\begin{thm}\label{thm:Kitaev.vanishing}
If $U,V\in\mathcal{L}^\times$ satisfy
\begin{equation}\label{eqn:Kitaev.class}
(U-\mathbf{1})(V-\mathbf{1})\in\mathcal{L}^1\quad\mathrm{and}\quad (V-\mathbf{1})(U-\mathbf{1})\quad \in\mathcal{L}^1,
\end{equation}
then $\det(UVU^{-1}V^{-1})=1$.
\end{thm}

\label{sec:introduction}
We write $\mathcal{L}\equiv \mathcal{L}(\mathcal{H})$ for the ring of bounded linear operators on a Hilbert space $\mathcal{H}$, and $\mathcal{L}^1\equiv \mathcal{L}^1(\mathcal{H})$ for the ideal of \emph{trace class} operators. The group of invertibles in $\mathcal{L}$ is denoted $\mathcal{L}^\times$, and $\mathbf{1}$ denotes the identity operator on $\mathcal{H}$.

The following Theorem is our main result.

\section{Introduction}\label{sec:introduction}
We write $\mathcal{L}\equiv \mathcal{L}(\mathcal{H})$ for the ring of bounded linear operators on a Hilbert space $\mathcal{H}$, and $\mathcal{L}^1\equiv \mathcal{L}^1(\mathcal{H})$ for the ideal of \emph{trace class} operators. The group of invertibles in $\mathcal{L}$ is denoted $\mathcal{L}^\times$, and $\mathbf{1}$ denotes the identity operator on $\mathcal{H}$.

The Fredholm determinant is conjugation invariant, and is a group homomorphism from the invertible determinant class operators to the non-zero complex numbers. From these algebraic properties alone, it is straightforward to show that $\det(UVU^{-1}V^{-1})=1$ whenever any of $U,V$ or $UV$ is determinant class. Nevertheless, in general, $\det(UVU^{-1}V^{-1})$ can take \emph{any} non-zero value \cite{AV,EF}, and it is closely related to the rich subject of traces of commutators \cite{HH}, as well as $K$-theory \cite{Brown}.
Now, Kitaev's condition \eqref{eqn:Kitaev.class} implies
\begin{equation}\label{eqn:Kitaev.implies.det.class}
\begin{aligned}
UVU^{-1}V^{-1}-\mathbf{1}&=(UV-VU)U^{-1}V^{-1}\\
&=\big((U-\mathbf{1})(V-\mathbf{1})-(V-\mathbf{1})(U-\mathbf{1})\big)U^{-1}V^{-1}\in\mathcal{L}^1,
\end{aligned}
\end{equation}
so that $\det(UVU^{-1}V^{-1})$ is automatically well-defined, but it is not at all obvious that it must be trivial. Furthermore, because Kitaev's condition occurs naturally in geometric and physics problems (see Section \ref{sec:quantized.traces}), the veracity of Theorem \ref{thm:Kitaev.vanishing} is of considerable general interest for applications.

\section{Proof of determinant trivialization theorem}\label{sec:proof}
For each $n\geq 1$, we write $M_n(-)$ for the $n\times n$ matrix ring with entries in $(-)$, and $\GL_n(\mathcal{L})$ for the group of invertibles in $M_n(\mathcal{L})$. We may identify $M_n(\mathcal{L})$ with $\mathcal{L}(\mathcal{H}^{\oplus n})$, and $M_n(\mathcal{L}^1)$ with the ideal $\mathcal{L}^1(\mathcal{H}^{\oplus n})$. Thus, there are Fredholm determinant homomorphisms
\[
{\det}^{(n)}:\{U\in \GL_n(\mathcal{L})\,:\,U-\mathbf{1}^{\oplus n}\in M_n(\mathcal{L}^1)\}\to\mathbb{C}^\times,\qquad n\geq 1,
\]
and they are compatible with $\det=\det^{(1)}$ in the sense that
\begin{equation}\label{eqn.det.stable}
{\det}^{(n)}\begin{pmatrix} U & 0 \\ 0 & \mathbf{1}^{\oplus (n-1)}\end{pmatrix} = {\det}^{(1)}\, U,\qquad n\geq 1.
\end{equation}

By combining the Steinberg relations with the Kitaev condition \eqref{eqn:Kitaev.class}, and exploiting the redundancy Lemma \ref{lem:redundant.elementary}, we can rewrite $\det(UVU^{-1}V^{-1})$ as the determinant of a \emph{single} commutator of \emph{elementary matrices}:
\begin{prop}\label{prop:algebraic.step}
Let $U,V\in\mathcal{L}^\times$ satisfy the Kitaev condition . Then
\begin{equation}\label{eqn:algebraic.simplification}
\det(UVU^{-1}V^{-1})={\det}^{(2)}\big(e_{12}(U-\mathbf{1})e_{21}(V-\mathbf{1})e_{12}(\mathbf{1}-U)e_{21}(\mathbf{1}-V)\big).
\end{equation}
\end{prop}
\begin{proof}
According to the identities \eqref{eqn:mult.commutator.stabalized}--\eqref{eqn:Whitehead.big.UV},
\begin{equation}\label{eqn:det.alg.calc.first.factorization}
\det(UVU^{-1}V^{-1})={\det}^{(3)} \big(d_{12}(U)d_{13}(V)d_{12}(U)^{-1}d_{13}(V)^{-1}\big),
\end{equation}
where
\begin{equation}\label{eqn:for.shuffling}
\begin{aligned}
d_{12}(U) & = e_{21}(U^{-1})e_{12}(\mathbf{1}-U)e_{21}(-\mathbf{1})e_{12}(\mathbf{1}-U^{-1})\\
d_{13}(V) & = e_{31}(V^{-1})e_{13}(\mathbf{1}-V)e_{31}(-\mathbf{1})e_{13}(\mathbf{1}-V^{-1}).
\end{aligned}
\end{equation}
We would like to swap the positions of $d_{12}(U)$ and $d_{13}(V)$ in \eqref{eqn:det.alg.calc.first.factorization}, in order to cancel $d_{12}(U)^{-1}d_{13}(V)^{-1}$. This entails moving the elementary matrix factors of $d_{12}(U)$ past those of $d_{13}(V)$. Because of \eqref{eqn:Steinberg.commutator.123}, we will pick up some commutator terms of the form $e_{ik}(ST)$ or $e_{ki}(-TS)$ along the way.
In general, a complicated collection of commutator terms will be acquired.

Fortunately, for the purposes of computing the determinant \eqref{eqn:det.alg.calc.first.factorization}, Lemma \ref{lem:redundant.elementary} tells us that we can discard any $e_{ik}(ST)$ or $e_{ki}(-TS)$ term with $ST\in\mathcal{L}^1$ or $TS\in \mathcal{L}^1$, since this kind of elementary matrix has trivial determinant. The Kitaev condition \eqref{eqn:Kitaev.class} enters here: Because $(U-\mathbf{1})(V-\mathbf{1})$ is trace class, for $i\neq k$,
\begin{align*}
&{\det}^{(3)}\big(W e_{ij}(U-\mathbf{1})e_{jk}(V-\mathbf{1}) W^\prime\big)\\
&={\det}^{(3)}\big(W e_{ik}((U-\mathbf{1})(V-\mathbf{1}))e_{jk}(V-\mathbf{1})e_{ij}(U-\mathbf{1}) W^\prime\big) & (\mathrm{Eq.}\,\eqref{eqn:Steinberg.commutator.123})\\
&={\det}^{(3)}\big(W e_{jk}(V-\mathbf{1})e_{ij}(U-\mathbf{1}) W^\prime\big) & (\mathrm{Lemma}\;\ref{lem:redundant.elementary})
\end{align*}
holds whenever $W,W^\prime$ are invertible operators such that the above determinants are defined. Similarly with $U$ and $V$ swapped. The above property is conveniently expressed as an extra ``commutativity'' relation,
\[
e_{ij}(U-\mathbf{1})e_{jk}(V-\mathbf{1})\simeq e_{jk}(V-\mathbf{1})e_{ij}(U-\mathbf{1}),\qquad i\neq k,
\]
\emph{where ``$\simeq$'' indicates equality after taking a determinant}.
Kitaev's condition \eqref{eqn:Kitaev.class} also implies
\[
(U-\mathbf{1})V=U-\mathbf{1}=V(U-\mathbf{1}),\qquad
(V-\mathbf{1})U=V-\mathbf{1}=U(V-\mathbf{1})\qquad \mathrm{mod}\;\mathcal{L}^1,
\]
The extra relations made available by Kitaev's condition \eqref{eqn:Kitaev.class} are summarized below:
\begin{equation}\label{eqn:Steinberg.Kitaev.extra.identity.2}
\begin{aligned}
e_{ij}(U-\mathbf{1})e_{jk}(V-\mathbf{1}) &\simeq e_{jk}(V-\mathbf{1})e_{ij}(U-\mathbf{1}), & i\neq k,\\
e_{ij}(V-\mathbf{1})e_{jk}(U-\mathbf{1}) &\simeq e_{jk}(U-\mathbf{1})e_{ij}(V-\mathbf{1}), & i\neq k;\\
e_{ij}((U-\mathbf{1})V)&\simeq e_{ij}(U-\mathbf{1})\simeq e_{ij}(V(U-\mathbf{1})),\\
e_{ij}((V-\mathbf{1})U)&\simeq e_{ij}(V-\mathbf{1})\simeq e_{ij}(U(V-\mathbf{1})).
\end{aligned}
\end{equation}
As Kitaev's condition \eqref{eqn:Kitaev.class} holds with $U$ replaced by $U^{-1}$ and/or $V$ replaced by $V^{-1}$, the same replacements can be made in \eqref{eqn:Steinberg.Kitaev.extra.identity.2}.

