator is $f$ and whose denominator is the identity $e_0 \in \tilde{S}^0_n$, and $n$-periodicallly performing swaps of adjacent columns. In particular,
any column vector $\begin{matrix} \bb \\ \aa \end{matrix}$ appearing in the two-line notation satisfies $\aa \leq \bb \leq \aa+n $.

As in the proof of Lemma~\ref{lem:conjugationLift}, the appearance of any $x \in \bbz$ in the top row is weakly to the left of $x$ in the bottom row.

Suppose $a<b<c<d$ are a witness against weak separation as
in \eqref{eq:wxyz}. We can uniquely lift these to linearly ordered numbers $a<b'<c'<d' \in \bbn$ such that $b' \in \{b,b+n\}$ and so on. Initially, the numbers $a,b',c',d'$ appear sorted in the order $a,\dots,b',\dots,c',\dots,d'$ in the denominator of the two-line notation. To reach $\gnec$, we perform a sequence of column swaps so that $\{a,c'\}$ and $\{b',d'\}$ are adjacent to each other in the bottom row.

For example, this might happen by starting with $a,\dots,b',\dots,c',\dots,d'$ in the bottom row, performing swaps until we reach $a,\dots,b',c',\dots,d$ with $b',c'$ adjacent, and then performing the swap that switches $b',c'$ yielding $a,\dots,c',b',\dots,d$. Once we have done this, the values $c',b'$ henceforth remain out of order in the bottom row, and in particular we would have $\ell(t_{(b',c')}r) < \ell(r)$. If we perform
a further sequence of swaps and arrive at the picture
$$\begin{matrix}\{b,d\} \\ \{a,c\}
\end{matrix} \cdots \begin{matrix}\{a,c\} \\ \{b,d\}
\end{matrix} $$
modulo $n$, we conclude that the picture in fact looks like
$$\begin{matrix}\{b',d'\} \\ \{a,c'\}
\end{matrix} \cdots \begin{matrix}\{a+n,c'+n\} \\ \{b',d'\}
\end{matrix} \cdots \begin{matrix}\{b'+n,d'+n\} \\ \{a'+n,c'+n\}
\end{matrix},
$$
using the fact $a$ in the top row appears left of $a$ in the bottom row, etc. The values $b',c'$ are also out of order in the numerator, i.e. we have $\ell(t_{b'c'}i) < \ell(i)$, as desired.

We have been considering the special case where $b'$ swaps past $c'$, but it is straightforward to see that it is necessary to perform at least one of the swaps ($d$ past $a$, $a$ past $b'$, $b'$ past $c'$, or $c'$ past $d'$) and the argument is identical.
\end{proof}

Now we prove (2) implies (1) in Theorem~\ref{thm:main}, i.e. that the condition $\ell(m) = \ell(f)$ is necessary for the necklace $\gnec$ to be weakly separated.

\begin{proof}[Proof of necessity]

We will show that if there exists $a<b$ such that $\ell(t_{ab}r) < \ell(r)$ and $\ell(t_{ab}i) < \ell(i)$, then we can chop the two-line notation
\eqref{eq:chordsnotationii} in two pieces as in \eqref{eq:wxyz} and \eqref{eq:wxyzb}. As in the above proof of sufficiency, we work with two-line notation for affine permutations. By assumption, the two-line notations
looks like
\begin{equation}
\begin{matrix}\cdots b \cdots a \cdots \cdots
\cdots \\
\cdots \cdots \cdots \cdots b \cdots a \cdots  \end{matrix}.
\end{equation}
The relative positions of the $a$ in the numerator and the $b$ in the denominator are not important for our argument.

We chop the necklace
as indicated by vertical bars
\begin{equation}
\begin{matrix}
\cdots b  \\
\cdots
\end{matrix}
\begin{vmatrix} \cdots a \cdots \cdots \\\ \cdots \cdots b \cdots
\end{vmatrix}
\begin{matrix}  \cdots \\
a \cdots
\end{matrix}
\end{equation}
(that is, just {\sl after} the $b$ in the top row and just before the $a$ in the bottom row).
Let $\mcb^- \subseteq (-\infty,a)$ be those values lying in within the vertical bars and in the bottom row of the two-line notation. Similarly, we let
$\mct^- \subseteq (-\infty,a)$ be those values lying within the vertical bars and in the top row of the two-line notation.

We claim that $\mcb^- \setminus \mct^-$ is nonempty. We have that $f^{-1}(a) \in \mcb^-$. Then the claim follows from noting that if $z \in \mct^-$ then there exists an element of $\mcb^-$ which is strictly less than $z$ (namely, the element $f^{-1}(z)$).

We can likewise set $\mcb^+:= (b,\infty) \cap \textnormal{ bottom row}$ and
$\mct^+:= (b,\infty) \cap \textnormal{ top row}$ (again, only considering those values within the vertical bars). Then $f(b) \in \mct^+$ and we claim that $\mct^+ \setminus \mcb^+$ is nonempty. This follows similarly from as above, noting that
if $z \in \mcb^+$ then there exists an element of $\mct^+$ which is strictly greater than $z$ (namely, the element $f(z)$).

Letting $z_- \in \mcb^- \setminus \mct^-$,
$z_+ \in \mct^+ \setminus \mcb^+$, we have
$z_- < a < b < z_+$ are a witness against weak separation.
\end{proof}

\subsection{Proof of \cref{thm:alignedIsQuasiEquiv}} \label{sec:reachablepf}
We prove \cref{thm:alignedIsQuasiEquiv} using the technology of plabic tilings introduced in \cite[Section 9]{OPS}. We briefly review the definition here.

Let $p_1, \dots, p_n \in \mathbb{R}^2$ be the vertices of a regular $n$-gon listed in clockwise order. For $I \subset [n]$, we use the notation $p(I):=\sum_{i \in I} p_i$ (where the summation here is summation of vectors in $\mathbb{R}^2$). As usual, we abbreviate $X \cup \{a\}$ as $Xa$.

Given a weakly separated collection $\mcc \subset \binom{[n]}{k}$, the associated plabic tiling $\mct(\mcc)$ is a 2-dimensional CW-complex embedded in $\bbr^2$. The vertices are $\{p(I): I \in \mcc\}$. Faces correspond to nontrivial black and white cliques in $\mcc$. For $X \in \binom{[n]}{k-1}$, the white clique $\mcw(X)$ consists of all $I \in \mcc$ which contain $X$. Similarly, for $X \in \binom{[n]}{k+1}$, the black clique $\mcb(X)$ consists of all $I \in \mcc$ which are contained in $X$. A clique is nontrivial if it has more than two elements. The elements of a white clique, for example, are $Xa_1, \dots, Xa_r$, where $a_1, \dots, a_r$ are cyclically ordered; we have edges between $p(Xa_i)$ and $p(Xa_{i+1})$ in $\mct(\mcc)$. The edges between vertices in a black clique are similar.

\begin{lem} \label{lem:dkkCondition}
Consider a pair of permutations $\iota \ler \pi$ and let
$\gnec = \gnec_{\bullet, \iota, \pi}$ be a Grassmannlike necklace with terms $(I_1, \dots, I_n)$. Let $u \to \pi(u)$ and $v \to \pi(v)$ be a pair of crossing chords and set $w:=\pi(u)$ and $x:=\pi(v)$. Then $\gnec$ does not contain a quadruple $I_a, I_{a+1}, I_b, I_{b+1}$ such that either $(I_a, I_{a+1})=(Xu, Xw)$ and $(I_b, I_{b+1})=(Xv, Xx)$, or $(I_a, I_{a+1})=(X\setminus u, X\setminus  w)$ and $(I_b, I_{b+1})=(X \setminus v, X \setminus x)$. \end{lem}

\begin{proof}
	We argue by contradiction. Suppose we have $(I_a, I_{a+1})=(Xu, Xw)$ and $(I_b, I_{b+1})=(Xv, Xx)$ for crossing chords $u \to w$ and $v \to x$. Since $u$ is removed from $I_a$, we have that $a=\rho^{-1}(u)$. We also have that $w=\pi(u)$. Since the chords $u \to w$ and the chords $v \to x$ cross, we know that one of $v, x$ is in the cyclic interval from $u$ to $\pi(u)$. Let's say it's $x$ (the other case is identical). That is, we have $x <_u \pi(u)$. By \cref{lem:notinpositroid}, this means that $I_{\rho^{-1}(u)} \setminus u \cup x= Xx$ is not in the matroid $\mcm_\pi$. But by \cref{thm:rightweak}, $\gnec$ is a unit necklace and so in particular, all terms are in $\mcm_\pi$, a contradiction.

	The other case is identical.
\end{proof}

For the remainder of this section, we fix a Grassmannlike necklace $\gnec=\gnec_{\bullet,\iota, \pi}$ with $\iota \ler \pi$. To this necklace $\gnec = (I_1,\dots,I_n)$ we associate the polygonal curve $\zeta(\gnec)$ with vertices $p(I_1),\dots,p(I_n)$ in that order. Since $\gnec$ can have repeated terms, the curve $\zeta(\gnec)$ can have self-intersections at vertices. However, the conclusion of \cref{lem:dkkCondition} says that $\zeta(\gnec)$ has no self-intersections involving a pair of crossing edges. Thus,  $\zeta(\gnec)$ is a union of simple closed polygonal curves meeting only at their vertices.

We have moreover that none of these simple closed curves encloses another. Indeed, if this were true, then using \cref{rmk:FGinside}, one can
replace each $p(I)$ on these curves by its corresponding $p(\rho^{-1}(I))$ to obtain a similar pair of nested simple curves in the polygonal curve for the Grassmann necklace $\rmci_\mu$. One knows that Grassmann necklaces do not have such nested polygonal curves, see \cite[Section 9]{OPS}.

Denote by $D^{\rm in}_\gnec$ (resp. $D^{\rm out}_\gnec$) those $k$-element subsets $I$ which are weakly separated from $\gnec$ and whose corresponding point~$p(I)$ are weakly enclosed by (resp. weakly outside of) the curve $\zeta(\gnec)$.

\begin{lemma}\label{lem:FGintersect}
If $\mathcal{C}$ is a maximal weakly separated collection containing $\gnec$, then
$\mathcal{C} \cap D^{\rm in}_\gnec$ coincides with $\targetF(G^\rho)$ for
(reduced) relabeled plabic graph $G^\rho$ whose trip permutation is $\pi$.
In particular, if $I \in D^{\rm in}_\gnec$, then $I \in \mcm_\pi$.
\end{lemma}

\begin{proof}The first assertion follows from the results mentioned in \cref{rmk:FGinside}, working one simple curve at a time. The second statement about positroids follows from our main commutative diagram \cref{prop:faceLabelsTorus}.
\end{proof}

\begin{proof}[Proof of \cref{thm:alignedIsQuasiEquiv}]
We abbreviate $\gnec = \gnec_{\rho,\bullet,\pi} = (I_1,\dots,I_n)$ and  $\gnec' = \gnec_{\sigma,\bullet,\pi}$. We let $\gnec = (I_1,\dots,I_n)$ and suppose the toggle takes place in position~$j$, with $I_j' \in \gnec'$ the new $k$-subset that arises by performing the toggle.

We first rule out the case that $I_{j-1} = I_{j+1}$. If $\iota_j=\rho_j$, then every relabeled plabic graph with boundary $\rho$ has a white lollipop at $\rho_j$. In this case, it is easy to find relabeled plabic graphs $G^\rho$ and $H^{\rho'}$ whose target seeds are identical (just move the white lollipop appropriately). Similarly, the $\iota_{j-1}=\rho_{j-1}$ case is easy, so we may assume $\iota_{j-1}, \rho_{j-1}, \iota_{j}, \rho_{j}$ are distinct.

For appropriate cyclically ordered indices $u < v < w < x$ and an appropriate subset $S \in \binom{[n]}{k-2}$ we have $I_{j-1}=Suv$, $I_{j}=Svx$, $I_{j+1}=Swx$, and $I'_j = Suw$. As in \cref{rmk:togglePluckerRel}, there is a three-term Pl\"ucker relation involving the Pl\"ucker coordinates $I_j$, $I'_j$, $I_{j-1}$,$I_{j+1}$, and the two ``extra'' terms $Sux, Svw$. By \cite[Lemma 5.1]{OS}, because both $I_j$ and $I'_j$ are weakly separated with $\gnec \setminus \{I_j,I'_j\}$, the collection $\gnec \cup Sux,Svw$ is weakly separated.

We next observe that $I_j$ appears only once in $\gnec$. This follows from the assumption that $\gnec'$ is weakly separated because $I_j$ and $I'_j$ are not weakly separated. Thus the vertex $p(I_j)$ has exactly two neighbors on the polygonal curve $\zeta(\gnec)$ and is not the location of a self-intersection of this curve. It also implies that neither $Sux$ nor $Svw$ resides in $\gnec$.

Next let $\mathcal{C}$ be a maximal weakly separated collection containing $\gnec \cup Sux,Svw$. Note that $p(I_j)$ has exactly four neighbors in the plabic tiling for $\mathcal{C}$. Three of these are included in the picture \eqref{eq:localpicture} below. The missing neighbor is $p(Svw)$ sitting ``above'' $p(I_j)$ and connected to it by a vertical edge. For brevity we replace the symbol $p(I)$ by the symbol $I$ in this picture.

Let $\mathcal{C}' = \mathcal{C} \setminus I_j \cup I'_j$ be the result of performing a square move at $I_j \in \mathcal{C}$. Note that $\gnec' \subset \mathcal{C}'$. By \cref{lem:FGintersect}, one has reduced plabic graphs $(G')^\rho$ and
$(H')^\sigma$, defined by $D^{\rm in}_\gnec \cap \mathcal{C} = \targetF((G')^\rho)$ and $D^{\rm in}_{\gnec'} \cap \mathcal{C}' = \targetF((H')^\sigma)$.

We make the following {\sl key claim}: the seeds $\Sigma^T_{(G')^\rho}$ and
$\Sigma^T_{(H')^\sigma}$ are quasi-equivalent. Since $\Sigma^T_{(G')^\rho}$ is mutation-equivalent to $\Sigma^T_{G^\rho}$ and $\Sigma^T_{(H')^\sigma}$ is mutation-equivalent to  $\Sigma^T_{H^\sigma}$, we would then have that
$\Sigma^T_{G^\rho}$ and $\Sigma^T_{H^\sigma}$ are related by a quasi-cluster transformation, completing the proof.

Let us establish the key claim. The coefficients group of these two seeds coincide by the unit necklace theorem. Their sets of mutable variables coincide, since the two plabic graphs differ only in their $j$th boundary face. What needs to be checked is the equality of exchange ratios in the two seeds.

It is not hard to see that every edges of $\zeta(\gnec)$ is either an edge of the plabic tiling for~$\mathcal{C}$ or cuts across a face of this tiling. For topological reasons, it follows that exactly one of the two terms in $Sux$ and $Svw$ resides in $D^{\rm in}_\gnec$. Suppose for concreteness that this is true of $Sux$. By the proof of \cref{prop:monomialtransform}, the other term $Svw \notin \mcm_\pi$. Recall that $Sux \notin \gnec$, thus it is a mutable variable in $\Sigma^T_{(G')^\rho}$.