Now specialize to the case $R=\mathcal{L}$ and $I=\mathcal{L}^1$. For $U,V\in \mathcal{L}^\times$ with determinant class $UVU^{-1}V^{-1}$, the above general definitions give
\begin{equation}\label{eqn:Brown.observation}
\begin{aligned}
\check{\det}\circ\partial\{u,v\}
&=\check{\det}\big[d_{12}(U)d_{13}(V)d_{12}(U)^{-1}d_{13}(V)^{-1}\big]_{K_1(\mathcal{L},\mathcal{L}^1)}\\
&=\check{\det}\left[\begin{pmatrix}UVU^{-1}V^{-1} & 0 & 0 \\ 0 & \mathbf{1} & 0 \\ 0 & 0 & \mathbf{1}\end{pmatrix}\right]_{K_1(\mathcal{L},\mathcal{L}^1)}\\
&=\det(UVU^{-1}V^{-1}).
\end{aligned}
\end{equation}
This $K$-theoretic nature of $\det(UVU^{-1}V^{-1})$ was first observed by Brown \cite{Brown}.

\subsection{Application to quantized traces}\label{sec:quantized.traces}\label{sec:application.quantized.traces}
Let $A,B\in\mathcal{L}$, and suppose
\begin{equation}\label{eqn:quantized.trace.hypothesis}
[A,B]\in \mathcal{L}^1\;\;\mathrm{and}\;\;(e^A-\mathbf{1})(e^B-\mathbf{1})\in\mathcal{L}^1.
\end{equation}
Then, as in \eqref{eqn:Kitaev.implies.det.class}, $(e^B-\mathbf{1})(e^A-\mathbf{1})$ is trace class as well. Thus, $e^A,e^B$ satisfy Kitaev's condition \eqref{eqn:Kitaev.class}, so Theorem \ref{thm:Kitaev.vanishing} guarantees that $\det(e^Ae^Be^{-A}e^{-B})=1$. By the Pincus--Helton--Howe formula \eqref{eqn:HH.formula}, we learn that the condition \eqref{eqn:quantized.trace.hypothesis} implies the trace quantization,
\[
\frac{1}{2\pi i}\mathrm{Tr}[A,B]\in\mathbb{Z}.
\]

This result was first suggested by A.~Kitaev in his influential paper \cite[pp.~56]{Kitaev}, for the purpose of demonstrating the integrality of certain trace formulae appearing in modern quantum physics (see Section \ref{sec:quantized.traces}). We shall therefore refer to \eqref{eqn:Kitaev.class} as the \emph{Kitaev condition} on an invertible pair $U,V\in \mathcal{L}^\times$.

\medskip
Recall that an operator $T\in\mathcal{L}$ is \emph{trace class} if for an(y) orthonormal basis $\mathcal{B}$, the sum $\sum_{e\in \mathcal{B}}\big\langle e\big||T|e\big\rangle$ is finite. In this case, its \emph{trace}
\[
\mathrm{Tr}(T)=\sum_{e\in \mathcal{B}}\big\langle e\big|Te\big\rangle \in\mathbb{C}
\]
is well-defined, independently of the choice of $\mathcal{B}$. An operator $U\in\mathcal{L}$ is \emph{determinant class} if $U-\mathbf{1}\in \mathcal{L}^1$, in which case, its \emph{(Fredholm) determinant} $\det(U)\in\mathbb{C}$ is well-defined. See \cite[Chapter 3]{Simon} for a detailed treatment. The identity
\[
\det(\exp(T))=\exp(\mathrm{Tr}(T)),\qquad T\in\mathcal{L}^1,
\]
shows that a trivial determinant is intimately related to integrality of a corresponding trace (up to a $2\pi i$ factor).

The Fredholm determinant is conjugation invariant, and is a group homomorphism from the invertible determinant class operators to the non-zero complex numbers. From these algebraic properties alone, it is straightforward to show that $\det(UVU^{-1}V^{-1})=1$ whenever any of $U,V$ or $UV$ is determinant class. Nevertheless, in general, $\det(UVU^{-1}V^{-1})$ can take \emph{any} non-zero value \cite{AV,EF}, and it is closely related to the rich subject of traces of commutators \cite{HH}, as well as $K$-theory \cite{Brown}.
Now, Kitaev's condition \eqref{eqn:Kitaev.class} implies
\begin{equation}\label{eqn:Kitaev.implies.det.class}
\begin{aligned}
UVU^{-1}V^{-1}-\mathbf{1}&=(UV-VU)U^{-1}V^{-1}\\
&=\big((U-\mathbf{1})(V-\mathbf{1})-(V-\mathbf{1})(U-\mathbf{1})\big)U^{-1}V^{-1}\in\mathcal{L}^1,
\end{aligned}
\end{equation}
so that $\det(UVU^{-1}V^{-1})$ is automatically well-defined, but it is not at all obvious that it must be trivial. Furthermore, because Kitaev's condition occurs naturally in geometric and physics problems (see Section \ref{sec:quantized.traces}), the veracity of Theorem \ref{thm:Kitaev.vanishing} is of considerable general interest for applications.

More than fifteen years after Kitaev's paper \cite{Kitaev} appeared, some progress was made in \cite{EF}, where Theorem \ref{thm:Kitaev.vanishing} was proved under the extra assumption that $(U^*-\mathbf{1})(V-\mathbf{1})$ and $(V-\mathbf{1})(U^*-\mathbf{1})$ are also trace class. Borel functional calculus played a key role in \cite{EF}, and it is unclear whether the techniques there can be improved to address the full \emph{adjoint-free} Theorem \ref{thm:Kitaev.vanishing}. Nevertheless, the result of \cite{EF} shows that there are no easy counterexamples to Theorem \ref{thm:Kitaev.vanishing}.

Our approach to Theorem \ref{thm:Kitaev.vanishing} is completely different, and we believe that it gets to the heart of the problem. Our proof is presented in Section \ref{sec:proof}, and comprises two main steps, both of which use the Kitaev condition \eqref{eqn:Kitaev.class} in crucial ways.
\begin{enumerate}
\item An \emph{algebraic} step, Prop.~\ref{prop:algebraic.step}, simplifies $\det(UVU^{-1}V^{-1})$ to $\det(E_1E_2E_1^{-1}E_2^{-1})$, where $E_1,E_2$ are so-called elementary matrices. This is the most novel part of the proof. Experts in algebraic $K$-theory may notice that the abstract concept of Steinberg symbols in $K_2$-theory is guiding our manipulations. We choose to keep the presentation concrete, and defer the $K$-theoretic interpretation to Section \ref{sec:K.perspective}.
\item An \emph{analytic} step computes $\det(E_1E_2E_1^{-1}E_2^{-1})=1$ using a formula of Pincus--Helton--Howe (Eq.~\eqref{eqn:HH.formula}), and Lidskii's trace theorem \cite{Lidskii}.
\end{enumerate}
Finally, the physical context for Theorem \ref{thm:Kitaev.vanishing} is discussed in Section \ref{sec:quantized.traces}.

The proof of Theorem~\ref{thm:Kitaev.vanishing} (given in Section~\ref{sec:proof}) has two main steps, both using the Kitaev condition \eqref{eqn:Kitaev.class} in crucial ways:
\begin{enumerate}
\item An \emph{algebraic} step (Prop.~\ref{prop:algebraic.step}) that “simplifies $\det(UVU^{-1}V^{-1})$ to $\det(E_1E_2E_1^{-1}E_2^{-1})$, where $E_1,E_2$ are so-called elementary matrices.” The text notes that “Steinberg symbols in $K_2$-theory [are] guiding our manipulations,” though the presentation is kept concrete.
\item An \emph{analytic} step that “computes $\det(E_1E_2E_1^{-1}E_2^{-1})=1$ using a formula of Pincus--Helton--Howe (Eq.~\eqref{eqn:HH.formula}), and Lidskii's trace theorem.”
\end{enumerate}
Additionally, the introduction observes that Kitaev's condition implies the commutator is determinant-class via \eqref{eqn:Kitaev.implies.det.class}, so $\det(UVU^{-1}V^{-1})$ is well-defined, though “it is not at all obvious that it must be trivial.”

The proof of \ref{thm:Kitaev.vanishing} (given later) has two main steps, both using the Kitaev condition
\begin{equation}\label{eqn:Kitaev.class}
(U-\mathbf{1})(V-\mathbf{1})\in\mathcal{L}^1\quad\mathrm{and}\quad (V-\mathbf{1})(U-\mathbf{1})\quad \in\mathcal{L}^1,
\end{equation}
in crucial ways:
\begin{enumerate}
\item An \emph{algebraic} step. We first prove the following proposition.
\begin{prop}\label{prop:algebraic.step}
Let $U,V\in\mathcal{L}^\times$ satisfy the Kitaev condition . Then
\begin{equation}\label{eqn:algebraic.simplification}
\det(UVU^{-1}V^{-1})={\det}^{(2)}\big(e_{12}(U-\mathbf{1})e_{21}(V-\mathbf{1})e_{12}(\mathbf{1}-U)e_{21}(\mathbf{1}-V)\big).
\end{equation}
\end{prop}
This “simplifies $\det(UVU^{-1}V^{-1})$ to $\det(E_1E_2E_1^{-1}E_2^{-1})$, where $E_1,E_2$ are so-called elementary matrices.” The text notes that “Steinberg symbols in $K_2$-theory [are] guiding our manipulations,” though the presentation is kept concrete.
\item An \emph{analytic} step that “computes $\det(E_1E_2E_1^{-1}E_2^{-1})=1$ using a formula of Pincus--Helton--Howe,
\begin{equation}\label{eqn:HH.formula}
\det(e^Ae^Be^{-A}e^{-B})=\exp(\mathrm{Tr}[A,B]).
\end{equation}
and Lidskii's trace theorem.”
\end{enumerate}
Additionally, the introduction observes that Kitaev's condition implies the commutator is determinant-class via \eqref{eqn:Kitaev.implies.det.class}, so $\det(UVU^{-1}V^{-1})$ is well-defined, though “it is not at all obvious that it must be trivial.”