Performing the square move at $I_j$ only changes the edges of the tiling involving $p(I_j)$ and its four neighbors. Of these, only $Sux$ is a mutable variable and has an exchange ratio. We obtain the seed $\Sigma^T_{(G')^\rho}$ by restricting the edges from the plabic tiling for $\mathcal{C}$ to the elements of $D^{\rm in}_\gnec $ and ignoring any arrows between elements in $\gnec$. The equality of exchange ratios is now a simple local check using the change of coordinates $\Delta_{I_j'} = \frac{\Delta_{I_{j-1}} \Delta_{I_{j+1}}}{\Delta_{I_j}}$ \eqref{eq:monomialtransform} which arises when performing the toggle, see the picture below.

	\begin{equation}\label{eq:localpicture}
		\begin{tikzpicture}
		\node (a)at (0,0) {$I_j$};
		\node (c) at (0,-1.5) {$Sux$};
		\node (d) at (1.5,0) {$I_{j+1}$};
		\node (e) at (-1.5,0) {$I_{j-1}$};
		\draw [->] (e)--(a);
		\draw [->] (d)--(a);
		\draw [->] (a)--(c);
		\draw[dotted] [->] (c)--(d);
		\draw[dashed] [->] (c)--(e);
		\begin{scope}[xshift = 5cm]
		\node (a)at (0,0) {$\frac{I_{j-1}I_{j+1}}{I_j}$};
		\node (c) at (0,-1.5) {$Sux$};
		\node (d) at (1.5,0) {$I_{j+1}$};
		\node (e) at (-1.5,0) {$I_{j-1}$};
		\draw [->] (a)--(e);
		\draw [->] (a)--(d);
		\draw [->] (c)--(a);
		\draw[dotted] [->] (d)--(c);
		\draw[dashed] [->] (e)--(c);
		\end{scope}
		\end{tikzpicture}
	\end{equation}
Here, the dotted ( resp. dashed) arrow is present on the left if and only if it is not present on the right. There may be other arrows between $Sux$ and other vertices, but these arrows in the same in both seeds. The calculation is not affected by simultaneously reversing all arrows in both pictures.
\end{proof}


TITLE: Prove: $h \in T_xM \iff \operatorname{dist}(x+\epsilon h,M) = o(\epsilon)$
QUESTION [2 upvotes]: Prove: $h \in T_xM \iff \operatorname{dist}(x+\epsilon h,M) = o(\epsilon)$
For $M$ a smooth manifold in $\mathbb{R}^n$, $h\in\mathbb{R}^n$, $x \in M$. I know that $T_xM = \operatorname{Im}(Df) = \ker(Dg) $ where $f$, $g$ are a chart and co-chart at point $x$, respectively. I just don't see how I could relate the definition to what I need to prove.
Any help would be appreciated!

REPLY [4 votes]: Here you go:
$ " \Rightarrow " $ Let $\gamma: \mathbb{R} \to M$ be a curve s.t. $\gamma(0)=x$ and $\gamma'(0)=h$. You can always find such a curve. Now you have:
$$ \operatorname{dist}(x+\epsilon h,M) \leq \operatorname{dist}(x+\epsilon h,\gamma(\epsilon))  $$
So what we want to show is that
$$ \lim_{\epsilon \to 0} \frac{\operatorname{dist}(x+\epsilon h,\gamma(\epsilon)) }{\epsilon}=0 $$
We can rewrite
\begin{eqnarray}
\frac{\operatorname{dist}(x+\epsilon h,\gamma(\epsilon)) }{\epsilon}&=&\frac{\Vert\gamma(\epsilon)-\epsilon\gamma'(0)-\gamma(0)  \Vert }{\epsilon}\\
&\leq & \left\Vert\frac{\gamma(\epsilon)-\gamma(0)   }{\epsilon}- \gamma'(0) \right\Vert
\end{eqnarray}
 Now letting $\epsilon \to 0$ we obtain that
$$ \lim_{\epsilon \to 0}\frac{\gamma(\epsilon)-\gamma(0)   }{\epsilon}= \gamma'(0)  $$
This shows our claim.
"$\Leftarrow$" We want to show that $h \in \ker(Dg_x)$. Choose a domain of definition $V \subset U$ for $g$ s.t. $\overline{V}\subset U$ and $\overline{V}$ is compact in $U$. By this we can assume that $g$ is Lipschitz continous with Lipschitz constant $L$.
Now let $\gamma(\epsilon):=x+\epsilon h$, notice that $\gamma(0)=x$ and $\gamma'(0)=h$. Furthermore let $\phi:\mathbb{R}\to M$ be the function that associates to $\epsilon$ one of the $y \in \overline{V} \cap M$ with $\operatorname{dist}(\gamma(\epsilon),M)=\operatorname{dist}(\gamma(\epsilon),y)$. Notice that as long as we restrict ourselfs to $\epsilon$ that are small enough this is well defined (You maybe want your $\epsilon$ so small that there is a ball $B$ around $\gamma(\epsilon)$ which contains $x$ and s.t. $B\subset V$ .). Also note that $\phi$ is not continous but just a function.
Now notice that $g(\gamma(0))=g(\phi(\epsilon))=0$. By the chain rule one gets:
\begin{eqnarray}
\Vert Dg_x(h) \Vert &=& \Vert Dg_x(\gamma'(0))\Vert\\
&=& \left\Vert \frac{d}{d\epsilon}g(\gamma(\epsilon))_{\vert\epsilon=0}\right\Vert\\
&=& \lim_{\epsilon\to 0}\left\Vert \frac{g(\gamma(\epsilon))-g(\gamma(0))}{\epsilon} \right\Vert\\
&=&\lim_{\epsilon\to 0}\frac{1}{\epsilon}\left\Vert g(\gamma(\epsilon))-g(\phi(\epsilon)) \right\Vert\\
&\leq&\lim_{\epsilon\to 0}\frac{1}{\epsilon} L \left\Vert \gamma(\epsilon)-\phi(\epsilon) \right\Vert\\
&=& L\lim_{\epsilon\to 0}\frac{\operatorname{dist}(\gamma(\epsilon),M)}{\epsilon}
\end{eqnarray}
But this is zero by assumption. So all in all we get $Dg_x(h)=0$ which finishes the proof.


TITLE: Minimum value of $S=\frac{3a}{b+c}+\frac{4b}{a+c}+\frac{5c}{a+b}$
QUESTION [3 upvotes]: Find Minimum value of Minimum value of $$S=\frac{3a}{b+c}+\frac{4b}{a+c}+\frac{5c}{a+b}$$
My Try:
we have $$S=3 \times \left (\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\right)+\frac{b}{a+c}+\frac{2c}{a+b}$$
By Standard $AM$ $GM$ inequality we know that
$$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b} \ge \frac{3}{2}$$
Is it possible to proceed here?

REPLY [8 votes]: For any positive numbers $\alpha, \beta, \gamma$, we have
$$\frac{\alpha^2 a}{b+c} + \frac{\beta^2 b}{c+a} + \frac{\gamma^2 c}{a+b}
= (a+b+c)\left(\frac{\alpha^2}{b+c} + \frac{\beta^2}{c+a} + \frac{\gamma^2}{a+b}\right) - \left(\alpha^2+\beta^2+\gamma^2\right)$$
By Engel's form of Cauchy Schwarz, we have
$$\frac{\alpha^2}{b+c} + \frac{\beta^2}{c+a} + \frac{\gamma^2}{a+b}
\ge \frac{(\alpha+\beta+\gamma)^2}{(b+c)+(c+a)+(c+a)}
= \frac{(\alpha+\beta+\gamma)^2}{2(a+b+c)}
$$
This leads to
$$\frac{\alpha^2 a}{b+c} + \frac{\beta^2 b}{c+a} + \frac{\gamma^2 c}{a+b}
\ge \frac{(\alpha+\beta+\gamma)^2}{2} - \left(\alpha^2 + \beta^2 + \gamma^2\right)$$
and the equality holds when and only when
$$\alpha : \beta : \gamma = b+c : c+a : a+b$$
Subsitute $\alpha,\beta,\gamma$ by $\sqrt{3},\sqrt{4}$ and $\sqrt{5}$
We obtain
$$\frac{3 a}{b+c} + \frac{4 b}{c+a} + \frac{5 c}{a+b}
\ge \frac{(\sqrt{3} + \sqrt{4} + \sqrt{5})^2}{2} - 12
= \sqrt{15} + \sqrt{20} + \sqrt{12} - 6$$
and the equality is achievable when
$$a : b : c = -\sqrt{3}+\sqrt{4}+\sqrt{5} : \sqrt{3} - \sqrt{4} + \sqrt{5} : \sqrt{3}+\sqrt{4} - \sqrt{5}$$
This means the minimum value of $S$ is $\sqrt{15} + \sqrt{20} + \sqrt{12} - 6$.


TITLE: Characteristic functions in a CW complex are related by a homeomorphism
QUESTION [2 upvotes]: The book "The Topology of CW Complexes" mentions the following result in passing, but does not provide a proof. Let $(X, \mathcal{C})$ be a CW complex, $B_k \subset \mathbb{R}^k$ be the origin-centered open unit-$k$-ball, and $\overline{B_k}$ be the corresponding closed $k$-ball. Let $f, g : \overline{B_k} \to X$ both be characteristic functions of the same cell $C \in \mathcal{C}$. Then there exists a homeomorphism $h : \overline{B_k} \to \overline{B_k}$ such that $f = g \circ h$. How can you prove this statement?
For a regular cell, the statement is of course trivial, so I am specifically interested in the non-regular case.
EDIT
In the light of the counter-example given by Eric Wofsey, I may have misunderstood what is said in the book. The direct quotes are as follows ($\dot{E}_n$ is the boundary of a closed cell $E_n$).
BEGIN QUOTES
Page 1:
A closed Euclidean $n$-cell $E^n$ is a homeomorphic image of the Euclidean n-cube $I^n$, the cartesian product of $n$ copies of the closed unit interval $I = \{t \in \mathbb{R} : 0 \leq t \leq 1\}$.
Page 6:
Let $X$ be a set. A cell structure on $X$ is a pair $(X, \Phi)$,
where $\Phi$ is a collection of maps of closed Euclidean cells into $X$
satisfying the following conditions.

If $\phi \in \Phi$ and $\phi$ has domain $E_n$, then $\phi$ is injective on $E_n - \dot{E}_n$.
The images $\{\phi(E_n - \dot{E}_n) : \phi \in \Phi\}$ partition $X$, i.e., they are disjoint and have union $X$.
If $\phi \in \Phi$ has domain $E_n$, then $\phi(\dot{E}_n) \subset \bigcup \{\psi(E_k - \dot{E}_k) : \psi \in \Phi \text{ has domain } E_k \text{ and } k \leq n - 1\}$

Page 7:
We say that two cell structures $(X, \Phi)$ and $(X, \Phi')$ are strictly equivalent if there is a one-to-one correspondence between $\Phi$ and $\Phi'$ such that a characteristic function with domain $E_n$ corresponds to a characteristic function with domain $E_n$, and corresponding functions differ only by a reparametrization of their domain. That is, if $\phi$ and $\phi'$ are corresponding functions of $\Phi$ and $\Phi'$, respectively, then $\phi' = \phi \circ h$, where $h: (E_n, \dot{E}_n) \to (E_n, \dot{E}_n)$ is a homeomorphism of pairs. We leave it to the reader to check that this is an equivalence relation on the collection of cell structures on the set X.
If $(X, \Phi)$ is a cell structure, let $\mathcal{S}_\Phi$ consist of all pairs $(\sigma^n, [\phi])$, where $\sigma^n = \phi(E^n)$ and $[\phi]$ is the strict equivalence class of $\phi \in \Phi$. For convenience, we will denote such a pair by $\sigma^n$ or $\phi(E^n)$ with the class of $\phi$ understood. Note that if $(X, \Phi)$ and $(X, \Phi')$ are strictly equivalent cell structures, then $\mathcal{S}_\Phi = \mathcal{S}_{\Phi'}$.
Page 8:
A cell complex on a set $X$ or a cellular decomposition
of a set $X$ is an equivalence class of cell structures $(X, \Phi)$ under
the equivalence relation of strict equivalence.
A cell complex on $X$ will be denoted by a pair $(X, \mathcal{S})$, where $\mathcal{S} = S_{\Phi}$ for some representative cell structure $(X, \Phi)$. The set $\mathcal{S}$ is called the set of (closed) cells of $(X, \mathcal{S})$.
Page 41:
A Hausdorff space $X$ is a CW complex with respect to a family of cells $\mathcal{S}$ provided:

the pair $(X,\mathcal{S})$ is a cell complex such that each cell $\sigma \in \mathcal{S}$ has a continuous characteristic function;
the space $X$ has the weak topology with respect to $\mathcal{S}$;
the cell complex $(X,\mathcal{S})$ is closure finite.

END QUOTES
I'm not sure where my misunderstanding is, but I suspect it has to do with that the strict equivalence is defined between cell-structures and not between characteristic functions.
EDIT 2
Referring to the discussions with Paul Frost below, the confusion arises because the definition in the book is more granular than the usual definition of a CW complex. Therefore, if characteristic functions are not related by a homeomorphism in this definition, then they are not the same CW complex.
In contrast, the usual definition of a CW complex requires only the existence of a characteristic function, which makes it possible to provide non-homeomorphically related characteristic functions for the same CW complex, as provided by the counter-example in the accepted answer.

REPLY [2 votes]: This is false.  If $k>1$, then there exists a map $f:\overline{B_k}\to\overline{B_k}$ which restricts to a homeomorphism $B_k\to B_k$ and maps $\partial B_k$ to itself but which is not injective on $\partial B_k$.  So, taking $X=\overline{B_k}$ with a CW-complex structure that has $B_k$ as a $k$-cell, both $f$ and $g=1_{\overline{B_k}}$ are possible characteristic functions for the $k$-cell.  Since $f$ is not a homeomorphism, there cannot exist any homeomorphism $h:\overline{B_k}\to\overline{B_k}$ such that $f=g\circ h$.
For an explicit example of such an $f$ for $k=2$, let $S=[0,1]\times[0,1]$ and $T=\{(x,y)\in\mathbb{R}^2:0\leq y\leq x\leq 1\}$ and define $f_0:S\to T$ by $f_0(x,y)=(x,xy)$.  Then $f_0$ is a homeomorphism on the interiors of $S$ and $T$ but collapses one of the sides of $S$ to a single point.  Composing with homeomorphisms $S,T\cong \overline{B_2}$ we get $f:\overline{B_2}\to\overline{B_2}$ which is a homeomorphism on the interior but collapses an arc on the boundary to a point.


TITLE: How to find the Laplace transform of $f(t)=t e^t$?
QUESTION [3 upvotes]: How to find the Laplace transform of the following function:
$$f(t)=t e^t$$
$$F(s)=\int_0^\infty (te^t e^{-st})dt$$
What method do I use to find the integral?

REPLY [4 votes]: To do it without integral (as in my comment in your other question), using properties of LT:

$$tf(t)\stackrel{\mathcal{L}}\longleftrightarrow-\frac{\mathrm d}{\mathrm ds}F(s)$$

Here $f(t)=e^t$ whose Laplace transform is $F(s)=\frac{1}{s-1}$.
The Laplace transform of $te^t$ becomes $$-\frac{\mathrm d}{\mathrm ds}\left(\frac{1}{s-1}\right)=\frac{1}{(s-1)^2}$$


TITLE: Where did I make a mistake in this "proof" that no complex structure $I$ can be parallel on a Kahler manifold.
QUESTION [1 upvotes]: Let $(M,I,h)$ be a Kahler manifold. We know that the complex structure $I$ must be parallel in the sense that $\nabla I = 0$, where $\nabla$ is the Levi-Civita connection on $M$.
I have understood this as follows: for every $Z,W \in  \mathcal{C}^{\infty}(TM) $, we have
$$
(\nabla_{Z} I)W = \nabla_{Z} (IW) = 0.
$$
But for every such $ Z $ and $  W $, we may take $ X = -IW $ such that $ IX = -I^{2}W = W $, $ X \in \mathcal{C}^{\infty}(TM) $, such that
$$ (\nabla_{Z}I)X = \nabla_{Z}(IX) = \nabla_{Z}W = 0. $$
Therefore $\nabla$ vanishes everywhere on $\mathcal{C}^{\infty}(TM) \times \mathcal{C}^{\infty}(TM) $.
What is the dumb thing that I am not seeing or understanding wrong?
Many thanks in advance.

REPLY [1 votes]: I think I noticed where my mistake is. In fact, we have
$$ \nabla I = [\nabla, I] $$
over $ I \in \text{End}(TM) $.


TITLE: Non-completeness of the space of bounded linear operators
QUESTION [5 upvotes]: If $X$ and $Y$ are normed spaces I know that the space $B(X,Y)$ of bounded linear functions from $X$ to $Y$,  is complete if $Y$ is complete.  Is there an example of a pair of normed spaces $X,Y$ s.t. $B(X,Y)$ is not complete?

REPLY [4 votes]: Let $X = \mathbb{R}$ with the Euclidean norm and let $Y$ be a normed space which is not complete.  You should find that $B(X, Y) \simeq Y$.


text:
\begin{document}

\begin{abstract}
We investigate a new family of locally harmonic Maass forms which correspond to periods of modular forms. They transform like negative weight modular forms and are harmonic apart from jump singularities along infinite geodesics. Our main result is an explicit splitting of the new locally harmonic Maass forms into a harmonic part and a locally polynomial part
that captures the jump singularities. As an application, we obtain finite rational formulas for suitable linear combinations of periods of meromorphic modular forms associated to positive definite binary quadratic forms.
\end{abstract}

\maketitle

\section{Introduction and statement of results}

\subsection{Locally harmonic Maass forms and cycle integrals} In the early 2000s, Zwegers \cite{zwegers} made the groundbreaking discovery that Ramanujan's mock theta functions, whose precise automorphic nature had been a long-standing conundrum, could be viewed as the holomorphic parts of \emph{harmonic weak Maass forms} of weight $1/2$ whose shadows are unary theta functions of weight $3/2$. The theory of harmonic weak Maass forms was developed systematically around the same time by Bruinier and Funke \cite{bruinierfunke}. Since then, it has become a vital area of research in number theory and has found many fascinating applications, for example to the singular theta correspondence and Borcherds products \cite{bruinierhabil, bruinierfunke}, the partition function and its variants \cite{ahlgrenandersen, bruinieronoalgebraic}, special values of $L$-functions of elliptic curves \cite{bruinieronoheegner}, and CM values of higher Green functions \cite{bringmannkanevonpippich, bruinierehlenyang, li}.

More recently, Bringmann, Kane, and Kohnen \cite{bringmannkanekohnen} constructed a new type of harmonic weak Maass forms which have jump singularities along certain geodesics in the upper half-plane $\H$, and hence are called \emph{locally harmonic Maass forms} (see also \cite{brikavia, crawford, crawfordfunke, hoevel}). Specifically, for $k \in \Z$ with $k \geq 2$ the authors of \cite{bringmannkanekohnen} associated to each non-square discriminant $D > 0$ the function ($\tau = u+iv \in \H$)
\[
\mathcal{F}_{1-k,D}(\tau) := \frac{(-1)^k D^{\frac{1}{2}-k}}{\binom{2k-2}{k-1}\pi}\sum_{Q = [a,b,c]\in \mathcal{Q}_{D}}\sgn\left(a|\tau|^{2}+bu+c\right)Q(\tau,1)^{k-1}\psi\left(\frac{D^{2}v}{|Q(\tau,1)|^{2}} \right),
\]
where $\mathcal{Q}_{D}$ denotes the set of all (positive definite if $D < 0$) integral binary quadratic forms $Q(x,y) = ax^{2}+bxy + cy^{2}$ of discriminant $D = b^{2}-4ac$, and $\psi(v) := \frac{1}{2}\int_{0}^{v}t^{k-\frac{3}{2}} (1-t)^{-\frac{1}{2}}dt$ is a special value of the incomplete $\beta$-function. The function $\mathcal{F}_{1-k,D}(\tau)$ transforms like a modular form of negative even weight $2-2k$ for $\Gamma := \SL_{2}(\Z)$, it is bounded at the cusp, and it is harmonic on $\H$ up to jump singularities along the exceptional set
\[
E_{D} :=\{\tau=u+iv \in \H \,:\, a|\tau|^{2}+bu+c = 0 \, , \, [a,b,c] \in \mathcal{Q}_{D}\}.
\]
Note that $E_{D}$ is a union of semi-circles centered at the real line if $D > 0$ is not a square. A special feature of the locally harmonic Maass form $\mathcal{F}_{1-k,D}(\tau)$ is the fact that its images under the two differential operators
\[
\xi_{2-2k} := 2iv^{2-2k}\overline{\frac{\partial}{\partial \overline{\tau}}}, \qquad \mathcal{D}^{2k-1} := \left(\frac{1}{2\pi i}\frac{\partial}{\partial \tau} \right)^{2k-1},
\]
are non-zero multiples of the weight $2k$ cusp form
\begin{align}\label{eq fkD}
f_{k,D}(\tau) := \frac{D^{k-\frac{1}{2}}}{\pi}\sum_{Q \in \mathcal{Q}_{D}}Q(\tau,1)^{-k}.
\end{align}
In contrast, it is impossible for a harmonic weak Maass form to map to a non-zero multiple of the same cusp form under both operators $\xi_{2-2k}$ and $\mathcal{D}^{2k-1}$. The cusp form $f_{k,D}(\tau)$ can be characterized by the fact that the Petersson inner product $\langle f,f_{k,D}\rangle$ of a cusp form $f(\tau)$ of weight $2k$ with $f_{k,D}(\tau)$ is a certain multiple of the $D$-th trace of cycle integrals
\[
\tr_{D}(f) := \sum_{Q \in \mathcal{Q}_{D}/\Gamma}\int_{\Gamma_{Q}\backslash S_{Q}}f(z)Q(z,1)^{k-1}dz
\]
of $f(\tau)$. Here $\Gamma_{Q}$ denotes the stabilizer of $Q$ in $\Gamma$ and $S_{Q}$ for $Q = [a,b,c]$ is the semi-circle consisting of all $\tau = u+iv \in \H$ with $a|\tau|^{2}+bu+c = 0$. In this sense, the locally harmonic Maass forms $\mathcal{F}_{1-k,D}(\tau)$ correspond to the (traces of) cycle integrals of cusp forms.

\subsection{Locally harmonic Maass forms and periods} In the present article we are concerned with the construction of a new family of locally harmonic Maass forms $\mathcal{H}_{1-k,n}(\tau)$, for $k \in \Z$ with $k \geq 2$ and integral $0 \leq n \leq 2k-2$, which correspond to the \emph{periods} of cusp forms, in a sense which will become apparent in a moment. In order to give the definition of $\mathcal{H}_{1-k,n}(\tau)$, for $k,\ell \in \Z$ with $k\geq 2$ we consider Petersson's Poincar\'e series ($z, \tau = u+iv \in \H$)
\begin{align}\label{eq Peterssons Poincare series}
H_{k,\ell}(z,\tau) := \sum_{M \in \Gamma}v^{k+\ell}(z-\tau)^{\ell-k}(z-\overline{\tau})^{-\ell-k}\Bigl|_{-2\ell,\tau}M,
\end{align}
with the usual slash operator applied in the $\tau$-variable. It transforms like a modular form of weight $2k$ in $z$ and of weight $-2\ell$ in $\tau$. We will be particularly interested in the case $\ell = k-1$. Then the function $H_{k,k-1}(z,\tau)$ is a meromorphic modular form of weight $2k$ in $z$ which has poles precisely at the $\Gamma$-translates of $\tau$ and decays like a cusp form towards $i\infty$. Furthermore, it transforms like a modular form of weight $2-2k$ in $\tau$ and is harmonic on $\H \setminus \Gamma z$. We can now make the following definition.

\begin{definition}
	For $k \in \Z$ with $k \geq 2$ and integral $0 \leq n \leq 2k-2$ we define the function
	\begin{align}\label{definition curlyH}
	\mathcal{H}_{1-k,n}(\tau):= \frac{(2i)^{2k-2}}{2\pi}\int_{0}^{\infty}H_{k,k-1}(iy,\tau)y^{n}dy.
	\end{align}
	If $\tau \in \bigcup_{M \in \Gamma}M(i \R_{+})$ then the integral is defined using the Cauchy principal value (see \cite{ls}, Section~3.5).
\end{definition}

In other words, the function $\mathcal{H}_{1-k,n}(\tau)$ is essentially the $n$-th period of the meromorphic modular form $z \mapsto H_{k,k-1}(z,\tau)$. In \cite{ls} we showed that the locally harmonic Maass form $\mathcal{F}_{1-k,D}(\tau)$ from \cite{bringmannkanekohnen} can be viewed as the $D$-th trace of cycle integrals of the function $z \mapsto H_{k,k-1}(z,\tau)$, which inspired the above definition of $\mathcal{H}_{1-k,n}(\tau)$.

It is clear from the construction that $\mathcal{H}_{1-k,n}(\tau)$ transforms like a modular form of weight $2-2k$ for $\Gamma$ and is harmonic on $\H \setminus\bigcup_{M \in \Gamma}M(i \R_{+})$. Further, it follows from Theorem~\ref{theorem splitting} below that $\mathcal{H}_{1-k,n}(\tau)$ is of moderate growth at the cusp and has jump singularities along the $\Gamma$-translates of the positive imaginary axis $i\R_{+}$. In particular, the function $\mathcal{H}_{1-k,n}(\tau)$ defines a locally harmonic Maass form of weight $2-2k$ for $\Gamma$ with exceptional set $E_{1} = \bigcup_{M \in \Gamma}M(i\R_{+})$ in the sense of \cite{bringmannkanekohnen}.

\begin{figure}[h]\label{figure}
  \centering
    \includegraphics[width=0.95\textwidth]{singularities.pdf}
		\caption{The set $E_1$ of singularities of $\mathcal{H}_{1-k,n}$.}
\end{figure}

We first determine the images of $\mathcal{H}_{1-k,n}(\tau)$ under the differential operators $\xi_{2-2k}$ and $\mathcal{D}^{2k-1}$. To state the result, we let $E_{2k}(\tau)$ be the usual normalized Eisenstein series of weight $2k$, and for $0 \leq n \leq 2k-2$ we let $R_{n}(\tau)$ be the weight $2k$ cusp form which is characterized by the fact that its Petersson inner product $\langle f,R_{n} \rangle$ with a cusp form $f(\tau)$ of weight $2k$ equals the $n$-th period
\begin{align}\label{definition periods}
r_{n}(f) := \int_{0}^{\infty}f(iy)y^{n}dy
\end{align}
of $f(\tau)$. Note that the periods satisfy the symmetry $r_n(f) = (-1)^k r_{2k-2-n}(f)$. The cusp form $R_n(\tau)$ is related to $\mathcal{H}_{1-k,n}(\tau)$ in the following way.

\begin{proposition}\label{proposition diffops} For $\tau \in \H \setminus \bigcup_{M \in \Gamma}M(i\R_{+})$ we have
\begin{align*}
\xi_{2-2k}\mathcal{H}_{1-k,n}(\tau) &= -R_{n}(\tau), \\
\mathcal{D}^{2k-1}\mathcal{H}_{1-k,n}(\tau) &= (-1)^{n+1}\frac{(2k-2)!}{(4\pi)^{2k-1}}R_{n}(\tau)  -\left((-1)^k\delta_{n = 0}+\delta_{n=2k-2}\right)\frac{(2k-2)!}{(2\pi)^{2k-1}}E_{2k}(\tau).
\end{align*}
\end{proposition}

We refer the reader to Section~\ref{section proof proposition diffops} for the proof of Proposition~\ref{proposition diffops}. The above proposition implies that $\mathcal{H}_{1-k,n}(\tau)$ can be written as a sum of the holomorphic and non-holomorphic Eichler integrals of $R_{n}(\tau)$ (and of $E_{2k}(\tau)$ if $n = 0$ or $n = 2k-2$) and a locally polynomial part, which captures the singularities of $\mathcal{H}_{1-k,n}(\tau)$. Recall that the holomorphic and non-holomorphic Eichler integrals of a cusp form $f(\tau) = \sum_{n=1}^{\infty}c_{f}(n)e(n\tau)$ of weight $2k$ (with $e(x):= e^{2\pi i x}$ for $x \in \C$) are defined by
\begin{align}\label{eq eichler integrals}
\begin{split}
\mathcal{E}_{f}(\tau) &:= \frac{(-2\pi i)^{2k-1}}{(2k-2)!}\int_{\tau}^{i\infty}f(z)(z-\tau)^{2k-2}dz= \sum_{n\geq 1}\frac{c_{f}(n)}{n^{2k-1}}e(n\tau), \\
f^{*}(\tau) &:= (-2i)^{1-2k}\int_{-\overline{\tau}}^{i\infty}\overline{f(-\overline{z})}(z+\tau)^{2k-2}dz = -\sum_{n\geq 1}\frac{\overline{c_{f}(n)}}{(4\pi n)^{2k-1}}\Gamma(2k-1, 4\pi nv)e(-n\tau),
\end{split}
\end{align}
where $\Gamma(s,x):=\int_{x}^\infty e^{-t} t^{s-1}dt$ is the incomplete Gamma function. They satisfy
\begin{align}\label{eq diffops eichler}
\xi_{2-2k}f^{*}(\tau) = f(\tau), \qquad \mathcal{D}^{2k-1}f^{*}(\tau) = 0, \qquad \xi_{2-2k}\mathcal{E}_{f}(\tau) = 0, \qquad \mathcal{D}^{2k-1}\mathcal{E}_{f}(\tau) = f(\tau).
\end{align}
By using the series expansions on the right-hand sides of \eqref{eq eichler integrals}, we can extend the definitions of the Eichler integrals to the Eisenstein series $E_{2k}(\tau)$. The main result of this work is the following splitting of $\mathcal{H}_{1-k,n}(\tau)$.