Question:
Let mathcal{H} be a Hilbert space, let \(\mathcal{L}(\mathcal{H})\) be the algebra of bounded linear operators on \(\mathcal{H}\), let \(\mathcal{L}^1(\mathcal{H})\) be the ideal of trace-class operators, let \(\mathcal{L}(\mathcal{H})^\times\) denote the invertible bounded operators, and let \(\mathbf{1}\) be the identity operator. Suppose \(U,V\in \mathcal{L}(\mathcal{H})^\times\) satisfy
\[
(U-\mathbf{1})(V-\mathbf{1})\in \mathcal{L}^1(\mathcal{H})
\quad\text{and}\quad
(V-\mathbf{1})(U-\mathbf{1})\in \mathcal{L}^1(\mathcal{H}).
\]
For an invertible operator of the form \(\mathbf{1}+K\) with \(K\in \mathcal{L}^1(\mathcal{H})\), write \(\det\) for its Fredholm determinant. Under these hypotheses, which statement about the multiplicative commutator \(UVU^{-1}V^{-1}\) holds?

The Fredholm determinant of the commutator is trivial: \(\det(UVU^{-1}V^{-1})=1\).

Choices:
B.
The multiplicative commutator is always determinant class under these hypotheses, and its Fredholm determinant is given by the trace formula
\[
\det(UVU^{-1}V^{-1})=\exp\!\big(\operatorname{Tr}((U-\mathbf{1})(V-\mathbf{1})-(V-\mathbf{1})(U-\mathbf{1}))\big).
\]
C.
The multiplicative commutator is determinant class, i.e.
\[
UVU^{-1}V^{-1}-\mathbf{1}\in \mathcal{L}^1(\mathcal{H}).
\]
D.
If \((U-\mathbf{1})(V-\mathbf{1})\in \mathcal{L}^1(\mathcal{H})\), then automatically
\[
\det(UVU^{-1}V^{-1})=1,
\]
without any need to assume \((V-\mathbf{1})(U-\mathbf{1})\in \mathcal{L}^1(\mathcal{H})\).
E.
Under these hypotheses the commutator itself is trace class relative to the identity in the stronger sense that
\[
UVU^{-1}V^{-1}=\mathbf{1},
\]
so in particular its Fredholm determinant equals \(1\).

analytic-step trace formula applied directly to U,V rather than logarithmic elementary-matrix commutator

drop the final trivialization conclusion and keep only determinant-class/well-definedness

two-sided Kitaev condition reduced to one-sided trace-class product hypothesis

replace determinant triviality of the commutator by actual operator triviality

The stem states the hypotheses and asks for the resulting conclusion, but it does not explicitly reveal that the determinant equals 1 or strongly telegraph choice A over the others.

This is close to a direct theorem-recall item: the hypotheses are presented in essentially theorem form and the correct option states the standard conclusion. However, the presence of nearby alternatives (weaker true statement, overstrong statement, and altered hypotheses) prevents it from being a pure tautological restatement.

The item requires some reasoning to distinguish the exact conclusion from a merely weaker consequence (C), an unjustified stronger claim (E), and a one-sided hypothesis variant (D). Still, for a student who recognizes the theorem, the answer is fairly immediate rather than deeply generative.

The distractors are mathematically plausible and target realistic failure modes: confusing determinant triviality with trace-class membership (C), overextending to a one-sided condition (D), inserting an appealing but unjustified trace formula (B), or strengthening determinant triviality to operator equality (E).

A solid, high-quality theorem-based MCQ with strong distractors and no answer leakage, but it remains somewhat recall-oriented and only moderately non-tautological.


For almost all irrational $\theta\in [0,1]$, if we denote by $q_k/p_k$ the sequence of convergents of the continued fraction expansion of $\theta$, then one has
$$\lim_{k\to \infty}\frac{\sigma_2(q_k/p_k)}{p_k^2}=\frac{16}{\pi^3}\sum_{n\ge 1 \textrm{ odd}}\frac{1}{n^3\sin(n\pi\theta)}.$$

\begin{Remark}
  {\em
The $\mathrm{SU}_2$-TQFT $\boV_p$ corresponds in Conformal Field Theory to the Wess-Zumino-Witten model at odd level $\ell=p-2$. Similar $\mathrm{SU}_2$-TQFTs exist at  even levels, that is, when $p$ is even. Their signature functions can again be computed using the Frobenius algebra formalism of Theorem~\ref{DM1}.  But we won't consider even level $\mathrm{SU}_2$-TQFTs in this paper.
}\end{Remark}

\section{A trigonometric formula for the signature in genus 2}\label{genre2}
Let us now start our investigation of the asymptotic behavior of the signature function. The first result of this paper is the following trigonometric formula for the signature in genus $2$.

\begin{Theorem}\label{sigma2}
  For $0<q<p$ coprime odd integers, we have
\begin{equation}\nonumber \sigma_2({{\textstyle\frac q p}})= \frac {1-p^2}{6p^2} +
  \frac 1 {4p^2}\sum_{n=1,\, \mathrm{ odd}}^{p-2}\frac{f(n,p,q)}{\sin^3(\frac{n\pi}{2p})\sin^2(\frac{qn\pi}{p})}
\end{equation}
where
\begin{eqnarray*} f(n,p,q)&=(3p-3)\sin(\frac{(2q-1)n\pi}{2p})+(p+1)\sin(\frac{(2q-3)n\pi}{2p})\\
&+(p-1)\sin(\frac{(2q+3)n\pi}{2p})+(3p+3)\sin(\frac{(2q+1)n\pi}{2p}).
\end{eqnarray*}
\end{Theorem}

\begin{proof}
By a finite Fourier transform over $\Z/2p\Z$, we can write for $m\in\Z$ :
\begin{equation}\label{FFT}
  (-1)^{\lfloor m/p\rfloor}=
  \frac 1 p
  \sum_{n=1,\text{ odd}}^{2p-1}\frac{\sin(\pi n(2m+1)/2p)}{\sin(\pi n/2p)}
  =\frac 1 p  \sum_{n=1,\text{ odd}}^{2p-1} \frac {t^{2m+1}-t^{-2m-1}}{t-t^{-1}}
    \Big|_{t=e^{\frac {i\pi n}{2p}}~.
      }
\end{equation}
In the sequel, we set $\O_p=\{1,3,\ldots,2p-1\}$ and $\O_p^+=\{1,3,\ldots,p\}$. Using Formula \eqref{FFT} for each sign $\epsilon_{j+1}=(-1)^{\lfloor q(j+1)/p\rfloor}$ and plugging it into the formula of Lemma \ref{triplepsilon}, we get
\begin{equation}\nonumber\sigma_2({\textstyle{\frac q p}})\,\,
  =\sum_{(j,k,\ell)\in T_p} \epsilon_{j+1}\epsilon_{k+1}\epsilon_{\ell+1}
  =\frac1 {p^3} \sum_{(a,b,c)\in \O_p^3}\psi(a,b,c)\end{equation}
where
\begin{equation}
\psi(a,b,c)=\!\!\!\!\sum\limits_{(j,k,l)\in T_p}\!\!\!\!
\frac{(x^{2q(j+1)+1}-x^{-2q(j+1)-1})(y^{2q(k+1)+1}-y^{-2q(k+1)-1})(z^{2q(\ell+1)+1}-z^{-2q(\ell+1)-1})}{(x-x^{-1})(y-y^{-1})(z-z^{-1})}\nonumber
\end{equation}
with $x=e^{\frac{i\pi a}{2p}}$,  $y=e^{\frac{i\pi b}{2p}}$ and $z=e^{\frac{i\pi c}{2p}}$.

The function $\psi$ is clearly symmetric in its three variables and satisfies also $\psi(2p-a,b,c)=\psi(a,b,c)$. We can hence rewrite $\sigma_2(\frac q p)$  as
\begin{equation}\label{eq4n}\sigma_2({\textstyle{\frac q p}})=
  \frac1 {p^3}
  \sum_{a,b,c\in \O_p^+} 2^{m_a+m_b+m_c}\psi(a,b,c)\end{equation}
where $m_a=1$ if $a\ne p$ and $m_p=0$.

In order to compute $\psi(a,b,c)$, we rewrite it as follows:
\begin{equation} \psi(a,b,c)= \sum\limits_{\epsilon,\mu,\nu=\pm 1}\epsilon\mu\nu \frac{ x^{\epsilon}y^{\mu}z^{\nu}F(x^{2q\epsilon},y^{2q\mu},z^{2q\nu})}{(x-x^{-1})(y-y^{-1})(z-z^{-1})}\label{eq5n}
\end{equation} where $F$ is the polynomial defined by $F(X,Y,Z)=\sum_{(j,k,l)\in T_p}X^{j+1}Y^{k+1}Z^{\ell +1}$.

The computation of $F(x^{2q\epsilon},y^{2q\mu},z^{2q\nu})$ is simplified by the following observation:  we have $$F(X,Y,Z)=X^pY^p F(X^{-1}, Y^{-1},Z)~,$$ since the involution $(j,k,\ell ) \mapsto (p-2-j,p-2-k,\ell)$ preserves the set $T_p$. Therefore, since $F$ is symmetric in its three variables,  whenever $F(X,Y,Z)$ is evaluated at numbers $X=x^{2q}$, $Y=y^{2q}$, and $Z=z^{2q}$
satisfying $X^p=Y^p=Z^p=-1$, then for  any $\epsilon,\mu,\nu=\pm 1$, we have that $F(X^\epsilon, Y^\mu,Z^\nu)$ is equal to either $F(X,Y,Z)$ (when $\epsilon\mu\nu =1$) or to  $F(X^{-1}, Y^{-1},Z^{-1})= \overline {F(X,Y,Z)}$ (when $\epsilon\mu\nu =-1$).
\begin{Remark}\label{rk33}{\em For later use, we note the following special case: when $F(x^{2q},y^{2q},z^{2q})$ happens to be real, then $F(x^{2q\epsilon},y^{2q\mu},z^{2q\nu})$
    does not depend at all on $\epsilon,\mu$, or $\nu$, and formula \eqref{eq5n} for $\psi(a,b,c)$ simply becomes $\psi(a,b,c)=F(x^{2q},y^{2q},z^{2q})$.}
    \end{Remark}

Let $p$ be an odd positive integer and $\mathbb{Q}(\zeta)$ be the cyclotomic field generated by a primitive root of unity $\zeta$ of order $2p$. The SU$_2$-TQFT constructs for any surface $S_g$ of genus $g$ a finite dimensional $\mathbb{Q}(\zeta)$-vector space $\boV_p(S_g)$ endowed with an Hermitian form $h_g$ defined over $\mathbb{Q}(\zeta)$. These vector spaces are part of a modular functor, meaning that they support a projective unitary action of the corresponding mapping class groups and satisfy a list of compatibilities when cutting the surfaces along simple closed curves.