\begin{theorem}\label{theorem splitting}
We have the decomposition
\begin{align}\begin{split}\label{split}
\mathcal{H}_{1-k,n}(\tau) &= \mathcal{P}_{1-k,n}(\tau) + (-1)^{n+1}\frac{(2k-2)!}{(4\pi )^{2k-1}}\mathcal{E}_{R_n}(\tau) - R_{n}^*(\tau) \\
 &\quad-((-1)^k\delta_{n=0}+\delta_{n=2k-2})\frac{(2k-2)!}{(2\pi)^{2k-1}}\mathcal{E}_{E_{2k}}(\tau),
 \end{split}
\end{align}
where $\mathcal{P}_{1-k,n}(\tau)$ is locally a polynomial on the connected components of $\H \setminus \bigcup_{M \in \Gamma}M(i\R_{+})$. It is explicitly given for $\tau\in\H \setminus \bigcup_{M \in \Gamma}M(i\R_{+})$ by
\begin{align*}
\mathcal{P}_{1-k,n}(\tau) &= r_n(E_{2k}) +\frac{(-i)^{n+1}}{n+1}\mathbb{B}_{n+1}(\tau)+\frac{i^{n+1}}{2k-1-n}\mathbb{B}_{2k-1-n}(\tau)
 \\
 &\quad +\frac{i^{1-n}}{2}\sum_{\substack{M = \left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix}\right)\in\Gamma \\ ac>0, \Re(M\tau)< 0}}(\tau^n -(-1)^n \tau^{2k-2-n})\Bigl |_{2-2k}M.
\end{align*}
Here $\mathbb{B}_{m}(\tau)$ is the $1$-periodic function on $\H$ which agrees with the Bernoulli polynomial $B_{m}(\tau)$ for $0 < u < 1$ and
$r_{n}(E_{2k})$ is the $n$-th period of the Eisenstein series $E_{2k}(\tau)$ given in \eqref{periodEisenstein}. \\
If there is a matrix $M\in\Gamma$ with $\Re(M\tau)=0$, then $\mathcal{P}_{1-k,n}(\tau)$ is given by the average value
\[
\lim_{\varepsilon\rightarrow 0+}\frac{1}{2}\big(\mathcal{P}_{1-k,n}(M^{-1}(M\tau - \varepsilon))+\mathcal{P}_{1-k,n}(M^{-1}(M\tau + \varepsilon))\big).
\]
\end{theorem}

\begin{remark}Note that the sum in the second line of $\mathcal{P}_{1-k,n}(\tau)$ is finite for every $\tau\in\H$ and vanishes if $\Im(\tau) > \frac{1}{2}$. Indeed, the condition $ac > 0$ for $M = \left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix}\right) \in \Gamma$ implies that $M^{-1}(i\R_+)$ is a semi-circle of radius $\frac{1}{2ac}$ centered at the real line, and $\Re(M \tau) < 0$ means that $\tau$ lies in the bounded component of $\H\setminus M^{-1}(i\R_+)$. See also Figure 1.
\end{remark}

Our proof of the above theorem is quite different from the proof of the analogous splitting of the locally harmonic Maass form $\mathcal{F}_{1-k,D}(\tau)$ given in \cite{bringmannkanekohnen}. As the main step, we will employ an explicit formula for the period polynomial of $R_{n}(\tau)$ due to Kohnen and Zagier \cite{kohnenzagierrationalperiods} in order to directly show the modularity of the expression on the right-hand side of \eqref{split}. We refer the reader to Section~\ref{section proof theorem splitting} for the details of the proof.

 \subsection{Periods of meromorphic modular forms} As an application of the above results, we study the periods of certain meromorphic modular forms. Namely, for a fixed positive definite quadratic form $P \in \mathcal{Q}_{d}$ of discriminant $d < 0$ we consider the function
 \begin{align}\label{definition fkP}
 f_{k,P}(z):= \frac{|d|^{k-\frac{1}{2}}}{\pi}\sum_{Q \in [P]}Q(z,1)^{-k},
 \end{align}
 where $[P]$ denotes the class of $P$ in $\mathcal{Q}_{d}/\Gamma$. We let
 \[
 f_{k,d}(z) := \sum_{P \in \mathcal{Q}_{d}/\Gamma}f_{k,P}(z) = \frac{|d|^{k-\frac{1}{2}}}{\pi}\sum_{Q \in \mathcal{Q}_{d}}Q(z,1)^{-k}.
 \]
 Notice the analogy with the definition of the cusp form $f_{k,D}(\tau)$ in \eqref{eq fkD} for $D > 0$. The function $f_{k,P}(z)$ transforms like a modular form of weight $2k$ for $\Gamma$ and decays like a cusp form towards $i\infty$. Moreover, it is meromorphic on $\H$ and has poles of order $k$ precisely at the $\Gamma$-translates of the CM point $\tau_{P}$ defined by $P(\tau_{P},1) = 0$.

 It was shown in \cite{anbs, anbms, ls} that certain linear combinations of geodesic cycle integrals of $f_{k,P}(z)$ are rational. Motivated by these results, in the present work we investigate the rationality of the periods
 \[
 r_{n}(f_{k,P}) = \int_{0}^{\infty}f_{k,P}(iy)y^{n}dy
 \]
 for $0 \leq n \leq 2k-2$, where the integral is defined using the Cauchy principal value as in \cite{ls}, Section~3.5, if a pole of $f_{k,P}$ lies on the positive imaginary axis. To get a first idea how these periods look, we consider the quadratic form $P = [1,1,1]$ of discriminant $d=-3$. In this case, there is only one class in $\mathcal{Q}_{-3}/\Gamma$, so we have $f_{k,[1,1,1]} = f_{k,-3}$. Numerical integration yields the following values of the periods of $f_{k,-3}$ for small values of $k$.
\begin{align*}
	 \renewcommand{\arraystretch}{1.2}
	\begin{array}{|c||c|c|c|c|c|c|c|}
	\hline
	n & 0 & 1 & 2 & 3 & 4 & 5 & 6\\
	 \hline \hline
	 r_n(f_{2,-3}) & -2.05670 & -2 & -2.05670 & - & - & - & -\\
	 \hline
	 r_n(f_{3,-3}) & -3.25653 & -1.5 & 0 & 1.5 & 3.25652 & - &  -\\
	\hline
	r_n(f_{4,-3}) & -6.76949 & -2.22222 & 0 & 0.66666 & 0 & -2.22222 & -6.76949\\
	\hline
	r_n(f_{5,-3}) &  -15.65457 & -4.08333 &0 & 0.66666 & 0 & -0.66666 & 0 \\
	\hline
	r_n(f_{6,-3}) &  -38.31573 & -8.36729 &0 & 0.89204 & 0 & -0.48637 & 0\\
	\hline
	r_n(f_{7,-3}) &  -97.17273 & -18.33333 &0 & 1.4 & 0 & -0.5 & 0\\
	\hline
	\end{array}
\end{align*}
We notice that the even periods of $f_{k,-3}(z)$ for $0 < n < 2k-2$ seem to vanish and the odd periods for $k \neq 6$ appear to be rational numbers, but the outer periods $r_{0}(f_{k,-3})$ and $r_{2k-2}(f_{k,-3})$ and the odd periods of $f_{6,-3}(z)$ do not seem to be particularly nice. In order to state the result explaining these observations, following \cite{kohnenzagierrationalperiods} we introduce the functions
\[
f_{k,P}^{+}(z) := \frac{1}{2}\left(f_{k,P}(z)+f_{k,P'}(z)\right), \qquad f_{k,P}^{-}(z):= \frac{i}{2}\left(f_{k,P}(z)-f_{k,P'}(z) \right),
\]
where we put $P' := [a,-b,c]$ for $P = [a,b,c]$.

\begin{theorem}\label{theorem rationality}
\begin{enumerate}
	\item For $0<n<2k-2$ we have
	\[
	r_n\left(f_{k,P}^{\varepsilon}\right)=0,
	\]
	where $\varepsilon$ is the sign of $(-1)^{n}$.
	\item The outer periods of $f_{k,P}^{+}(z)$ are given by
	\[
	r_0(f_{k,P}^{+}) = (-1)^{k}r_{2k-2}(f_{k,P}^{+}) = -\frac{|d|^{k-\frac12}\zeta_P(k)}{2(2k-1)|\overline{\Gamma}_P|\zeta(2k)},
	\]
	where $\zeta_{P}(s) := \sum_{(x,y) \in \Z^2 \setminus \{(0,0)\}}P(x,y)^{-s}$ is the Epstein zeta function associated to $P$, and $\overline{\Gamma}_P$ denotes the stabilizer of $P$ in $\overline{\Gamma} := \Gamma/\{\pm 1\}$.
	\item Let $a_{n} \in \Q$ for odd $0 < n < 2k-2$ be coefficients such that
	\[
	\sum_{\substack{0 < n < 2k-2 \\ n \text{ odd}}}a_{n}R_{n}(\tau) = 0
	\]
	in $S_{2k}$. Then the linear combination
	\[
	\sum_{\substack{0 < n < 2k-2 \\ n \text{ odd}}}a_{n}r_{n}(f_{k,P})
	\]
	of odd periods of $f_{k,P}(z)$ is rational. Similarly, if a rational linear combination of the cusp forms $R_n$ with even $0 < n < 2k-2$ vanishes in $S_{2k}$, then the corresponding linear combination of even periods of $f_{k,P}(z)$ is in $i\Q$.
\end{enumerate}
\end{theorem}

The proof of Theorem~\ref{theorem rationality} will be given in Section~\ref{section proof rationality}. A nice feature of the proof is that it also yields an exact rational formula for the linear combinations of periods of $f_{k,P}(z)$ considered in item (3) of Theorem~\ref{theorem rationality}, compare Theorem~\ref{theorem rn formula} below. In contrast, the algebraic nature of the special values $\zeta_P(k)$ appearing in the outer periods of $f_{k,P}^{+}(z)$ is more mysterious. For example, for $k = 3$ and $P = [1,1,1]$ we have $\zeta_P(3) = 6\zeta(3)L(\chi_{-3},3) = \frac{8\pi^3}{27\sqrt{3}}\zeta(3)$ (compare \cite{zagierzetafunctions}, Proposition 3), which is expected to be transcendental.

We will now give an example to illustrate item (3) of Theorem~\ref{theorem rationality}.

\begin{example}
	We first notice that for $k = 2,3,4,5,$ and $7$, the space of cusp forms of weight $2k$ is trivial. Hence, Theorem~\ref{theorem rationality} shows that in these cases the odd periods of $f_{k,P}(z)$ are rational and the even periods for $0 < n < 2k-2$ are in $i\Q$ for all positive definite forms $P$, in accordance with the numerical values given in the above table. More generally, Cohen \cite{cohen} has found relations between the cusp forms $R_{n}(\tau)$ in any weight, for example
	\[
	\sum_{\substack{0 < n \leq j-1 \\ n \text{ odd}}}\binom{j}{n}(-1)^{\frac{n-1}{2}}R_{n}(\tau)+\sum_{\substack{j \leq n < 2k-2 \\ n \text{ odd}}}\binom{2k-2-j}{n-j}(-1)^{\frac{n-1}{2}}R_{n}(\tau) = 0
	\]
	for all $0 \leq j \leq 2k-2$. Relations of this kind can be proved using the Eichler-Shimura isomorphism. From Theorem~\ref{theorem rationality} we obtain that the linear combinations
	\[
	\sum_{\substack{0 < n \leq j-1 \\ n \text{ odd}}}\binom{j}{n}(-1)^{\frac{n-1}{2}}r_{n}(f_{k,P})+\sum_{\substack{j \leq n < 2k-2 \\ n \text{ odd}}}\binom{2k-2-j}{n-j}(-1)^{\frac{n-1}{2}}r_{n}(f_{k,P})
	\]
	of odd periods of $f_{k,P}(z)$ are rational for all $0 \leq j \leq 2k-2$ and all $P$. For instance, for $k = 6, d = -3$, and $j = 4$, we find that the linear combination
	\[
	10r_{1}(f_{6,-3}) - 24r_{3}(f_{6,-3}) + 6r_{5}(f_{6,-3})
	\]
	is rational. Indeed, plugging in the numerical values given in the above table, we find that this linear combination of periods is numerically close to the integer $-108$. Using our exact formula for the periods of $f_{k,P}(z)$ given in Theorem~\ref{theorem rn formula} below one can show that this really is the correct value.
\end{example}

We will explain the results from Theorem~\ref{theorem rationality} by using the following connection between the periods of $f_{k,P}(z)$ and special values of (derivatives of) the locally harmonic Maass forms $\mathcal{H}_{1-k,n}(\tau)$.

\begin{proposition}\label{proposition special value}
	We have
	\[
	r_{n}(f_{k,P}) = \frac{(-1)^{k-1}|d|^\frac{k-1}{2}}{2^{k-2}(k-1)!|\overline{\Gamma}_P|}R_{2-2k}^{k-1}\mathcal{H}_{1-k,n}(\tau_{P}),
	\]
	where $\tau_{P} \in \H$ denotes the CM point characterized by $P(\tau_{P},1) = 0$, and $R_{2-2k}^{k-1} := R_{-2}\circ \dots \circ R_{2-2k}$ is an iterated version of the Maass raising operator $R_{\kappa} := 2i\frac{\partial}{\partial \tau}+\kappa v^{-1}$.
\end{proposition}

The proof of the above proposition can be found in Section~\ref{section proof rationality}. In Theorem~\ref{theorem rn formula} below we will apply the iterated raising operator $R_{2-2k}^{k-1}$ to the splitting of $\mathcal{H}_{1-k,n}(\tau)$ from Theorem~\ref{theorem splitting}. Together with Proposition~\ref{proposition special value} we obtain an explicit formula for the periods $r_{n}(f_{k,P})$, which we will then use to prove Theorem~\ref{theorem rationality}.

Eventually, we remark that the methods of this work can be used to study the rationality of periods of certain linear combinations of the meromorphic modular forms $f_{k,P}(z)$, similar to \cite{ls}. For example, in analogy to Theorem~2.4 from \cite{ls}, one can show that certain linear combinations of Hecke-translates of $f_{k,P}(z)$ have rational periods.

We start with a section on the necessary preliminaries. In the remaining sections, we give the proofs of the above results.

\section{Preliminaries}\label{section preliminaries}

\subsection{Derivatives of Eichler integrals}

The holomorphic and the non-holomorphic Eicher integrals defined in \eqref{eq eichler integrals} are related by the iterated raising operator $R_{2-2k}^{k-1} = R_{-2}\circ \dots \circ R_{2-2k}$, with $R_{\kappa} = 2i\frac{\partial}{\partial \tau} + \kappa v^{-1}$, as follows.

\begin{proposition}\label{proposition eichler integral relations}
		For any holomorphic modular form $f \in M_{2k}$ we have the relation
		\[
		R_{2-2k}^{k-1}f^*(\tau) = -\frac{(2k-2)!}{(4\pi)^{2k-1}}\overline{R_{2-2k}^{k-1}\mathcal{E}_f(\tau)}.
		\]
\end{proposition}

\begin{proof}
	Using the Fourier expansions of the Eichler integrals given in \eqref{eq eichler integrals} we see that the claim is equivalent to
\begin{align}\label{eq eichler integrals simplified}
R_{2-2k}^{k-1}\left(\Gamma(2k-1,4\pi n v)e(-n\tau)\right) = (2k-2)! \overline{R_{2-2k}^{k-1}e(n\tau)}
\end{align}
for all $n \geq 1$. Following \cite{bruinierhabil}, Section 1.3, we consider the function
\[
\mathcal{W}_{\kappa,s}(y) := |y|^{-\kappa/2}W_{\frac{\kappa}{2}\sgn(y),s-\frac{1}{2}}(|y|), \qquad (\kappa \in \R, \ s \in \C, \ y \in \R \setminus \{0\}),
\]
where $W_{\nu,\mu}(y)$ is the usual $W$-Whittaker function. At $s = 1-\frac{\kappa}{2}$ it simplifies to
\[
\mathcal{W}_{\kappa,1-\frac{\kappa}{2}}(y) = \begin{cases}
e^{-y/2}, & \text{if $y > 0$}, \\
e^{-y/2}\Gamma(1-\kappa,|y|), & \text{if $y < 0$}.
\end{cases}
\]
In particular, \eqref{eq eichler integrals simplified} is equivalent to
\begin{align}\label{eq eichler integrals simplified 2}
R_{2-2k}^{k-1}\left(\mathcal{W}_{2-2k,k}(-4\pi n v)e(-nu) \right) = (2k-2)! \overline{R_{2-2k}^{k-1}\left(\mathcal{W}_{2-2k,k}(4\pi n v)e(nu)\right)}.
\end{align}
On the other hand, using (13.4.33) and (13.4.31) in \cite{abramowitz}, we obtain the formula
\begin{align*}
&R_{\kappa}\left(\mathcal{W}_{\kappa,s}(4\pi m v)e(mu)\right) \\
&\quad = \begin{cases}
-4\pi |m|\left(s+\frac{\kappa}{2}\right)\left(s-\frac{\kappa}{2}-1\right)\mathcal{W}_{\kappa+2,s}(4\pi m v)e(mu), & \text{if $m < 0$}, \\
-4\pi |m|\mathcal{W}_{\kappa+2,s}(4\pi m v)e(mu), & \text{if $m > 0$},
\end{cases}
\end{align*}
for $u+iv \in \H$ and $m \in \R \setminus \{0\}$. This easily implies \eqref{eq eichler integrals simplified 2} and finishes the proof.
\end{proof}

\subsection{Periods of cusp forms and non-cusp forms}\label{section prelims periods}

We recall from \cite{kohnenzagierrationalperiods} that the period polynomial of a cusp form $f \in S_{2k}$ is defined by
\[
r_f(\tau) := \int_0^{i\infty} f(z)(z - \tau)^{2k-2}dz = \sum_{n = 0}^{2k-2}i^{-n+1}\binom{2k-2}{n}r_n(f)\tau^{2k-2-n}.
\]
The even period polynomial $r_f^+(\tau)$ of $f$ is defined as $-i$ times the even part of $r_f(\tau)$, and the odd period polynomial $r_f^-(\tau)$ of $f$ is the odd part of $r_f(\tau)$, such that $r_f(\tau) = r_f^-(\tau)+ir_f^+(\tau)$. It is well-known that the errors of modularity of the Eichler integrals of $f$ can be expressed in terms of the period function of $f$ as
\begin{align}\label{holomorphiceichlertrafo}
	\mathcal{E}_{f}(\tau)\Bigl|_{2-2k}(I-S) &= \frac{(-2\pi i )^{2k-1}}{(2k-2)!}r_{f}(\tau)
\end{align}
and
\begin{align}\label{nonholomorphiceichlertrafo}
	f^*(\tau)\Bigl|_{2-2k}(I-S) &= (-2i)^{1-2k}r_{f}^c(\tau),
\end{align}
	where $r_f^c(\tau) := \overline{r_f(\overline{\tau})}$ is the polynomial whose coefficients are the complex conjugates of the coefficients of $r_f(\tau)$.

For $0 \leq n \leq 2k-2$ the periods of a (not necessarily cuspidal) modular form $f \in M_{2k}$ are defined by
\[
r_n(f) := \frac{n!}{(2\pi)^{n+1}}L_f(n+1),
\]
where $L_f(s)$ denotes the usual $L$-function associated to $f$. For a cusp form $f \in S_{2k}$ this agrees with the definition~\eqref{definition periods}. Note that the functional equation of the $L$-function of $f$ implies the symmetry $r_n(f) = (-1)^k r_{2k-2-n}(f)$.

The periods of the normalized Eisenstein series $E_{2k}$ are given by
\begin{align}\label{periodEisenstein}
r_{n}(E_{2k}) = \begin{dcases}
-\frac{\pi \zeta(2k-1) }{(2k-1)\zeta(2k)}, & \text{if $n=0$,}\\
0, & \text{if $0<n<2k-2$ is even,}\\
(-1)^{\frac{n-1}{2}}\frac{2k B_{n+1}B_{2k-1-n}}{B_{2k}(n+1)(2k-1-n)}, & \text{if $0<n<2k-2$ is odd,}\\
(-1)^{k-1}\frac{\pi \zeta(2k-1) }{(2k-1)\zeta(2k)}, & \text{if $n =2k-2$,}
\end{dcases}
\end{align}
 compare p.240 of \cite{kohnenzagierrationalperiods}. Following \cite{zagierperiods}, we define the corresponding period function by
\begin{equation}\label{EisenSplit}
r_{E_{2k}}(\tau) :=\frac{1}{2k-1}\left(\tau^{2k-1}+\frac{1}{\tau}\right) + \sum_{n=0}^{2k-2}i^{1-n}\binom{2k-2}{n}r_{n}(E_{2k})\tau^{2k-2-n} = r^-_{E_{2k}}(\tau)+ir^+_{E_{2k}}(\tau)
\end{equation}
with
\begin{align}\label{EisenEven}
r^+_{E_{2k}}(\tau) := r_{0}(E_{2k})(\tau^{2k-2}-1)
\end{align}
and
\begin{align}\label{EisenPeriodOddPoly}
r^-_{E_{2k}}(\tau)
:=\frac{2k(2k-2)!}{B_{2k}}\sum_{\substack{-1\leq n\leq 2k-1, \\ \text{$n$ odd}}}\frac{B_{n+1}B_{2k-1-n}}{(n+1)!(2k-1-n)!}\tau^{2k-2-n}.
\end{align}
Then we also have
\begin{align}\label{EisenEichlerTrafo}
\mathcal{E}_{E_{2k}}(\tau)\Bigl|_{2-2k}(I-S) = \frac{(-2\pi i )^{2k-1}}{(2k-2)!}r_{E_{2k}}(\tau).
 \end{align}

\subsection{The cusp forms $R_{n}(\tau)$}\label{section period polynomial Rn}

Recall that $R_n(\tau)$ denotes the unique cusp form of weight $2k$ for $\Gamma$ which satisfies the inner product formula $\langle f,R_n \rangle = r_n(f)$ for every cusp form $f(\tau) \in S_{2k}$. We will need the following well-known series representation of $R_n(\tau)$.

	\begin{proposition}\label{proposition Rn}
		For $0 < n < 2k-2$ the cusp form $R_n(\tau) \in S_{2k}$ is given by
		\[
R_{n}(\tau) = \frac{2^{2k-3}}{i^{2k-1-n}\binom{2k-2}{n}\pi}\sum_{M \in \Gamma}\tau^{-n-1}\Bigl|_{2k}M.
		\]
		Moreover, for $n = 0$ or $n = 2k-2$ it can be constructed as
		\[
		R_{2k-2}(\tau) = (-1)^k R_{0}(\tau) = 2^{2k-1}\sum_{m=1}^\infty P_{2k,m}(\tau),
		\]
		where $P_{2k,m}(\tau) := \sum_{M \in \Gamma_\infty \backslash \Gamma}e(m\tau)|_{2k}M$ with $\Gamma_\infty := \{\pm\left(\begin{smallmatrix}1 & n \\ 0 & 1 \end{smallmatrix} \right): n\in \Z\}$ is the usual cuspidal Poincar\'e series of weight $2k$.
	\end{proposition}

	\begin{proof}
		The above series representation of $R_n(\tau)$ for $0 < n < 2k-2$ was first given by Cohen in \cite{cohen}. We also refer the reader to the Lemma in Section 1.2 of \cite{kohnenzagierrationalperiods} for a proof.

		For $n = 2k-2$ and $f(\tau) = \sum_{m=1}^\infty c_f(m)e(m\tau) \in S_{2k}$ we have
		\[
		\left\langle f,2^{2k-1}\sum_{m=1}^\infty  P_{2k,m}\right\rangle = \frac{(2k-2)!}{(2\pi)^{2k-1}}\sum_{m=1}^{\infty}\frac{c_f(m)}{m^{2k-1}} = \frac{(2k-2)!}{(2\pi)^{2k-1}}L_f(2k-1) = r_{2k-2}(f) = \langle f,R_{2k-2} \rangle
		\]
		by the usual Petersson inner product formula for $P_{2k,m}(\tau)$. In other words, this means that $2^{2k-1}\sum_{m=1}^\infty P_{2k,m}(\tau)$ converges weakly to $R_{2k-2}(\tau)$, which implies $2^{2k-1}\sum_{m=1}^\infty P_{2k,m}(\tau) = R_{2k-2}(\tau)$ since $S_{2k}$ is finite-dimensional.
	\end{proof}

We state the explicit formulas of Kohnen and Zagier \cite{kohnenzagierrationalperiods} for the even and odd period polynomials of the cusp forms $R_n(\tau)$, in a form that is convenient for our purposes.

\begin{theorem}[\cite{kohnenzagierrationalperiods}, Theorem 1']\label{Periods R_n}
For $0\leq n\leq 2k-2$ even, the odd period polynomial of $R_n(\tau)$ is given by
\begin{multline*}
\left(\frac{i}{2}\right)^{2k-2}ir^-_{R_n}(\tau)
=i^{n+1}\left(\frac{B_{n+1}(\tau)}{n+1}- \frac{B_{2k-1-n}(\tau)}{2k-1-n}\right)\Bigl|_{2-2k}(I-S)\\
+i^{n+1}(\tau^n-\tau^{2k-2-n})+i(\delta_{n=0}+(-1)^k\delta_{n=2k-2})r^-_{E_{2k}}(\tau),
\end{multline*}
and for $0< n< 2k-2$ odd, the even period polynomial of $R_n(\tau)$ is given by
\begin{multline*}
\left(\frac{i}{2}\right)^{2k-2}r^+_{R_n}(\tau)
=-i^{n+1}\left(\frac{B_{n+1}(\tau)}{n+1}+ \frac{B_{2k-1-n}(\tau)}{2k-1-n}\right)\Bigl|_{2-2k}(I-S)  \\
-i^{n+1}(\tau^{n} + \tau^{2k-2-n})+r_n(E_{2k})(\tau^{2k-2}-1).
\end{multline*}
\end{theorem}

\section{The proof of Proposition~\ref{proposition diffops}}\label{section proof proposition diffops}

First note that the Poincar\'e series defined in \eqref{eq Peterssons Poincare series} satisfy the differential equations
\begin{align*}
R_{-2\ell,\tau}H_{k,\ell}(z,\tau) &=  (k-\ell)H_{k,\ell-1}(z,\tau), \\
L_{-2\ell,\tau}H_{k,\ell}(z,\tau) &= (k+\ell)H_{k,\ell+1}(z,\tau),
\end{align*}
which can be checked by a direct computation. Furthermore, by Bol's identity the iterated derivative $\mathcal{D}^{2k-1}$ can be expressed in terms of the iterated raising operator by
\[
\mathcal{D}^{2k-1} = -\frac{1}{(4\pi)^{2k-1}}R_{2-2k}^{2k-1},
\]
compare equation (56) in Zagier's part of \cite{zagier123}. In particular, we find
\begin{align*}
\xi_{2-2k,\tau}H_{k,k-1}(z,\tau) &= v^{-2k}\overline{L_{2-2k,\tau}H_{k,k-1}(z,\tau)} = (2k-1)v^{-2k}\overline{H_{k,k}(z,\tau)}, \\
\mathcal{D}_\tau^{2k-1}H_{k,k-1}(z,\tau) &= -\frac{1}{(4\pi)^{2k-1}}R_{2-2k, \tau}^{2k-1}H_{k,k-1}(z,\tau)=-\frac{(2k-1)!}{(4\pi)^{2k-1}}H_{k,-k}(z,\tau).
\end{align*}
For $\tau \in \C\setminus i\R_{\geq 0}$ we will need the evaluation
\begin{align}\label{yintegraleval}
\int_{0}^{\infty} \frac{y^ndy}{(iy-\tau)^{2k}} = \frac{i^{n+1}}{2k-1}\binom{2k-2}{n}^{-1}\tau^{n+1-2k},
\end{align}
which can be shown directly for $\tau = iv$ with $v < 0$ and then follows by analytic continuation for all $\tau \in \C \setminus i\R_{\geq 0}$.
For $0 < n < 2k-2$ we can now compute
\begin{align*}
\xi_{2-2k}\mathcal{H}_{1-k,n}(\tau) &=(2k-1) \frac{(2i)^{2k-2}}{2\pi}v^{-2k}\overline{\sum_{M\in\Gamma}\left(v^{2k}\int_{0}^{\infty} \frac{y^ndy}{(iy-\overline{\tau})^{2k}}\right)\Bigl|_{-2k,\tau}M}\\
&=(-i)^{n+1}\frac{(2i)^{2k-2}}{2\pi}\binom{2k-2}{n}^{-1}v^{-2k}\overline{\sum_{M\in\Gamma}v^{2k}\overline{\tau}^{n+1-2k}\Bigl|_{-2k,\tau}M} \\
&=(-i)^{n+1}\frac{(2i)^{2k-2}}{2\pi}\binom{2k-2}{n}^{-1}\sum_{M\in\Gamma}\tau^{n+1-2k}\Bigl|_{2k,\tau}M\\
&= (-1)^{k-1}R_{2k-2-n}(\tau)= - R_n(\tau),
\end{align*}
and
\begin{align*}
\mathcal{D}^{2k-1}\mathcal{H}_{1-k,n}(\tau) &=(-1)^k\frac{(2k-1)!}{2(2\pi)^{2k}}\sum_{M\in\Gamma}\left(\int_{0}^{\infty}  \frac{y^ndy}{(iy-\tau)^{2k}}\right)\Bigl|_{2k,\tau}M\\
&=i^{n+1}(-1)^k\frac{(2k-2)!}{2(2\pi)^{2k}}\binom{2k-2}{n}^{-1}\sum_{M\in\Gamma}\tau^{n+1-2k}\Bigl|_{2k,\tau}M\\
& = (-1)^{n+1} \frac{(2k-2)!}{(4\pi)^{2k-1}}(-1)^k R_{2k-2-n}(\tau) = (-1)^{n+1}\frac{(2k-2)!}{(4\pi)^{2k-1}}R_{n}(\tau),
\end{align*}
where we used the series representation of $R_n(\tau)$ from Proposition~\ref{proposition Rn}.