The dimension of $\boV_p(S_g)$, given by the celebrated Verlinde formula, has received considerable attention since the nineties. On the other hand, taking $0<q<p$ two coprime odd integers, one can embed $\mathbb{Q}(\zeta)$ into $\mathbb{C}$ by sending $\zeta$ to $e^{i\pi q/p}$. The Hermitian form $h_g$ has a signature that we denote by
$$\sigma_g(\textstyle{\frac{q}{p}})=\Sign\Big(\boV_p(S_g)\!\!\underset{\zeta=e^{i\pi q/p}}{\otimes}\!\!\mathbb{C},\,h_g\Big)$$
and which has been very little studied. It was shown in \cite{DM} that for a fixed $\zeta=e^{i\pi q/p}$, these signatures have cut and paste properties very similar to those of the dimensions, which allows to encode them in a single Frobenius algebra $V_{\frac q p}$. Later it was proved in \cite{S2B} that $V_{\frac q p}$ is closely related to the hyperbolic geometry of the two-bridge knot $K(p,q)$. The purpose of this paper is to study signatures in a different direction, by analyzing the asymptotic behavior of $\sigma_{g}(\frac{q}{p})$ when $\frac{q}{p}\to \theta$ for a fixed $\theta\in [0,1]$.

If we take $q=1$, the Hermitian form $h_g$ becomes positive definite,
so that $\sigma_g(\frac 1 p)$ coincides with the dimension of $\boV_p(S_g)$. The problem of computing the asymptotic behavior of the dimensions when $p\to \infty$ is well-known, and related to considerations of semi-classical analysis on character varieties $\mathcal{M}_g=\Hom(\pi_1(S_g),\SU_2)/\SU_2$. This led Witten \cite{Witten} to show that for $g>1$,
$$\lim_{p\to \infty}\frac{\dim \boV_p(S_g)}{p^{3g-3}}=\operatorname{Vol}(\mathcal{M}_g)=2(2\pi^2)^{1-g}\zeta(2g-2).$$

In genus $0$ and $1$, the hermitian form is positive definite for all values of $q$, so let us consider the first non trivial case $g=2$. In this case, one can write a simple formula for $\sigma_2(\frac q p)$:
\begin{equation}\label{defsigma2}
 \sigma_2({\textstyle{\frac{q}{p}}})=\sum_{(j,k,\ell)\in \Delta_p}
 (-1)^{\lfloor {jq}/{p}\rfloor +\lfloor {kq}/{p}\rfloor+\lfloor {\ell q}/{p}\rfloor}
\end{equation}
 where the sum runs over the set $\Delta_p$ of triples of integers satisfying
$$\begin{cases}0<j,k,\ell<p, \\
j<k+\ell,k<j+\ell,\ell<j+k,\\
j+k+\ell<2p,\textrm{ and } j+k+\ell\textrm{ odd}.
\end{cases}$$

Our main result is the following

Our main result is the following

Figure \ref{fig:sigma2} suggests that the convergence is rapid: the graph of the map on the right hand side is shown in green whereas red dots represent the points $(q/p,\sigma_2(q/p)/p^2)$ for $0<q<p<31$.

We briefly review the properties of the convergents $q_k/p_k$ of $\theta$ that we will need. See {\em e.g.} Perron \cite{P} or Khinchin \cite{Kh1} for more details\footnote{Unfortunately, \cite{Kh1} writes $p_k$ for our $q_k$ and {\em vice versa.}}. Each irrational $\theta\in {\mathbb R}$ has a unique regular continued fraction expansion $$\theta = [a_0; a_1, a_2, \ldots ] = a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\dots}} $$ where regular means that $a_0\in \Z$ and $a_i\in \Z_{\geq 1}$ for $i\geq 1$. Note that in our case $a_0=0$ since
$\theta \in [0,1]\setminus \Q$.
The convergents
$$\frac{q_k}{p_k} = [a_0; a_1, a_2, \ldots, a_k] = a_0+\cfrac{1}{a_1+\cfrac{1}{\dots+ \frac{1}{a_k}}} $$ can also be defined recursively by
\begin{eqnarray} q_k&=a_k q_{k-1} + q_{k-2}, \ \ \ \ q_0=a_0, \ \ \ \ q_{-1}=1 \label{rec5}\\
 p_k&=a_k p_{k-1} + p_{k-2}, \ \ \ \ p_0=1, \ \ \ \ p_{-1}=0 \label{rec6}
\end{eqnarray}
The following classical facts are well-known.
\begin{Proposition}\label{qpfacts}
 For all irrational $\theta$, the convergents $q_k/p_k$ of $\theta$ satisfy the following properties:
 \begin{enumerate}
 \item[(i)] One has $1=p_0\leq p_1<p_2<\ldots$, {\em i.e.} the denominators $p_k$ are positive integers and moreover strictly increasing from $p_1$ onwards.
 \item[(ii)] For all $k\geq 1$, $q_k$ and $p_k$ are coprime.
 \item[(iii)] $\lim_{k\rightarrow\infty} q_k/p_k = \theta$.
 \item[(iv)]
 For all $i\geq 1$, we have
 \begin{equation}\nonumber
 \frac 1 {p_i(p_i+p_{i+1})} <
 \Big|\theta - \frac {q_i}{p_i} \Big| < \frac 1 {p_ip_{i+1}} < \frac 1 {p_i^2}~.
 \end{equation}
 \item[(v)] {\em [Lagrange]}\footnote{For a proof, see Perron \cite[\S 15, Satz 2.17]{P}.} For all $1\leq i<k$, and any $n,\ell \in\Z$ with $1\leq n<p_{i+1}$, we have
 \begin{equation}\nonumber
 \Big| n\frac {q_k}{p_k} -\ell \Big|
 \geq \Big|p_i\frac {q_k}{p_k} - q_i\Big| \geq \frac{1}{p_i + p_{i+1}} ~.
 \end{equation}
 \end{enumerate}
 \end{Proposition}

\begin{proof}[\bf{Proof of Theorem~\ref{asympto}}] We start from the trigonometric formula for $\sigma_2(\frac{q}{p})$ given in Theorem~\ref{sigma2}. The first observation is that the term $(1-p^2)/(6p^2)$ in this formula stays bounded as $p\rightarrow \infty$ and so does not contribute to the limit of $\sigma_2(\frac{q_k}{p_k})/ p_k^2$. Thus it suffices to show
 \begin{equation}\label{eq4}
 \lim_{k\rightarrow \infty} \frac{1}{4p_k^4} \sum_{n=1,\textrm{ odd}}^{p_k-2}\frac{f(n,p_k,q_k)}{\sin^3({n\pi}/{2p_k})\sin^2({q_kn\pi}/{p_k})} =\frac{16}{\pi^3}\sum_{n\ge 1, \textrm{odd}}\frac{1}{n^3\sin(n\pi\theta)}
 \end{equation} with $f(n,p,q)$ given as in Theorem~\ref{sigma2}.

Let $S^{(1)}(q_k,p_k,\theta)$ denote the sum $S^{(1)}(q_k,p_k)$ but with $\sin(n\pi\theta)$ in the denominator in place of $\sin({q_kn\pi}/{p_k})$. Using (\ref{sin-ineq}), we see that the $n$-th summand of $S^{(1)}(q_k,p_k,\theta)$ is bounded from above by $1/(n^3\sin(n\pi\theta))$. So we can apply the Lebesgue dominated convergence theorem to this sum. We get
 $$\lim_{k\rightarrow \infty}S^{(1)}(q_k,p_k,\theta) =\frac{8}{\pi^3}\sum_{n\ge 1, \textrm{odd}}\frac{1}{n^3\sin(n\pi\theta)}~.$$
 Thus to prove Lemma~\ref{3.3} it suffices to show that the difference $d_k=S^{(1)}(q_k,p_k) -S^{(1)}(q_k,p_k,\theta)$ goes to zero as $k\rightarrow \infty$. We have
 $$d_k= \frac{1}{p_k^3} \sum_{n=1,\textrm{ odd}}^{p_k-2}\frac{\cos (n\pi/2p_k)}{\sin^3({n\pi}/{2p_k}) }\Biggl (\frac {1} {\sin({q_kn\pi}/{p_k})} - \frac {1} {\sin(n\pi\theta)}\Biggr)~.$$ We can bound the denominators from below using (\ref{sin-ineq}), (\ref{sinn-inequ}), and the trivial lower bound $|\sin({q_kn\pi}/{p_k})| \geq \sin (\pi/p_k) \geq 2/p_k$. Thus for all $\epsilon >0$ we have