However, for $n = 0$ or $n = 2k-2$ the series representation of $R_n(\tau)$ used above does not converge, so we need to proceed differently. By the symmetries $\mathcal{H}_{1-k,n}(\tau) = (-1)^k \mathcal{H}_{1-k,2k-2-n}(\tau)$ and $R_{n}(\tau) = (-1)^k R_{2k-2-n}(\tau)$ it suffices to treat the case $n = 2k-2$. First, we write
\begin{align*}
\xi_{2-2k}\mathcal{H}_{1-k,2k-2}(\tau) &= 2(2k-1) \frac{(2i)^{2k-2}}{2\pi}v^{-2k}\overline{\sum_{M\in\Gamma_\infty\backslash\Gamma}\left(v^{2k}\int_{0}^{\infty} \sum_{m \in \Z} \frac{y^{2k-2}dy}{(iy-\overline{\tau}+m)^{2k}}\right)\Bigl|_{-2k,\tau}M}.
\end{align*}
Using the Lipschitz formula
\begin{align}\label{eq Lipschitz formula}
\sum_{m \in \Z}(z + m)^{-2k} = \frac{(2\pi i )^{2k}}{(2k-1)!}\sum_{m = 1}^\infty m^{2k-1}e(mz),
\end{align}
which is valid for $z \in \H$, we can rewrite the integral as
\begin{align*}
\int_{0}^{\infty} \sum_{m \in \Z} \frac{y^{2k-2}dy}{(iy-\overline{\tau}+m)^{2k}} &= \frac{(2\pi i)^{2k}}{(2k-1)!}\int_{0}^{\infty}y^{2k-2} \sum_{m = 1}^{\infty}m^{2k-1}e^{-2\pi m y}e(-m\overline{\tau})dy \\
&= \frac{2\pi (-1)^k}{2k-1}\sum_{m=1}^\infty e(-m \overline{\tau}).
\end{align*}
Putting everything together, we obtain
\begin{align*}
\xi_{2-2k}\mathcal{H}_{1-k,2k-2}(\tau) &= -2^{2k-1}v^{-2k}\overline{\sum_{M \in \Gamma_\infty\backslash \Gamma}\left(v^{2k}\sum_{m=1}^\infty e(-m \overline{\tau})\right)\Bigl|_{-2k,\tau}M} \\
&= -2^{2k-1}\sum_{m=1}^\infty \sum_{M \in \Gamma_\infty \backslash \Gamma}e(m \tau)\Bigl|_{2k,\tau}M \\
&= -2^{2k-1}\sum_{m=1}^\infty P_{2k,m}(\tau) = - R_{2k-2}(\tau),
\end{align*}
where we used Proposition~\ref{proposition Rn} in the last equality.

To compute $\mathcal{D}^{2k-1}\mathcal{H}_{1-k,2k-2}(\tau)$, we write as before
\begin{align*}
\mathcal{D}^{2k-1}\mathcal{H}_{1-k,2k-2}(\tau) &= -\frac{(2i)^{2k-2}(2k-1)!}{\pi (4\pi)^{2k-1}}\sum_{M \in \Gamma_\infty\backslash\Gamma}\left(\int_0^\infty \sum_{m \in \Z}\frac{y^{2k-2}}{(iy-\tau+m)^{2k}} dy\right)\Bigl|_{2k,\tau}M.
\end{align*}
It now suffices to prove the identity
\begin{align}\label{eq D identity}
\int_0^\infty \sum_{m \in \Z}\frac{y^{2k-2}}{(iy-\tau+m)^{2k}}dy = \frac{2^{2k-1}\pi}{(2i)^{2k-2}(2k-1)}\cdot\frac{1}{1-e(\tau)}
\end{align}
for $\tau \in \H$ with $0 < u < 1$, since we can write $\frac{1}{1-e(\tau)} = 1+\sum_{m=1}^\infty e(m\tau)$ and then finish the computation of $\mathcal{D}^{2k-1}\mathcal{H}_{1-k,2k-2}(\tau)$ in the same way as for the $\xi_{2-2k}$-image, using Proposition~\ref{proposition Rn}. Note that we cannot directly apply the Lipschitz formula \eqref{eq Lipschitz formula} to prove the identity \eqref{eq D identity} since $iy-\tau$ has negative imaginary part for $0 < y < v$. However, since both sides of \eqref{eq D identity} are holomorphic functions on the vertical strip consisting of all $\tau \in \C$ with $0 < u < 1$, it is enough to prove \eqref{eq D identity} for all $\tau$ in this strip with $v < 0$ and then use analytic continuation. Under the assumption $v < 0$ we can apply the Lipschitz formula~\eqref{eq Lipschitz formula} and obtain similarly as before
\begin{align*}
	\int_0^\infty \sum_{m \in \Z}\frac{y^{2k-2}}{(iy-\tau+m)^{2k}}dy
	&= \frac{(2\pi i)^{2k}}{(2k-1)!} \int_0^\infty y^{2k-2}\sum_{m=1}^\infty m^{2k-2}e^{-2\pi m y}e(-m\tau)dy \\
	&= \frac{(2\pi i)^{2k}}{(2k-1)!}\cdot \frac{(2k-2)!}{(2\pi)^{2k-1}}\sum_{m=1}^\infty e(-m\tau) \\
	&= -\frac{2^{2k-1}\pi}{(2i)^{2k-2}(2k-1)}\left(\frac{1}{1-e(-\tau)}-1 \right) \\
	&= \frac{2^{2k-1} \pi}{(2i)^{2k-2}(2k-1)}\cdot\frac{1}{1-e(\tau)},
\end{align*}
	which yields \eqref{eq D identity} and concludes the computation of $\mathcal{D}^{2k-1}\mathcal{H}_{1-k,2k-2}(\tau)$. This finishes the proof of Proposition~\ref{proposition diffops}.

\section{The proof of Theorem~\ref{theorem splitting}}\label{section proof theorem splitting}

	In this section we prove Theorem~\ref{theorem splitting}, i.e., the decomposition of the locally harmonic Maass form $\mathcal{H}_{1-k,n}(\tau)$ into a sum of a locally polynomial part and Eichler integrals of the cusp forms $R_n(\tau)$. For brevity, we let
 \begin{align*}
 \widetilde{\mathcal{H}}_{1-k,n}(\tau) &:= \mathcal{P}_{1-k,n}(\tau) + (-1)^{n+1}\frac{(2k-2)!}{(4\pi )^{2k-1}}\mathcal{E}_{R_n}(\tau) - R_{n}^*(\tau) \\
 &\quad-((-1)^k\delta_{n=0}+\delta_{n=2k-2})\frac{(2k-2)!}{(2\pi)^{2k-1}}\mathcal{E}_{E_{2k}}(\tau)
 \end{align*}
 be the expression on the right-hand side of Theorem~\ref{theorem splitting}. Then the theorem is equivalent to the identity $\mathcal{H}_{1-k,n}(\tau) = \widetilde{\mathcal{H}}_{1-k,n}(\tau)$. In order to prove this identity, we show that $\widetilde{\mathcal{H}}_{1-k,n}(\tau)$ is a locally harmonic Maass form of weight $2-2k$ with the same singularities as $\mathcal{H}_{1-k,n}(\tau)$ and the same images under the differential operators $\xi_{2-2k}$ and $\mathcal{D}^{1-2k}$. This implies that the difference $\mathcal{H}_{1-k,n}(\tau)-\widetilde{\mathcal{H}}_{1-k,n}(\tau)$ is a polynomial on $\H$ which is also modular of negative weight $2-2k$, and hence vanishes identically.

 We first show that $\widetilde{\mathcal{H}}_{1-k,n}(\tau)$ is modular. To this end, we study the modularity properties of the local polynomial $\mathcal{P}_{1-k,n}(\tau)$. We start by rewriting $\mathcal{P}_{1-k,n}(\tau)$ in a more convenient form.

\begin{lemma}\label{lemma alternative polynomial}
For $\tau \in \H$ we have
\begin{align*}
\mathcal{P}_{1-k,n}(\tau) &= r_n(E_{2k}) +\frac{(-i)^{n+1}}{n+1}B_{n+1}(\tau)+\frac{i^{n+1}}{2k-1-n}B_{2k-1-n}(\tau)
 \\
 &\quad +\frac{(-i)^{n+1}}{2}\left(\tau^n-(-1)^n\tau^{2k-2-n}\right)+\frac{i^{1-n}}{4}\sum_{M\in\Gamma}((\sgn(u)-\sgn(M))\tau^n)\Bigl |_{2-2k}M,
\end{align*}
where
\[
\sgn\left(\begin{pmatrix} a & b\\ c& d \end{pmatrix}\right):=\begin{cases}
\sgn(ac), & \text{if $ac\neq 0$,}\\
\sgn(bd), & \text{otherwise.}
\end{cases}
\]
\end{lemma}

\begin{proof}
For simplicity, we sketch the computations in the case that $\tau\notin\bigcup_{M \in \Gamma}M(i\R_{+})$ and leave the general case to the reader. We first rewrite
\begin{multline*}
\sum_{\substack{M = \left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix}\right)\in\Gamma \\ ac>0, \Re(M\tau)> 0}}(\tau^n -(-1)^n \tau^{2k-2-n})\Bigl |_{2-2k}M = \sum_{\substack{M = \left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix}\right)\in\Gamma \\ ac>0, \Re(M\tau)> 0}}\tau^n \Bigl |_{2-2k}M - \tau^n|_{2-2k}SM \\
=\sum_{\substack{M = \left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix}\right)\in\Gamma \\ ac \Re(M\tau) < 0 }}\sgn(M)\tau^n \Bigl |_{2-2k}M
=-\frac12\sum_{\substack{M = \left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix}\right)\in\Gamma \\ ac \neq 0 }}\left((\sgn(u)-\sgn(M))\tau^n\right) \Bigl |_{2-2k}M.
\end{multline*}
Here we used that $ac>0$ if and only if $SM = \left(\begin{smallmatrix}\alpha & \beta \\ \gamma & \delta \end{smallmatrix}\right)$ has $\alpha\gamma <0$ and $\sgn(\Re(SM\tau))\neq \sgn(\Re(M\tau))$.

Using $\frac{B_m(\tau+1)}{m} = \frac{B_m(\tau)}{m} + \tau^{m-1}$ we write
\begin{align*}
\frac{\mathbb{B}_m(\tau)}{m} -\frac{B_m(\tau)}{m} &=
\begin{cases}-\sum_{j=1}^{\lfloor u \rfloor}(\tau-j)^{m-1},& \text{if $u> 0$,} \\
\sum_{j=0}^{\lfloor -u\rfloor } (\tau+j)^{m-1}, & \text{if $u<0$,}
\end{cases} \\
&= \frac12\tau^{m-1}+\frac12\sum_{j\in\Z}(\sgn(j)-\sgn(u+j))(\tau+j)^{m-1} \\
&=\frac12\tau^{m-1}+\frac14\sum_{\substack{M = \left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix}\right)\in\Gamma \\ c = 0 }}(\sgn(M)-\sgn(u))\tau^{m-1}\Bigl |_{2-2k}M.
\end{align*}
Hence, we find after a short computation
\begin{align*}
\Biggl(\frac{\mathbb{B}_{n+1}(\tau)}{n+1}&-(-1)^n\frac{\mathbb{B}_{2k-1-n}(\tau)}{2k-1-n}\Biggr)-\Biggl(\frac{B_{n+1}(\tau)}{n+1}-(-1)^n\frac{B_{2k-1-n}(\tau)}{2k-1-n}\Biggr) \\
&=\frac12\left(\tau^n -(-1)^n\tau^{2k-2-n}\right)- \frac14\sum_{\substack{M = \left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix}\right)\in\Gamma \\ ac = 0 }}(\sgn(u)-\sgn(M))\tau^{n}\Bigl |_{2-2k}M.
\end{align*}
Gathering everything together, we obtain the formula given in the lemma.
\end{proof}

We can now state the transformation law for the locally polynomial part $\mathcal{P}_{1-k,n}(\tau)$.

\begin{lemma}\label{lemma trafo P}
For $\tau \in \H$ we have
\begin{align*}
\mathcal{P}_{1-k,n}(\tau)\Bigl|_{2-2k}(I-T) = 0
\end{align*}
and
\begin{align*}
\mathcal{P}_{1-k,n}(\tau)\Bigl|_{2-2k}(I-S) = \begin{dcases}
-\left(\frac{i}{2}\right)^{2k-2}r^+_{R_n}(\tau),&\text{if $n$ is odd,} \\
\begin{aligned} &-\left(\frac{i}{2}\right)^{2k-2}ir^-_{R_n}(\tau)  \\
&\quad+i(\delta_{n=0} +(-1)^k \delta_{n=2k-2})r_{E_{2k}}(\tau),
\end{aligned}&\text{if $n$ is even.}
\end{dcases}
\end{align*}
\end{lemma}

\begin{proof}
For the $T$-transformation, we write
$$
\left(\sum_{M\in\Gamma}\left((\sgn(u)-\sgn(M))\tau^n\right)\Bigl |_{2-2k}M\right)\Bigl|_{2-2k}T =
\sum_{M\in\Gamma}\left((\sgn(u)-\sgn(MT^{-1}))\tau^n\right)\Bigl |_{2-2k}M.
$$
Now one can show that the only matrices with $\sgn(MT^{-1})\neq \sgn(M)$ are $M\in\pm \{I, S, T, ST\}$. Therefore we get
\begin{align*}
\Biggl(&\sum_{M\in\Gamma}((\sgn(u)-\sgn(M))\tau^n)\Bigl |_{2-2k}M\Biggr)\Bigl|_{2-2k}(I-T)\\
&= 2\sum_{M\in\{I, S, T, ST\}}((\sgn(MT^{-1})-\sgn(M))\tau^n)\Bigl |_{2-2k}M\\
&= 2(-\tau^n + (-1)^n\tau^{2k-2-n} - (\tau+1)^n + (-1)^n(\tau+1)^{2k-2-n}).
\end{align*}
The Bernoulli polynomials satisfy $
\frac{B_m(\tau+1)}{m} = \frac{B_m(\tau)}{m} + \tau^{m-1}$,
so all in all we obtain
\begin{align*}
&\mathcal{P}_{1-k,n}(\tau)\Bigl|_{2-2k}(I-T) = \frac12 (-i)^{n+1}\Bigl(-2\tau^n +2(-1)^n\tau^{2k-2-n} + \tau^n -(\tau+1)^n -(-1)^n\tau^{2k-2-n} \\
&\qquad+ (-1)^n(\tau+1)^{2k-2-n}+\tau^n -(-1)^n\tau^{2k-2-n}+(\tau+1)^n -(-1)^n(\tau+1)^{2k-2-n} \Bigr) = 0.
\end{align*}

For the $S$-transformation, note that
\begin{align*}
\left(\sum_{M\in\Gamma}\left((\sgn(u)-\sgn(M))\tau^n\right)\Bigl |_{2-2k}M\right)\Bigl|_{2-2k}S
&=\sum_{M\in\Gamma}\left((\sgn(u)-\sgn(MS^{-1}))\tau^n\right)\Bigl |_{2-2k}M \\
&= \sum_{M\in\Gamma}\left((\sgn(u)-\sgn(M))\tau^n\right)\Bigl |_{2-2k}M,
\end{align*}
since we have $\sgn(MS^{-1}) = \sgn(M)$ for all $M\in\Gamma$. Furthermore, if $n$ is odd, then $(-i)^{n+1} = i^{n+1}$ and Theorem \ref{Periods R_n} gives
\begin{align*}
&\Bigl(r_n(E_{2k}) +\frac{(-i)^{n+1}}{n+1}B_{n+1}(\tau)+\frac{i^{n+1}}{2k-1-n}B_{2k-1-n}(\tau) \\
&\qquad+\frac{(-i)^{n+1}}{2}(\tau^n-(-1)^n \tau^{2k-2-n}\Bigr)\Bigl|_{2-2k}(I-S) \\
&= -r_n(E_{2k})(\tau^{2k-2}-1) +i^{n+1}\left(\frac{B_{n+1}(\tau)}{n+1}+\frac{B_{2k-1-n}(\tau)}{2k-1-n}\right)\Bigl|_{2-2k}(I-S) \\
&\qquad +i^{n+1}(\tau^n+\tau^{2k-2-n}) \\
&=-\left(\frac{i}{2}\right)^{2k-2}r^+_{R_n}(\tau).
\end{align*}
If $n$ is even, then $(-i)^{n+1} = -i^{n+1}$ and Theorem~\ref{Periods R_n} yields
\begin{align*}
&\Bigl( r_n(E_{2k}) +\frac{(-i)^{n+1}}{n+1}B_{n+1}(\tau)+\frac{i^{n+1}}{2k-1-n}B_{2k-1-n}(\tau) \\
&\qquad +\frac{(-i)^{n+1}}{2}(\tau^n-(-1)^n\tau^{2k-2-n})\Bigr)\Bigl|_{2-2k}(I-S) \\
&=  -r_n(E_{2k})(\tau^{2k-2}-1) +i^{n+1}\left(-\frac{B_{n+1}(\tau)}{n+1}+\frac{B_{2k-1-n}(\tau)}{2k-1-n}\right)\Bigl|_{2-2k}(I-S) \\
&\qquad -i^{n+1}(\tau^n-\tau^{2k-2-n}) \\
&=-i\left(\frac{i}{2}\right)^{2k-2}r^-_{R_n}(\tau)+i(\delta_{n=0} +(-1)^k \delta_{n=2k-2})r_{E_{2k}}(\tau).
\end{align*}
In the last step we used that $r_n(E_{2k}) =0$ for even $n$ unless $n=0$ or $n=2k-2$, as well as \eqref{EisenSplit} and \eqref{EisenEven}. Pulling everything together, we obtain the stated result.
\end{proof}

Combining the above results, we obtain the modularity of the function $\widetilde{\mathcal{H}}_{1-k,n}(\tau)$ defined in the beginning of this section.

\begin{proposition}\label{proposition modularity Htilde}
	The function $\widetilde{\mathcal{H}}_{1-k,n}(\tau)$ transforms like a modular form of weight $2-2k$ for $\Gamma$.
\end{proposition}

\begin{proof}
	Since the Eichler integrals are one-periodic by definition and the local polynomial $\mathcal{P}_{1-k,n}(\tau)$ is one-periodic by Lemma~\ref{lemma trafo P}, the same is true for $\widetilde{\mathcal{H}}_{1-k,n}(\tau)$.