\begin{proof}[\bf{Proof of Lemma~\ref{3.4}}]
 We assume that $\mu(\theta)=2$ and that $\theta$ satisfies the conditions
 in Theorem~\ref{ThKL}.
 Recall $S^{(2)}(q_k,p_k)$, defined in (\ref{sum2}), is a sum from $n=1$ to $n=p_k-2$. We cut
it into two parts, as follows. Put
\begin{equation}\label{floor} N(k)= \lfloor 3k/4 \rfloor
\end{equation} and write $$S^{(2)}(q_k,p_k) = S^{(2.1)}(q_k,p_k)
+S^{(2.2)}(q_k,p_k)~,$$ where $S^{(2.1)}(q_k,p_k)$ is the sum from
$n=1$ to $n=p_{N(k)+1}-1$ and $S^{(2.2)}(q_k,p_k)$ is the sum from
$n=p_{N(k)+1}$ to $p_k-2$. The following two lemmas provide upper bounds for these two sums.
\begin{Lemma}\label{3.9} We have $$|S^{(2.1)}(q_k,p_k) |\leq \frac
{p_{N(k)}} {p_k} F_7(k)$$ where $F_7(k) ={\mathcal O}(k^7)$.
\end{Lemma}
\begin{Lemma}\label{3.10} We have $$|S^{(2.2)}(q_k,p_k)| \leq \frac{1}{2} \zeta(2)
\frac {p_{k}} {p_{N(k)+1}^2}~.$$
\end{Lemma}
Assuming these two lemmas for the moment, since $\lim_{m\rightarrow\infty} p_m^{1/m}= L>1$ by the Khinchin-L{\'e}vy Theorem~\ref{ThKL}(ii), $p_k$ grows like $L^k$ as $k\rightarrow \infty$. It is not hard to deduce from this that both $ S^{(2.1)}(q_k,p_k)$ and $ S^{(2.2)}(q_k,p_k)$ converge to zero.~\footnote{Here is how to see this for $ S^{(2.1)}(q_k,p_k)$. Theorem~\ref{ThKL}(ii) gives that for all $\epsilon>0$ there exist $m_0(\epsilon)$ so that $L^{1-\epsilon} \leq p_m^{1/m} \leq L^{1+\epsilon}$ holds for for all $m\geq m_0(\epsilon)$. Thus also $L^{m-m\epsilon} \leq p_m\leq L^{m+m\epsilon}$ for all $m\geq m_0(\epsilon)$. Applying now the first inequality with $m=k$ and the second one with $m=N(k)=\lfloor 3k/4\rfloor$ and taking $\epsilon$ small enough, it follows that $p_{N(k)}/p_k \leq L^{-\delta k}$ for some $\delta>0$ as soon as $k\geq 2 m_0(\epsilon)$. Thus $p_{N(k)} F_7(k)/ {p_k} \leq k^7 L^{-\delta k}$ for $k$ big enough, proving that the limit as $k\rightarrow\infty$ is zero, as asserted. }
This proves Lemma~\ref{3.4}.
\end{proof}
It remains to prove Lemmas~\ref{3.9} and~\ref{3.10}. Let us first deal
with Lemma~\ref{3.10} whose proof is easy.
\begin{proof}[\bf{Proof of Lemma~\ref{3.10}}] Since $n\geq p_{N(k)+1}$
 in this sum,
 we can bound the
$\sin^2(n\pi/2p_k)$ term in the denominator from below by
$p_{N(k)+1}^2/p_k^2$. Thus
\begin{equation}\label{sum3}|S^{(2.2)}(q_k,p_k)| \leq \frac 1 {p_k
p_{N(k)+1}^2}\sum_{n=1,\textrm{ odd}}^{p_k-2} \frac 1
{\sin^2(q_kn\pi/p_k)}~.\end{equation}

\section{The higher genus case}\label{sec5}
In this section, we briefly discuss how the asymptotic behaviour of the genus $2$ signature might generalize to higher genus. Numerical experiments (see below) indicate that
 $\sigma_{g,n}(\frac{q}{p};\lambda)$ might grow like $p^{\max\{g,2g-2\}}$ when $\frac q p$ goes to an irrational $\theta\in[0,1]$. So we ask the following question.
\begin{Question}\label{51}
 Given $g,n\in \Z_{\geq 0}$ and $n$-tuples of integers $\lambda=(\lambda_1,\ldots,\lambda_n)\in \Z^n_{\geq 0}$,
 is there a function $F_{g;\lambda}:
 [0,1]
 \to \R$ defined almost everywhere satisfying the following:
 For almost all irrational $\theta\in[0,1]$,
 if we write $\frac{q_k}{p_k}$ for the convergents of $\theta$, one has
$$ \lim_{k\to\infty}\frac{\sigma_{g,n}(\frac{q_k}{p_k};\lambda)}{p_k^{\max\{g,2g-2\}}}=F_{g;\lambda}(\theta)$$
\end{Question}
Theorem \ref{asympto} shows that the answer is yes for $g=2$ and $n=0$. Let us
compare this to the asymptotics of the dimension
of the TQFT vector spaces
which
is well-known: it has a semi-classical interpretation for which we refer to \cite[Section 3]{Witten}. Precisely:
$$ \lim_{p\to\infty}\frac{\dim \boV_p(S_g)}{p^{\max\{g,3g-3\}}}=\operatorname{Vol}(\mathcal{M}_g)=2(2\pi^2)^{1-g}\zeta(2g-2)$$
where the last equality holds only for $g>0$ and $\mathcal{M}_g$ is the character variety of representations of $\pi_1(S_g)$ into SU$_2$, endowed with its Liouville measure. We have no conceptual explanation why the order of growth $3g-3$
for the dimension
should be replaced by $2g-2$
for the signature.

Figure \ref{fig:sigma2}  suggests that the convergence is rapid: the graph of the map on the right hand side is shown in green whereas red dots represent the points $(q/p,\sigma_2(q/p)/p^2)$ for $0<q<p<31$.

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=10cm]{genre2.png}
\caption{Graph of the signature function $\sigma_2$}\label{fig:sigma2}
\end{center}
\end{figure}

The proof of this theorem involves several steps and occupies Sections~\ref{genre2} and~\ref{sec2}. We first apply in Section~\ref{genre2} a finite Fourier transform in Equation \eqref{defsigma2} to transform this expression into a sum of evaluations of a rational function at roots of unity indexed by $\Delta_p$. We then apply Brion's theorem to get an explicit trigonometric sum computing $\sigma_2(q/p)$ (see Theorem \ref{sigma2} for the precise statement): this step already presents miraculous cancellations for which we do not have any conceptual explanation.

Then in Section~\ref{sec2}, we analyse the asymptotic behavior of the trigonometric sum in the most technical part of the paper. In order to control its denominators, we need to impose that the coefficients of the continued fraction expansion of $\theta$ do not grow too quickly. We use for that  some classical results of Khinchin on metric Diophantine approximation and a variation on a technique due to Zaremba.

In higher genus one can also write down a formula for
  $\sigma_g(q/p)$ as the sum of signs indexed by lattice points inside some polytope (see \cite[Remark 4.12]{BHMV}~\footnote{Our $\boV_p$ corresponds to the $V_{2p}$ of \cite{BHMV}.}.) But the signs are much more complicated than in Formula \eqref{defsigma2} which prevents us from generalizing our theorem. We briefly discuss in Section~\ref{sec5} how the asymptotics might look like for general surfaces including marked points, and illustrate this by some numerical experiments.

In the last section of the paper, we discuss some modular properties of the signature. We observe that the function
$$\Lambda(\theta)=\frac{16}{\pi^3}\sum_{n\ge 1\textrm{ odd}}\frac{1}{n^3\sin(n\pi\theta)}$$
is the boundary value of an Eichler integral of a modular form of weight $4$ for the level $2$ group $\Gamma(2)$. In addition to the obvious property $\Lambda(\theta+1)=-\Lambda(\theta)$, it satisfies the following transformation law:
\begin{equation}\label{Lambdamod}
\Lambda\Big(\frac{\theta}{2\theta+1}\Big)(2\theta+1)^2-\Lambda(\theta)=2\theta^2+2\theta+1.
\end{equation}
This functional equation suggests that a similar integral version holds for signatures as stated in the following conjecture, checked by computer for all $0<q<p<100$.
\begin{Conjecturestar}
For any odd coprime integers $0<q<p$ one has
$$\sigma_2\Big(\frac{q}{2q+p}\Big)-\sigma_2\Big(\frac{q}{p}\Big)=2q^2+2qp+p^2-1.$$
\end{Conjecturestar}

This formula is reminiscent of the reciprocity formula of Dedekind sums: in fact we will make this analogy more precise in Section~\ref{SectionS}. We prove that the signature at $\zeta=e^{i\pi q/p}$ of  the TQFT-vector space associated to the 4 times punctured sphere (with  colors $(p-1)/2$  at the marked points)  is actually equal to a 2-smoothed version, $S(q/p)$,  of the Dedekind sum $s(q,p)$. It is then classical that  this function $S(q/p)$ is the boundary value of a modular form (this time of weight $1/2$) and that it satisfies the following transformation law

$$S\Big(\frac{q}{2q+p}\Big)-S\Big(\frac{q}{p}\Big)=1.$$
We provide an elementary proof of these facts for the benefit of the non-specialist reader.

This very satisfactory state of things would hopefully generalize to higher genus as well. However, even in genus 2, a precise (non asymptotic) relation between the signature $\sigma_g(q/p)$
and the Eichler integral is still lacking. Such a relation would presumably lead to a proof  of the conjectured transformation law for $\sigma_2$.

To end this introduction, we would like to mention the parallel paper \cite{MY} in which the first author and S. Yoon compute the asymptotics of the function
$\sigma_g(q/p)$, but for sequences of the form $\frac{q_n}{p_n}=\frac{a+bn}{c+dn}$ where $\left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right)\in \Gamma(2)$. This regime is somewhat orthogonal to the one described in the present paper. The techniques used and the results are also very different. A synthesis of the two papers has yet to be made.
\vskip 8pt

{\bf Acknowledgements:} It is our pleasure to thank Pierre Charollois for early discussions on many aspects of this paper, Gaëtan Chenevier for his expertise on modular forms,
and Don Zagier for very helpful explanations about Eichler integrals and quantum modular forms.