	Next, we show the invariance of $\widetilde{\mathcal{H}}_{1-k,n}(\tau)$ under $S$. Using \eqref{holomorphiceichlertrafo} and \eqref{nonholomorphiceichlertrafo}, and the fact that $R_n(\tau)$ has real Fourier coefficients, we compute
\begin{align*}
\left((-1)^{n+1}\frac{(2k-2)!}{(4\pi )^{2k-1}}\mathcal{E}_{R_n}(\tau) - R_{n}^*(\tau)\right)\Bigl|_{2-2k}(I-S)
&=\left(\frac{i}{2}\right)^{2k-2}
\begin{cases}
r_{R_n}^+(\tau), & \text{if $n$ is odd,}\\
ir_{R_n}^-(\tau),& \text{if $n$ is even.}
\end{cases}
\end{align*}
Combining this with Lemma~\ref{lemma trafo P} (and \eqref{EisenEichlerTrafo} if $n = 0$ or $n = 2k-2$), we see that
\[
\widetilde{\mathcal{H}}_{1-k,n}(\tau)\Bigl|_{2-2k}(I-S) = 0.
\]
This finishes the proof.
\end{proof}

Next, we determine the singularities of $\mathcal{H}_{1-k,n}(\tau)$ and $\widetilde{\mathcal{H}}_{1-k,n}(\tau)$. Here, we say that a function $f$ has a \emph{singularity of type $g$} at a point $\tau_{0}$ if there exists a neighbourhood $U$ of $\tau_{0}$ on which $f-g$ is harmonic.

\begin{lemma}\label{jumps}
The functions $\mathcal{H}_{1-k,n}(\tau)$ and $\widetilde{\mathcal{H}}_{1-k,n}(\tau)$ are harmonic on $\H\setminus \bigcup_{M \in \Gamma}M(i\R_{+})$. At a point $\tau_{0} \in \H$ they have a singularity of type
	\[
 \frac{i^{1-n}}{4}\sum_{\substack{M\in\Gamma \\  \Re(M\tau_0)=0}}\left(\sgn(u)\tau^{n}\right)\Bigl|_{2-2k}M.
	\]
\end{lemma}

\begin{proof}
We start with the function $\widetilde{\mathcal{H}}_{1-k,n}(\tau)$. Note that the weight $2-2k$ invariant Laplace operator can be written as $\Delta_{2-2k} = -\xi_{2k}\circ \xi_{2-2k}$ and that $\xi_{2k}$ annihilates holomorphic functions. Hence, the action \eqref{eq diffops eichler} of $\xi_{2-2k}$ on Eichler integrals and the fact that $\mathcal{P}_{1-k,n}(\tau)$ is a polynomial on each connected component of $\H\setminus \bigcup_{M \in \Gamma}M(i\R_{+})$ imply that $\widetilde{\mathcal{H}}_{1-k,n}(\tau)$ is harmonic on this set. The singularities of $\widetilde{\mathcal{H}}_{1-k,n}(\tau)$ come from the sum in the locally polynomial part $\mathcal{P}_{1-k,n}(\tau)$ and are given by the stated formula by Lemma~\ref{lemma alternative polynomial}.

Now we consider the function $\mathcal{H}_{1-k,n}(\tau)$. Since the function $\tau \mapsto H_{k,k-1}(z,\tau)$ is harmonic on $\H \setminus \Gamma z$, the function $\mathcal{H}_{1-k,n}(\tau)$ is harmonic on $\H \setminus \bigcup_{M \in \Gamma}M(i\R_{+})$. To determine the singularities, we keep $\tau_{0}\in\H$ fixed and consider the function
\[
G_{\tau_0}(z, \tau) := \sum_{\substack{M\in\Gamma \\ \Re(M\tau_0) = 0}} \left(\frac{v^{2k-1}}{(z-\tau)(z-\overline{\tau})^{2k-1}}\right)\Big|_{2-2k, \tau} M.
\]
Note that $\Re(M\tau_0)=0$ means that $\tau_0$ lies on the geodesic $M^{-1}(i\R_+)$, hence the above sum is finite (see also Figure 1 in the introduction). Furthermore, the function $G_{\tau_0}(z,\tau)$ is meromorphic in $z$ and harmonic in $\tau$ on $\H\setminus\Gamma z$. We split $\mathcal{H}_{1-k,n}(\tau)$ into
\[
\mathcal{H}_{1-k,n}(\tau)  =\frac{(2i)^{2k-2}}{2\pi}\left(\int_0 ^\infty (H_{k,k-1}(iy,\tau) - G_{\tau_0}(iy,\tau))y^ndy + \int_0 ^\infty G_{\tau_0}(iy,\tau)y^ndy\right).
\]
The function
\[
z\mapsto H_{k,k-1}(z,\tau) - G_{\tau_0}(z,\tau)
\]
is meromorphic and has no singularities near $\tau_0$ and the function
\[
\tau \mapsto \int_0 ^\infty (H_{k,k-1}(iy,\tau) - G_{\tau_0}(iy,\tau))y^ndy
\]
is harmonic in a neighborhood of $\tau_0$. For the second summand we compute for any $\tau\notin E_1$
\[
\int_0 ^\infty G_{\tau_0}(iy,\tau)y^ndy =(-i)^{n+1}\sum_{\substack{M\in\Gamma \\ \Re(M\tau_0)=0}}\left(v^{2k-1}\int_0 ^{i\infty} \frac{z^ndz}{(z-\tau)(z-\overline{\tau})^{2k-1}}\right)\Big|_{2-2k, \tau} M.
\]
We now evaluate the inner integral for fixed $\tau$. If we shift the path of integration to the left towards $-\infty$, we pick up a residue $2\pi i\frac{\tau^n}{(2iv)^{2k-1}}$ if $\Re(\tau) <0$. If we shift towards $\infty$, we pick up a residue of $-2\pi i\frac{\tau^n}{(2iv)^{2k-1}}$ if $\Re(\tau) >0$. Thus we get
\begin{multline*}
\int_0 ^\infty G_{\tau_0}(iy,\tau)y^ndy =(-i)^{n+1}\sum_{\substack{M\in\Gamma \\ \Re(M\tau_0)=0}}\Bigl(\frac{v^{2k-1}}{2}\left(\int_{-\infty}^0-\int_0 ^\infty\right)\frac{z^ndz}{(z-\tau)(z-\overline{\tau})^{2k-1}} \\
-\sgn(u)\pi i\frac{\tau^n}{(2i)^{2k-1}}\Bigr)\Big|_{2-2k, \tau} M.
\end{multline*}
The function
\[
\tau\mapsto \left(\int_{-\infty}^0-\int_0 ^\infty\right)\frac{z^ndz}{(z-\tau)(z-\overline{\tau})^{2k-1}}
\]
is harmonic on $\H$, so it does not contribute to the singularity. The sum over the signed terms yields the claimed singularity.
\end{proof}

We can now finish the proof of Theorem~\ref{theorem splitting}. It follows from Proposition~\ref{proposition modularity Htilde} and Lemma~\ref{jumps} that $\widetilde{\mathcal{H}}_{1-k,n}(\tau)$ is a locally harmonic Maass form of weight $2-2k$ with the same singularities as $\mathcal{H}_{1-k,n}(\tau)$, i.e., the difference $\mathcal{H}_{1-k,n}(\tau) - \widetilde{\mathcal{H}}_{1-k,n}(\tau)$ transforms like a modular form of weight $2-2k$ and is harmonic on all of $\H$. Furthermore, it follows from \eqref{eq diffops eichler} and Proposition~\ref{proposition diffops} that $\mathcal{H}_{1-k,n}(\tau) - \widetilde{\mathcal{H}}_{1-k,n}(\tau)$ is annihilated by $\xi_{2-2k}$ and $\mathcal{D}^{2k-1}$, which implies that it is a polynomial on $\H$. But the only one-periodic polynomials on $\H$ are the constant functions, and the only constant function which transforms like a modular form of non-zero weight is the constant 0 function. This shows $\mathcal{H}_{1-k,n}(\tau) = \widetilde{\mathcal{H}}_{1-k,n}(\tau)$ and concludes the proof of Theorem~\ref{theorem splitting}.

\section{The proof of Theorem~\ref{theorem rationality} and Proposition~\ref{proposition special value}}\label{section proof rationality}

Let $P \in \mathcal{Q}_d$ be a positive definite binary quadratic form and let $\tau_P \in \H$ be the associated CM point defined by $P(\tau_P,1) = 0$. For simplicity, we assume throughout this section that $\tau_P$ does not lie on any $\Gamma$-translate of the imaginary axis $i\R_+$ and leave the necessary adjustments in the general case to the reader.

We start with the proof of Proposition~\ref{proposition special value}. We can write
\[
P(z,1) = \frac{\sqrt{|d|}}{2\Im(\tau_P)}(z-\tau_P)(z-\overline{\tau}_P),
\]
which implies
\begin{align*}
H_{k,0}(z,\tau_P) &= \sum_{M \in \Gamma}j(M,z)^{-2k}\left(\frac{(Mz - \tau_P)(Mz - \overline{\tau}_P)}{\Im(\tau_P)} \right)^{-k} \\
&= 2^{-k}|d|^{\frac{k}{2}}\sum_{M \in \Gamma}j(M,z)^{-2k}P(Mz,1)^{-k} = 2^{1-k}|d|^\frac{1-k}{2}|\overline{\Gamma}_P|\pi f_{k,P}(z).
\end{align*}
Using $(k-1)!H_{k,0}(z,\tau) = R_{2-2k,\tau}^{k-1}H_{k,k-1}(z,\tau)$ we obtain
\begin{align*}
r_n(f_{k,P}) &= \int_0^\infty f_{k,P}(iy)y^n dy  \\
&= \frac{2^{k-1}|d|^\frac{k-1}{2}}{|\overline{\Gamma}_P| \pi }\int_0^\infty H_{k,0}(iy,\tau_P)y^n dy  \\
&= \frac{2^{k-1}|d|^\frac{k-1}{2}}{(k-1)!|\overline{\Gamma}_P| \pi }\int_0^\infty R_{2-2k,\tau}^{k-1}H_{k,k-1}(iy,\tau)\big|_{\tau = \tau_P}y^n dy  \\
&=\frac{(-1)^{k-1}|d|^\frac{k-1}{2}}{2^{k-2}(k-1)!|\overline{\Gamma}_P|}R_{2-2k}^{k-1}\mathcal{H}_{1-k,n}(\tau_P).
\end{align*}
This finishes the proof of Proposition~\ref{proposition special value}.

Before we come to the proof of Theorem~\ref{theorem rationality}, we give a general formula for the periods $r_n(f_{k,P})$ of the meromorphic modular forms $f_{k,P}(z)$.

\begin{theorem}\label{theorem rn formula}
	Assume that $\tau_P \in \H \setminus \bigcup_{M \in \Gamma}M(i\R_{+})$ and $0 < \Re(\tau_P) < 1$. Then we have the formula
	\begin{align*}
	r_n(f_{k,P}) &=  \frac{|d|^{\frac{k-1}{2}}}{|\overline{\Gamma}_P|}\bigg(\frac{(-1)^{k-1}}{2^{k-2}(k-1)!}R_{2-2k}^{k-1}\mathcal{P}_{1-k,n}(\tau_P)  \\
	&\qquad \qquad \quad +\frac{(-1)^{k}}{2^{k-2}(k-1)!}\left((-1)^{n+1}\overline{R_{2-2k}^{k-1}R_n^*(\tau_P)}+ R_{2-2k}^{k-1}R_n^*(\tau_P)\right) \\
	&\qquad \qquad \quad +(\delta_{n=0}+(-1)^k \delta_{n=2k-2})\frac{2^{k+1}(2k-2)!}{(4\pi)^{2k-1}(k-1)!}R_{2-2k}^{k-1}\mathcal{E}_{E_{2k}}(\tau_P)\bigg),
	\end{align*}
	where $R_{2-2k}^{k-1}\mathcal{P}_{1-k,n}(\tau)$ is explicitly given by
	\begin{align*}
	R_{2-2k}^{k-1}\mathcal{P}_{1-k,n}(\tau) &= (-v)^{1-k}\frac{(2k-2)!}{(k-1)!}r_n(E_{2k}) \\
	&\quad + \frac{(-i)^{n+1}}{n+1}R_{2-2k}^{k-1}B_{n+1}(\tau) + \frac{i^{n+1}}{2k-1-n}R_{2-2k}^{k-1}B_{2k-1-n}(\tau) \\
	&\quad + \frac{i^{1-n}}{2}\sum_{\substack{M \in \Gamma \\ ac > 0, \Re(M\tau) \leq 0}}\left(R_{2-2k}^{k-1}\tau^{n}-(-1)^n R_{2-2k}^{k-1}\tau^{2k-2-n} \right)\bigg|_0 M.
	\end{align*}
\end{theorem}

\begin{proof}
First, by Proposition~\ref{proposition special value} we have
\[
r_n(f_{k,P}) = \frac{(-1)^{k-1}|d|^\frac{k-1}{2}}{2^{k-2}(k-1)!|\overline{\Gamma}_P|}R_{2-2k}^{k-1}\mathcal{H}_{1-k,n}(\tau_P).
\]
Furthermore, by Theorem~\ref{theorem splitting} the locally harmonic Maass form $\mathcal{H}_{1-k,n}(\tau)$ has the splitting
\begin{align*}
\mathcal{H}_{1-k,n}(\tau) &= \mathcal{P}_{1-k,n}(\tau) + (-1)^{n+1}\frac{(2k-2)!}{(4\pi )^{2k-1}}\mathcal{E}_{R_n}(\tau) - R_{n}^*(\tau) \\
 &\quad-((-1)^{k}\delta_{n=0}+\delta_{n=2k-2})\frac{(2k-2)!}{(2\pi)^{2k-1}}\mathcal{E}_{E_{2k}}(\tau).
\end{align*}
By Proposition~\ref{proposition eichler integral relations}, we can rewrite
\[
\frac{(2k-2)!}{(4\pi)^{2k-1}}R_{2-2k}^{k-1}\mathcal{E}_{R_n}(\tau) = -\overline{R_{2-2k}^{k-1}R_n^*(\tau)}.
\]
Finally, using the formula $R_{\kappa}v^j = (j+\kappa)v^{j-1}$ we can compute the action of the iterated raising operator on the constant $r_n(E_{2k})$ in the locally polynomial part $\mathcal{P}_{1-k,n}(\tau)$. This finishes the proof.
\end{proof}