The proof “involves several steps and occupies Sections~\ref{genre2} and~\ref{sec2}.” First, “we apply in Section~\ref{genre2} a finite Fourier transform in Equation \eqref{defsigma2} to transform this expression into a sum of evaluations of a rational function at roots of unity indexed by $\Delta_p$.” Next, “we then apply Brion's theorem to get an explicit trigonometric sum computing $\sigma_2(q/p)$ (see Theorem \ref{sigma2} for the precise statement),” a step that “already presents miraculous cancellations.” Then, “in Section~\ref{sec2}, we analyse the asymptotic behavior of the trigonometric sum in the most technical part of the paper.” To “control its denominators,” one “need[s] to impose that the coefficients of the continued fraction expansion of $\theta$ do not grow too quickly,” using “classical results of Khinchin on metric Diophantine approximation and a variation on a technique due to Zaremba.”

The proof “involves several steps and occupies” the part starting at

\begin{Remark}
  {\em
The $\mathrm{SU}_2$-TQFT $\boV_p$ corresponds in Conformal Field Theory to the Wess-Zumino-Witten model at odd level $\ell=p-2$. Similar $\mathrm{SU}_2$-TQFTs exist at  even levels, that is, when $p$ is even. Their signature functions can again be computed using the Frobenius algebra formalism of Theorem~\ref{DM1}.  But we won't consider even level $\mathrm{SU}_2$-TQFTs in this paper.
}\end{Remark}

\section{A trigonometric formula for the signature in genus 2}\label{genre2}
Let us now start our investigation of the asymptotic behavior of the signature function. The first result of this paper is the following trigonometric formula for the signature in genus $2$.

\begin{Theorem}\label{sigma2}
  For $0<q<p$ coprime odd integers, we have
\begin{equation}\nonumber \sigma_2({{\textstyle\frac q p}})= \frac {1-p^2}{6p^2} +
  \frac 1 {4p^2}\sum_{n=1,\, \mathrm{ odd}}^{p-2}\frac{f(n,p,q)}{\sin^3(\frac{n\pi}{2p})\sin^2(\frac{qn\pi}{p})}
\end{equation}
where
\begin{eqnarray*} f(n,p,q)&=(3p-3)\sin(\frac{(2q-1)n\pi}{2p})+(p+1)\sin(\frac{(2q-3)n\pi}{2p})\\
&+(p-1)\sin(\frac{(2q+3)n\pi}{2p})+(3p+3)\sin(\frac{(2q+1)n\pi}{2p}).
\end{eqnarray*}
\end{Theorem}

and the later section \ref{sec2}.

First, “we apply” a finite Fourier transform in the defining expression
\begin{equation}\label{defsigma2}
  \sigma_2({\textstyle{\frac{q}{p}}})=\sum_{(j,k,\ell)\in \Delta_p}
 (-1)^{\lfloor {jq}/{p}\rfloor +\lfloor {kq}/{p}\rfloor+\lfloor {\ell q}/{p}\rfloor}
\end{equation}
to transform this expression into a sum of evaluations of a rational function at roots of unity indexed by $\Delta_p$.

Next, “we then apply Brion's theorem to get an explicit trigonometric sum computing $\sigma_2(q/p)$,” namely the formula stated in
\begin{Theorem}\label{sigma2}
  For $0<q<p$ coprime odd integers, we have
\begin{equation}\nonumber \sigma_2({{\textstyle\frac q p}})= \frac {1-p^2}{6p^2} +
  \frac 1 {4p^2}\sum_{n=1,\, \mathrm{ odd}}^{p-2}\frac{f(n,p,q)}{\sin^3(\frac{n\pi}{2p})\sin^2(\frac{qn\pi}{p})}
\end{equation}
where
\begin{eqnarray*} f(n,p,q)&=(3p-3)\sin(\frac{(2q-1)n\pi}{2p})+(p+1)\sin(\frac{(2q-3)n\pi}{2p})\\
&+(p-1)\sin(\frac{(2q+3)n\pi}{2p})+(3p+3)\sin(\frac{(2q+1)n\pi}{2p}).
\end{eqnarray*}
\end{Theorem}

a step that “already presents miraculous cancellations.” Then, “we analyse the asymptotic behavior of the trigonometric sum in the most technical part of the paper.” To “control its denominators,” one “need[s] to impose that the coefficients of the continued fraction expansion of $\theta$ do not grow too quickly,” using “classical results of Khinchin on metric Diophantine approximation and a variation on a technique due to Zaremba.”

Question:
For a reduced rational number \(q/p\in(0,1)\), define
\[
\sigma_2\!\left(\frac qp\right)=\sum_{(a,b,c)\in \Delta_p}(-1)^{\lfloor aq/p\rfloor+\lfloor bq/p\rfloor+\lfloor cq/p\rfloor},
\]
where \(\Delta_p\) is the set of triples \((a,b,c)\) of integers such that
\[
0<a,b,c<p,\qquad a<b+c,\quad b<a+c,\quad c<a+b,
\]
\[
a+b+c<2p,\qquad a+b+c\text{ is odd}.
\]
Now let \(\theta\in[0,1]\) be irrational, and let \(q_k/p_k\) be the convergents of the regular continued-fraction expansion of \(\theta\). Which statement holds for Lebesgue-almost every irrational \(\theta\in[0,1]\)?

The limit exists and is given by
\[
\lim_{k\to\infty}\frac{\sigma_2(q_k/p_k)}{p_k^2}
=\frac{16}{\pi^3}\sum_{n\ge 1\,\mathrm{odd}}\frac{1}{n^3\sin(n\pi\theta)}.
\]

Choices:
B.
The limit exists and is given by
\[
\lim_{k\to\infty}\frac{\sigma_2(q_k/p_k)}{p_k^2}
=\frac{16}{\pi^3}\sum_{n\ge 1}\frac{1}{n^3\sin(n\pi\theta)}.
\]
C.
For Lebesgue-almost every irrational \(\theta\in[0,1]\), the limit
\[
\lim_{k\to\infty}\frac{\sigma_2(q_k/p_k)}{p_k^2}
\]
exists.
D.
The limit exists and is given by
\[
\lim_{k\to\infty}\frac{\sigma_2(q_k/p_k)}{p_k^2}
=\frac{16}{\pi^3}\sum_{n\ge 1\,\mathrm{odd}}\frac{1}{n^3\,|\sin(n\pi\theta)|}.
\]
E.
For every irrational \(\theta\in[0,1]\), if \(q_k/p_k\) are the convergents of its regular continued-fraction expansion, then
\[
\lim_{k\to\infty}\frac{\sigma_2(q_k/p_k)}{p_k^2}
=\frac{16}{\pi^3}\sum_{n\ge 1\,\mathrm{odd}}\frac{1}{n^3\sin(n\pi\theta)}.
\]

odd-index restriction in the trigonometric expansion

explicit identification of the limiting value by the odd-series formula

signed denominator \(\sin(n\pi\theta)\) replaced by absolute value, destroying the exact Fourier-cancellation structure

almost-everywhere metric Diophantine hypothesis strengthened to all irrationals

The stem does not explicitly reveal the correct limit formula. It defines the quantity and asks for the asymptotic statement; the correct answer is not directly embedded in the wording.

The item is close to a theorem-recall question: one option is essentially the exact stated limiting theorem, with distractors formed by small perturbations. It is not a pure tautology, but it is only a mild reformulation of a known result.

Some reasoning is needed to compare quantifiers, normalization, and the odd-index restriction, but the question mostly tests recognition/recall of the theorem rather than genuine derivation or generative mathematical reasoning.

The distractors are plausible and mathematically meaningful: removing the odd restriction, weakening to mere existence, overstrengthening 'almost all' to 'every,' and changing the normalization to p_k^3 are all distinct and aligned with common failure modes.

A reasonably well-constructed theorem-recognition MCQ with strong distractors and little direct answer leakage, but it only moderately avoids tautology and does not strongly test generative reasoning.


The relationship between the geometry of manifolds and the analytic properties of functions on manifolds is a defining theme of geometric analysis. Our direction starts with the Liouville theorems for harmonic functions on \(\mathbb{R}^n\) and Yau's generalization.

Yau proved in \cite{yau1975harmonic} that a bounded harmonic function on a complete manifold with nonnegative Ricci curvature is a constant. In 1974, he conjectured that a more general result should hold: on a complete manifold \(M\) with nonnegative Ricci curvature, the space \(\mathcal{H}_d(M)\) of harmonic functions with polynomially bounded growth should have finite dimension. Colding and Minicozzi proved his conjecture in \cite{colding1997harmonic}.

A natural generalization is to try to show this result for solutions of the heat equation. However, the heat equation is very flexible compared to the Laplace equation, and since there are bounded solutions to the heat equation, no Liouville theorem is possible in general.

Despite this, if we restrict attention to specifically \textit{ancient} solutions of the heat equation, that is, solutions which are defined for all time going back to \(-\infty\), then Liouville theorems actually do become possible. Indeed, in \cite{colding2021optimal}, Colding and Minicozzi generalize \cite{colding1997harmonic} to show that the space \(\mathcal{P}_d(M)\) of ancient solutions of the heat equation with polynomially bounded growth also has finite dimension. In (\cite{colding2020complexity}, \cite{colding2019search}, \cite{colding2019liouville}), Colding and Minicozzi show how the spaces \(\mathcal{P}_d(M)\) are relevant to geometric flows.

Continuing to more types of equations, Wang and Zhu recently generalized the result of \cite{colding1997harmonic} to biharmonic functions \cite{wang2025qualitativebehaviorbiharmonicfunctions}, i.e., functions \(u: M \rightarrow \mathbb{R}\) solving
\begin{equation*}
    \Delta\Delta u = 0.
\end{equation*}

This equation is also more flexible than the Laplace equation (indeed, any harmonic function is biharmonic), and we cannot prove as general a Liouville theorem as for harmonic functions. To find a Liouville theorem, rather than restricting attention to a subclass of biharmonic functions as in \cite{colding2021optimal}, Wang and Zhu instead restrict attention to a subclass of manifolds with polynomial volume growth and Ricci curvature bounded below at infinity.

Our goal in this paper is to generalize Wang and Zhu's result to ancient solutions of the biharmonic heat equation, following the strategy of Colding and Minicozzi in \cite{colding2021optimal}. Our main result is
\begin{theorem*}
    Let \(M\) be a complete Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Let \(u: M \times (-\infty, 0] \rightarrow \mathbb{R}\) be an ancient solution of
    \begin{equation*}
        \partial_t u(x, t) + \Delta\Delta u(x, t) = 0
    \end{equation*}
    such that \(|u(x, t)|\) and \(|\nabla u(x, t)|\) have polynomially bounded growth in the heat balls \(B_R(x) \times [-R^4, 0]\). The space of all such solutions \(u(x, t)\) is finite dimensional.
\end{theorem*}

\subsection{Definitions and Notation}
We now give more precise definitions and statements. Given a manifold \(M\) and an interval \(I \subset \mathbb{R}\), a function \(u: M \times I \rightarrow \mathbb{R}\) satisfies the biharmonic heat equation if
\begin{equation}\label{first instance of biharmonic heat equation}
    \partial_t u(x, t) + \Delta\Delta u(x, t) = 0.
\end{equation}
We will call such a function ``bicaloric'' for brevity. A bicaloric function \(u\) is ancient if it can be defined on an interval extending infinitely backwards in time, i.e. for \(t \in (-\infty, 0]\). We say that \(u \in \mathcal{P}_{d, d'}(M)\) for \(d, d' > 0\) if \(\partial_t u + \Delta\Delta u = 0\), \(u\) is ancient, and for some constants \(C, C' > 0\),
\begin{equation}\label{definition of polynomially bounded growth in introduction}
    \sup_{B_R(p) \times [-R^4, 0]}|u(x, t)| \le C(1 + R)^d, \quad \sup_{B_R(p) \times [-R^4, 0]}|\nabla u(x, t)| \le C'(1 + R)^{d'}
\end{equation}
for any \(p \in M\) and \(R > 0\). We similarly say that \(u \in \mathcal{H}_{d, d'}(M)\) if \(\Delta\Delta u = 0\) and the same bounds in (\ref{definition of polynomially bounded growth in introduction}) hold, where we take the supremum over only the ball \(B_R(p)\).

A manifold \(M\) is said to have polynomial volume growth if there are constants \(C, d_V > 0\) and some \(p \in M\) such that \(\operatorname{Vol}(B_R(p)) \le C(1 + R)^{d_V}\) for all \(R> 0\). Furthermore, we say that the Ricci curvature tensor is bounded below quadratically with constant \(K\) if for some \(p \in M\) and all \(R > 0\),
\begin{equation}
    \sup_{v \in TB_{R}(p)}\frac{\Ric(v, v)}{|v|^2} \ge -\frac{K}{R^2}.
\end{equation}

With these definitions, our main results are more precisely stated as

Yau proved in \cite{yau1975harmonic} that a bounded harmonic function on a complete manifold with nonnegative Ricci curvature is a constant. In 1974, he conjectured that a more general result should hold: on a complete manifold \(M\) with nonnegative Ricci curvature, the space \(\mathcal{H}_d(M)\) of harmonic functions with polynomially bounded growth should have finite dimension. Colding and Minicozzi proved his conjecture in \cite{colding1997harmonic}.

This equation is also more flexible than the Laplace equation (indeed, any harmonic function is biharmonic), and we cannot prove as general a Liouville theorem as for harmonic functions. To find a Liouville theorem, rather than restricting attention to a subclass of biharmonic functions as in \cite{colding2021optimal}, Wang and Zhu instead restrict attention to a subclass of manifolds with polynomial volume growth and Ricci curvature bounded below at infinity.

Our goal in this paper is to generalize Wang and Zhu's result to ancient solutions of the biharmonic heat equation, following the strategy of Colding and Minicozzi in \cite{colding2021optimal}. Our main result is
\begin{theorem*}
 Let \(M\) be a complete Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Let \(u: M \times (-\infty, 0] \rightarrow \mathbb{R}\) be an ancient solution of
 \begin{equation*}
 \partial_t u(x, t) + \Delta\Delta u(x, t) = 0
 \end{equation*}
 such that \(|u(x, t)|\) and \(|\nabla u(x, t)|\) have polynomially bounded growth in the heat balls \(B_R(x) \times [-R^4, 0]\). The space of all such solutions \(u(x, t)\) is finite dimensional.
\end{theorem*}

\subsection{Definitions and Notation}
We now give more precise definitions and statements. Given a manifold \(M\) and an interval \(I \subset \mathbb{R}\), a function \(u: M \times I \rightarrow \mathbb{R}\) satisfies the biharmonic heat equation if
\begin{equation}\label{first instance of biharmonic heat equation}
 \partial_t u(x, t) + \Delta\Delta u(x, t) = 0.
\end{equation}
We will call such a function ``bicaloric'' for brevity. A bicaloric function \(u\) is ancient if it can be defined on an interval extending infinitely backwards in time, i.e. for \(t \in (-\infty, 0]\). We say that \(u \in \mathcal{P}_{d, d'}(M)\) for \(d, d' > 0\) if \(\partial_t u + \Delta\Delta u = 0\), \(u\) is ancient, and for some constants \(C, C' > 0\),
\begin{equation}\label{definition of polynomially bounded growth in introduction}
 \sup_{B_R(p) \times [-R^4, 0]}|u(x, t)| \le C(1 + R)^d, \quad \sup_{B_R(p) \times [-R^4, 0]}|\nabla u(x, t)| \le C'(1 + R)^{d'}
\end{equation}
for any \(p \in M\) and \(R > 0\). We similarly say that \(u \in \mathcal{H}_{d, d'}(M)\) if \(\Delta\Delta u = 0\) and the same bounds in (\ref{definition of polynomially bounded growth in introduction}) hold, where we take the supremum over only the ball \(B_R(p)\).

A manifold \(M\) is said to have polynomial volume growth if there are constants \(C, d_V > 0\) and some \(p \in M\) such that \(\operatorname{Vol}(B_R(p)) \le C(1 + R)^{d_V}\) for all \(R> 0\). Furthermore, we say that the Ricci curvature tensor is bounded below quadratically with constant \(K\) if for some \(p \in M\) and all \(R > 0\),
\begin{equation}
 \sup_{v \in TB_{R}(p)}\frac{\Ric(v, v)}{|v|^2} \ge -\frac{K}{R^2}.
\end{equation}

With these definitions, our main results are more precisely stated as

\begin{equation}\label{definition of polynomially bounded growth in introduction}
    \sup_{B_R(p) \times [-R^4, 0]}|u(x, t)| \le C(1 + R)^d, \quad \sup_{B_R(p) \times [-R^4, 0]}|\nabla u(x, t)| \le C'(1 + R)^{d'}
\end{equation}

A manifold \(M\) is said to have polynomial volume growth if there are constants \(C, d_V > 0\) and some \(p \in M\) such that \(\operatorname{Vol}(B_R(p)) \le C(1 + R)^{d_V}\) for all \(R> 0\). Furthermore, we say that the Ricci curvature tensor is bounded below quadratically with constant \(K\) if for some \(p \in M\) and all \(R > 0\),
\begin{equation}
 \sup_{v \in TB_{R}(p)}\frac{\Ric(v, v)}{|v|^2} \ge -\frac{K}{R^2}.
\end{equation}

Combining this with Wang and Zhu's result \cite{wang2025qualitativebehaviorbiharmonicfunctions}, we have the following corollary:

\begin{corollary}
 Let \(M\) be a Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Then for \(k, \ell \ge 0\) the spaces \(\mathcal{P}_{4k, 4\ell}(M)\) are finite dimensional.
\end{corollary}