Note that the Fourier expansion of the raised Eichler integrals appearing in Theorem~\ref{theorem rn formula} can be computed using Proposition~\ref{proposition eichler integral relations} and \eqref{raising and derivative}. In order to understand the algebraic nature of the expressions appearing in $R_{2-2k}^{k-1}\mathcal{P}_{1-k,n}(\tau_P)$, the following formula will be useful.

\begin{lemma}\label{lemma raising taupower}
	For $k \in \N$, $0 \leq \ell \leq 2k-1$, and $\tau = u+iv \in \H$ we have the formula
	\begin{align*}
	R_{2-2k}^{k-1}\tau^\ell &= (-v)^{1-k}\sum_{j=0}^{\min\{k-1,\ell\}}\binom{\ell}{j}\frac{(2k-2-j)!}{(k-1-j)!}(-2)^j\sum_{\substack{\alpha = 0 \\ \ell-\alpha \text{ even }}}^{\ell-j}\binom{\ell-j}{\alpha}u^{\alpha}(iv)^{\ell-\alpha} \\
&\quad + i\delta_{\ell=2k-1}(-1)^{k-1}2^{2k-2}(k-1)!v^k.
	\end{align*}
\end{lemma}

\begin{proof}
We first apply the formula
\begin{align}\label{raising and derivative}
R_{2-2k}^{k-1}f(\tau) = (-v)^{1-k}(k-1)!\sum_{j=0}^{k-1}\frac{(-2iv)^j}{j!}\binom{2k-2-j}{k-1}\frac{\partial^j}{\partial \tau^j}f(\tau),
\end{align}
which holds for any smooth function $f: \H \to \C$ (see equation (56) in Zagier's part of \cite{zagier123}). This yields
\begin{align}\label{eq raising taupower}
R_{2-2k}^{k-1}\tau^\ell = (-v)^{1-k}\sum_{j=0}^{\min\{k-1,\ell\}}\binom{\ell}{j}\frac{(2k-2-j)!}{(k-1-j)!}(-2iv)^j\tau^{\ell-j}.
\end{align}
On the other hand, using $R_{\kappa}v^j = (j+\kappa)v^{j-1}$ we compute
\begin{align*}
R_{2-2k}^{k-1}\tau^\ell &= R_{2-2k}^{k-1}(\overline{\tau}+2iv)^\ell = R_{2-2k}^{k-1} \sum_{j=0}^{\ell}\binom{\ell}{j}(2iv)^j\overline{\tau}^{\ell-j} \\
&= (-v)^{1-k}\sum_{j=0}^{\ell}\binom{\ell}{j}(2iv)^j\overline{\tau}^{\ell-j} (2k-2-j)(2k-3-j)\cdots (k-j).
\end{align*}
Note that $(2k-2-j)(2k-3-j)\cdots (k-j)$ equals $\frac{(2k-2-j)!}{(k-1-j)!}$ for $0 \leq j \leq k-1$, it vanishes for $k \leq j \leq 2k-2$, and it equals $(-1)^{k-1}(k-1)!$ if $j = 2k-1$, which can occur only if $\ell = 2k-1$. Hence we obtain
\begin{align*}
R_{2-2k}^{k-1}\tau^\ell &= (-v)^{1-k}\sum_{j=0}^{\min\{k-1,\ell\}}\binom{\ell}{j}\frac{(2k-2-j)!}{(k-1-j)!}(2iv)^j\overline{\tau}^{\ell-j}+\delta_{\ell=2k-1}(2i)^{2k-1}(k-1)!v^{k}.
\end{align*}
Comparing this with \eqref{eq raising taupower}, we see that
\[
R_{2-2k}^{k-1}\tau^\ell - \frac{1}{2}\delta_{\ell=2k-1}(2i)^{2k-1}(k-1)!v^k
\]
is real. Thus, expanding $\tau^{\ell-j}$ in \eqref{eq raising taupower} using the binomial theorem, we obtain
\begin{align*}
R_{2-2k}^{k-1}\tau^\ell
&= (-v)^{1-k}\sum_{j=0}^{\min\{k-1,\ell\}}\binom{\ell}{j}\frac{(2k-2-j)!}{(k-1-j)!}(-2)^j\sum_{\alpha = 0}^{\ell-j}\binom{\ell-j}{\alpha}u^{\alpha}(iv)^{\ell-\alpha} \\
&= (-v)^{1-k}\sum_{j=0}^{\min\{k-1,\ell\}}\binom{\ell}{j}\frac{(2k-2-j)!}{(k-1-j)!}(-2)^j\sum_{\substack{\alpha = 0 \\ \ell-\alpha \text{ even }}}^{\ell-j}\binom{\ell-j}{\alpha}u^{\alpha}(iv)^{\ell-\alpha} \\
&\quad + i\delta_{\ell=2k-1}(-1)^{k-1}2^{2k-2}(k-1)!v^k.
\end{align*}
Here we used that the terms with odd $\ell-\alpha$ would be purely imaginary and hence cannot occur, apart from possibly the summand for $\alpha = 0$ in the case $\ell = 2k-1$, which is a purely imaginary multiple of $v^k$ and hence has to be equal to $i(-1)^{k-1}2^{2k-2}(k-1)!v^k$. This finishes the proof.
\end{proof}

We obtain the following rationality result for the special values $R_{2-2k}^{k-1}\mathcal{P}_{1-k,n}(\tau_P)$.

\begin{lemma}\label{lemma P rational}
	We have
	\begin{align*}
|d|^{\frac{k-1}{2}}R_{2-2k}^{k-1}\mathcal{P}_{1-k,n}(\tau_P) \in \begin{cases}\Q, & \text{if $0 < n < 2k-2$ is odd,} \\
i\Q, & \text{if $0 < n < 2k-2$ is even.}
\end{cases}
\end{align*}
\end{lemma}

\begin{proof}
If we write $P=[a,b,c]$ then the corresponding CM point is given by
\[
\tau_P = -\frac{b}{2a}+i\frac{\sqrt{|d|}}{2a}.
\]
Noting that $B_m(\tau) = \sum_{\ell=0}^m \binom{m}{\ell}B_{m-\ell}\tau^\ell$, we see from Lemma~\ref{lemma raising taupower} that
\begin{align}\label{eq rational 1}
|d|^{\frac{k-1}{2}}R_{2-2k}^{k-1}B_m(\tau_P) \in \Q
\end{align}
for every $0 \leq m \leq 2k-2$, and
\begin{align}\label{eq rational 2}
|d|^{\frac{k-1}{2}}R_{2-2k}^{k-1}B_{2k-1}(\tau_P) = C + i|d|^{\frac{k-1}{2}}(-1)^{k-1}2^{2k-2}(k-1)!v_P^k
\end{align}
for some constant $C \in \Q$. By the same lemma we see that
\begin{align}\label{eq rational 3}
|d|^{\frac{k-1}{2}}R_{2-2k}^{k-1}\tau^\ell \in \Q
\end{align} for every $0 \leq \ell \leq 2k-2$ and every CM point $\tau$ of discriminant $d$. Note that $M\tau_P$ is a CM point of discriminant $d$ for every $M \in \Gamma$. Recall from Section~\ref{section prelims periods} that $r_n(E_{2k})$ vanishes for even $0 < n < 2k-2$ and is rational for odd $n$. Summarizing, we obtain the stated rationality result.
\end{proof}

Now we come to the proof of Theorem~\ref{theorem rationality}. First, it follows from the definition of $f_{k,P}(z)$ that
\[
f_{k,P'}(iy) = \overline{f_{k,P}(iy)},
\]
where we put $P' = [a,-b,c]$ for $P = [a,b,c]$. This implies
\begin{align}\label{eq real imag}
r_n(f_{k,P}^{+}) = \Re(r_n(f_{k,P})), \qquad r_n(f_{k,P}^{-}) = -\Im(r_n(f_{k,P})).
\end{align}

It is now easy to see from Theorem~\ref{theorem rn formula} and Lemma~\ref{lemma P rational} that $r_n(f_{k,P})$ for $0 < n < 2k-2$ is a real number if $n$ is odd, and a purely imaginary number if $n$ is even. Therefore, \eqref{eq real imag} implies $r_n(f_{k,P}^\varepsilon) = 0$ for $0 < n < 2k-2$, where $\varepsilon$ is the sign of $(-1)^n$. This concludes the proof of the first part of Theorem~\ref{theorem rationality}.

Concerning the period $r_0(f_{k,P}^+) = \Re(r_0(f_{k,P}))$, we obtain from Theorem~\ref{theorem rn formula} and equations \eqref{eq rational 1}--\eqref{eq rational 3} the formula
\begin{align*}
&r_0(f_{k,P}^+) = -\frac{2^{k-1}|d|^{\frac{k-1}{2}}}{(2k-1)|\overline{\Gamma}_P|\zeta(2k)} \\
&\times\bigg(2v_P^k \zeta(2k) + 2^{3-2k}v_P^{1-k}\pi \zeta(2k-1)\binom{2k-2}{k-1}-\frac{4(2k-1)!\zeta(2k)}{(4\pi)^{2k-1}(k-1)!} \Re(R_{2-2k}^{k-1}\mathcal{E}_{2k} (\tau_P))\bigg).
\end{align*}
Here we used the evalution of $r_0(E_{2k})$ given in Section~\ref{section prelims periods}. Writing out the Fourier expansion of $R_{2-2k}^{k-1}\mathcal{E}_{2k}(\tau_P)$ explicitly using the formula \eqref{raising and derivative} and the well-known expansion
\[
E_{2k}(\tau) = 1+\frac{(2\pi i)^{2k}}{(2k-1)!\zeta(2k)}\sum_{n=1}^\infty \sigma_{2k-1}(n)e(n\tau),
\]
and then using the formula for the Epstein zeta function given in equation (2.4) in \cite{smartepstein} (note that the quadratic form $Q$ in \cite{smartepstein} is normalized to have to discriminant $-4$), we see that the expression in the second line in the above formula for $r_0(f_{k,P}^+)$ equals $2^{-k}|d|^{\frac{k}{2}}\zeta_P(k)$. This proves the second part of Theorem~\ref{theorem rationality}.

	In order to prove the rationality statement in the third item in Theorem~\ref{theorem rationality}, we note that the assumption $\sum_n a_n R_n(\tau) = 0$ implies that the raised Eichler integrals of the cusp forms $R_n(\tau)$ in Theorem~\ref{theorem rn formula} cancel out in the linear combination $\sum_n a_n r_n(f_{k,P})$. In particular, $\sum_n a_n r_n(f_{k,P})$ is given by a rational linear combination of the values $|d|^{\frac{k-1}{2}}R_{2-2k}^{k-1}\mathcal{P}_{1-k,n}(\tau_P)$. Now Lemma~\ref{lemma P rational} implies the third item in Theorem~\ref{theorem rationality}. This finishes the proof.

config: arxiv


TITLE: Computing the convex hull of lattice points
QUESTION [4 upvotes]: Consider a set of $n$ points in $\mathbb Z^2$. It is known that their convex hull can be computed in time $O(n\log n)$, or even $O(n\log h)$ where $h$ is the number of points in the convex hull. These bounds are in given (for instance) in the model of algebraic decision trees.
What is the bit-complexity of computing the convex hull of $n$ points in $\mathbb Z^2$, whose both coordinates are bounded in absolute value by $M$?
I am interested on the one hand in the dependence of the complexity in $M$, and on the other hand in the possible gain in complexity one obtains using the fact that the input points have integer coordinates. In my problem, I see $M$ as a large parameter, that is $\log(M)$ has the same order of magnitude as $n$.
Note that I am especially interested in the problem with the additional requirement that the points on the convex hull have to be listed in the order of the (say) counterclockwise rotation, beginning for instance at the leftmost point.
I am also interested in the same questions in higher dimensions.

REPLY [10 votes]: The time is essentially the same as that for sorting numbers in the range $1,\dots M$.
Graham scan (in the version where the points are sorted by their x-coordinates (with ties broken by y coordinates) rather than the one the textbooks unaccountably use in which they are sorted radially) can find convex hulls in linear time after the sorting step. So for your problem in which the coordinates are small integers, all the standard bounds of integer sorting apply. In particular, radix sort can achieve time $O(n(1+\frac{\log M}{\log n}))$ (e.g. linear time whenever $M$ is at most polynomial in $n$), and a randomized sorting algorithm of Han & Thorup 2002 can achieve time $O(n\sqrt{\log\log n})$. These are in models of computation in which machine words hold integers of some word length that is big enough to store the numbers $n$ and $M$ and in which one can do arithmetic and table lookups in constant time.
Conversely, you could use a convex hull algorithm to sort a set of numbers $x_i$, by finding the hull of the points $(x_i,x_i^2)$. So if you could do convex hulls any faster than this, you could also improve the times for integer sorting. (The range $M$ of $y$-coordinates of these points would be the square of the range of the numbers, but since all the time bounds involve logarithms of this range, squaring $M$ makes no difference in terms of the eventual $O$-notation.)


TITLE: Arithmetic in cyclotomic field
QUESTION [0 upvotes]: I know that the basis of  $\mathbb{Z}(\zeta_9)$  is: $\{1,\zeta_9,\zeta_9^2,\zeta_9^3,\zeta_9^4,\zeta_9^5\}$ .
If I have for example:  $3+104\zeta_9+\zeta_9^7$ .
Is it a cyclotomic integer in  $\mathbb{Z}(\zeta_9)$ ?
Can I reduce  $\zeta_9^7$ to a linear combination of  $\zeta_9,\zeta_9^2,\zeta_9^3,\zeta_9^4,\zeta_9^5$  to be inside the basis of the ring?
Thank you

REPLY [3 votes]: The fact that $\{1,\zeta_9,\zeta_9^2,\zeta_9^3,\zeta_9^4,\zeta_9^5\}$ is a basis for $\Bbb Z(\zeta_9)$ automatically means that any element of $\Bbb Z(\zeta_9)$, including $\zeta_9^7$, can be written as a $\Bbb Z$-linear combination of $1,\zeta_9,\zeta_9^2,\zeta_9^3,\zeta_9^4,\zeta_9^5$. (Note that the $1$ needs to be included.)
The fact that the minimal polynomial of $\zeta_9$ is $x^6+x^3+1$, so that $\zeta_9^6 = -\zeta_9^3-1$, helps us do this concretely: we have
$$
\zeta_9^7 = \zeta_9 \zeta_9^6 = \zeta_9 (-\zeta_9^3-1) = -\zeta_9^4-\zeta_9.
$$