Since there are \(d + 1\) vectors \((1, t_i, \ldots, t_i^{d + 1})\), they span \(\mathbb{R}^{d + 1}\), and so there are constants \(b_i^j\) such that
 \begin{equation}\label{bdefinitions}
 e_j = b_i^j(1, t_i, \ldots, t_i^{d}).
 \end{equation}
 It now follows that
 \begin{equation}\label{expression of p in b}
 p_j(x) = b_i^ju(x, t_i), \quad \nabla p_j(x) = b_i^j\nabla u(x, t_i),
 \end{equation}
 and we conclude that \(p_j\) can grow at most polynomially of degree \(4k\) and \(\nabla p_j\) can grow at most polynomially of degree \(4\ell\). Because \(p_j\) vanishes when \(j > k\) and \(\nabla p_j\) vanishes when \(j > \ell + 1\), we have
 \begin{equation}
 u = p_0 + t p_1 + \ldots + t^{k}p_{k} \quad \text{and} \quad \nabla u = \nabla p_0 + t\nabla p_1 + \ldots + t^{\ell}\nabla p_{\ell},
 \end{equation}
 it follows that
 \begin{equation}
 \begin{split}
 |u(x, t)| \le C(1 + |t|^k + |x|^{4k}) \quad \text{and} \quad |\nabla u(x, t)| \le C(1 + |t|^{\ell} + |x|^{4\ell})
 \end{split}
 \end{equation}
 From equation (\ref{bdefinitions}) we have
 \begin{equation}
 \begin{split}
 \sum_i b_i^ju(x, R^4t_i) & = \sum_i\sum_{m}b_i^jp_{m}(x)R^{4j}t_i^{m} = R^{4j}\sum_mp_m(x)\left(\sum_ib_{i}^jt_i^m\right) \\ & = \sum R^{4j}\sum_m p_m(x)\delta_{mj} \\ & = R^{4j}p_j(x).
 \end{split}
 \end{equation}
 Similarly,
 \begin{equation}
 \sum_i b_i^j\nabla u(x, R^4t_i) = R^{4j}\nabla p_j(x)
 \end{equation}
 Thus
 \begin{equation}
 \begin{split}
 |R^{4j}p_j(x)| & = \left|\sum_i b_i^ju(x, R^4t_i)\right| \le A\sum_i\left|u(x, R^4t_i)\right| \\ & \le A(1 + |x|^{4k} + \sum_i|Rt_i|^{4d}) \le AR^{4k}
 \end{split}
 \end{equation}
 and similarly
 \begin{equation}
 \begin{split}
 |R^{4j}\nabla p_j(x)| & \le A'(1 + |x|^{4\ell} + \sum_i|Rt_i|^{4\ell}) \le A'R^{4\ell},
 \end{split}
 \end{equation}
 so that \(|p_j(x)| \le A_jR^{4(k - j)}\) and \(|p_j(x)| \le A_j'R^{4(\ell - j)}\).
\end{proof}
\begin{proof}[Proof of Theorem \ref{dimension bound theorem}]
 Choose some \(u \in {P}_{4k, 4\ell}(M)\) and suppose \(u = p_0(x) + tp_1(x) + \cdots + t^dp_d(x)\), as in lemma \ref{coefficientbound}. Then \(\Delta\Delta p_{d} = 0\) and for \(j < d\), \(\Delta\Delta p_j = -(j+1)p_{j + 1}\). Thus, there a linear map \(\Psi_0: \mathcal{P}_{4k, 4\ell} \rightarrow \mathcal{H}_{4k, 4\ell}\) defined by \(\Psi_0u = p_{d}\) (here we use the coefficient estimate in (\ref{coefficients in polynomial expression bounds})). If we let \(\mathcal{K}_0 = \ker \Psi_0\) we find
 \begin{equation}
 \dim \mathcal{P}_{4k, 4\ell} \le \dim \mathcal{K}_0 + \dim \mathcal{H}_{4k, 4\ell}
 \end{equation}
 If \(u \in \mathcal{K}_0\), then \(p_{d} = 0\) and \(\Delta\Delta p_{d - 1} = -dp_{d} = 0\), and so we have a map \(\Psi_1: \mathcal{K}_0 \rightarrow \mathcal{H}_{4(k - 1), 4(\ell - 1)}\) defined by setting \(\Psi_1u = p_{d - 1}\). Letting \(\ker \Psi_1 = \mathcal{K}_1\) then
 \begin{equation}
 \dim \mathcal{K}_0 \le \dim \mathcal{K}_1 + \dim \mathcal{H}_{4k, 4\ell}
 \end{equation}
 When \(k \le \ell + 1\), we can repeat this \(k\) times to get
 \begin{equation}
 \dim \mathcal{P}_{4k, 4\ell}(M) \le \sum_{i = 0}^{k} \dim \mathcal{H}_{4(k - i), 4(\ell - i)}(M)
 \end{equation}
 When \(k > \ell + 1\), we have from lemma \ref{coefficientbound} that \(\nabla p_{\ell + 1} = 0\), and so \(p_{\ell + 1} = p_d\) is a constant. Thus in this case \(p_d\) lies in a one dimensional subspace of \(H_{4k, 4\ell}(M)\), so that
 \begin{equation}
 \dim \mathcal{P}_{4k, 4\ell} = \dim \mathcal{K}_0 + 1.
 \end{equation}
 We then iterate the same argument as before, \(\ell\) times, to get the second inequality in (\ref{equation in the first statement of the main theorem}).

\end{proof}
\subsection{Bicaloric Polynomials in \(\mathbb{R}^n\)}
Now we consider polynomially bounded solutions to the biharmonic heat equation in \(\mathbb{R}^n\).
\begin{proposition}\label{bicaloric functions in Rn are polynomials}
 Let \(u \in \mathcal{P}_{k, \ell}(\mathbb{R}^n)\). Then \(u\) is a polynomial in \(x_i\) and \(t\).
\end{proposition}
\begin{proof}
 As before, this follows from the reverse Poincaré estimate and the fact that the operators \(\partial_{x_i}\), \(\partial_t\), and \(\Delta\) commute in \(\mathbb{R}^n\).
\end{proof}
\begin{corollary}
 If \(u \in \mathcal{P}_{k, \ell}(\mathbb{R}^n)\), then there is some \(d\) such that \(u \in \mathcal{P}_{d, d-1}(\mathbb{R}^n)\).
\end{corollary}\label{fewer caloric spaces in R^n}
\begin{proof}
 This follows the same way as before.
\end{proof}
Given a monomial in \(x_i\) and \(t\), we define its biparabolic degree as follows: for \(t^{n_0}\prod x_i^{n_i}\), the biparabolic degree is \(4n_0 + \sum n_i\). The degree of a polynomial is then the maximal degree of the monomials summing to it. Let \(\mathcal{A}_j^n\) be the set of homogeneous polynomials in \(\mathbb{R}^n\) of biparabolic degree \(j\). We have
\begin{equation}
 \mathcal{A}_d^n = A_d^n \oplus t A_{d - 4}^n \oplus t^2A_{d - 8}^n \oplus \cdots
\end{equation}
\begin{lemma}\label{dim for bicaloric polynomials}
 For \(d > 0\) we have \(\dim(\mathcal{P}_{d, d-1}(\mathbb{R}^n) \cap \mathcal{A}_d^n) = \dim A_d^n\) and
 \begin{equation}
 \dim \mathcal{P}_{d, d-1}(\mathbb{R}^n) = \sum_{j = 0}^d\dim A_j^n.
 \end{equation}
\end{lemma}
\begin{proof}
 Both \(\partial_t\) and \(\Delta\Delta\) map \(\mathcal{A}_d^n\) to \(\mathcal{A}_{d-4}^n\). We note that for \(u \in \mathcal{A}_{d-4}^n\) we have
 \begin{equation}
 (\partial_t + \Delta\Delta)\left(tu - \frac{1}{2}t^2(\partial_t + \Delta\Delta)u + \frac{1}{6}t^3(\partial_t + \Delta\Delta)^2 u - \cdots\right) = u,
 \end{equation}
 so that \(\partial_t + \Delta\Delta\) is surjective. Thus,
 \begin{equation}
 \dim (\mathcal{P}_{d, d-1}(\mathbb{R}^n) \cap \mathcal{A}_d^n) = \dim \mathcal{A}_d^n - \dim\mathcal{A}_{d-4}^n = \dim A_d^n
 \end{equation}
 since \(\mathcal{P}_{d, d-1}(\mathbb{R}^n) \cap \mathcal{A}_d^n\) is the kernel of \(\partial_t + \Delta\Delta\) restricted to \(\mathcal{A}_d^n\). Summing gives the second claim.

\end{proof}
Now finally we show that the estimate in Theorem \ref{dimension bound theorem} is sharp in \(\mathbb{R}^n\).
\begin{corollary}\label{sharp in Rn}
 For positive integers \(d > 0\)
 \begin{equation}
 \dim \mathcal{P}_{4d, 4d - 1}(\mathbb{R}^n) = \sum_{i = 0}^{d} \dim \mathcal{H}_{4(d - i), 4(d - i) - 1}(\mathbb{R}^n).
 \end{equation}
\end{corollary}
\begin{proof}
 From lemmas \ref{dim for bicaloric polynomials} and \ref{dim for biharmonic polynomials} we have
 \begin{equation}
 \begin{split}
 \dim \mathcal{P}_{4d, 4d-1}(\mathbb{R}^n) & = \sum_{j = 0}^{4d}\dim A_j^n = \sum_{j = 0}^d(\dim A_{4j}^n + \dim A_{4j - 1}^n + \dim A_{4j - 2}^n + \dim A_{4j - 3}^n) \\ & = \sum_{j = 0}^d\dim \mathcal{H}_{4d, 4d - 1}(\mathbb{R}^n).
 \end{split}
 \end{equation}
 Setting \(k = \ell = d\) in Theorem \ref{dimension bound theorem} and noting \(\mathcal{P}_{4d, 4d}(\mathbb{R}^n) = \mathcal{P}_{4d, 4d-1}(\mathbb{R}^n)\) and \(\mathcal{H}_{4d, 4d}(\mathbb{R}^n) = \mathcal{H}_{4d, 4d-1}(\mathbb{R}^n)\) shows that the estimate in Theorem \ref{dimension bound theorem} is sharp.

Combining this with Wang and Zhu's result \cite{wang2025qualitativebehaviorbiharmonicfunctions}, we have the following corollary:

\begin{corollary}
    Let \(M\) be a Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Then for \(k, \ell \ge 0\) the spaces \(\mathcal{P}_{4k, 4\ell}(M)\) are finite dimensional.
\end{corollary}

\subsection{Harmonic and biharmonic functions}

Biharmonic functions arise in several variational problems. Just as minimizing \(\int|\nabla u|^2\) leads one to the Laplace and heat equations, minimizing \(\int |\Delta u|^2\) leads to the biharmonic and biharmonic heat equations.

In general, fourth order elliptic operators arise naturally when taking variations involving second order objects, one major example being variations of metrics in conformal geometry (see \cite{chang1995extremal}, \cite{Lin1998classification}). They also arise in the study of the Willmore energy. For an immersed surface \(\phi: M^2 \rightarrow \mathbb{R}^3\), the Willmore energy is defined as
\begin{equation}\label{Willmore functional}
    \mathcal{W}(\phi) = \int_{M}H^2\,dA
\end{equation}
where \(dA\) is the induced volume element and \(H\) is the mean curvature \cite{willmore2000surfaces}. In studying critical points of this functional one arrives at the Euler-Lagrange equation
\begin{equation}\label{euler-lagrange equation}
    \Delta H + 2H(H^2 - K) = 0,
\end{equation}
a fourth order elliptic operator. The biharmonic heat equation similarly arises when studying the gradient flow of the Willmore energy (\cite{kuwert2002gradient}, \cite{lamm2005biharmonic}). Ancient solutions to heat equations often appear when doing blowup analysis of general solutions to a variational problem. See \cite{kuwert2004removable} for blowup analysis of singularities of Willmore flows. We also again reference (\cite{colding2020complexity}, \cite{colding2019search}, \cite{colding2019liouville}) for more on how ancient solutions to heat equations with polynomially bounded growth are relevant to geometric flows.

Although both arise from variational problems, biharmonic functions in general differ significantly from harmonic functions, because no maximum principle holds for biharmonic functions. This limits the kinds of estimates we can find for biharmonic functions. In particular, the usual pointwise derivative estimates one can find for harmonic functions on a ball cannot be found for a biharmonic function.

On the bright side, energy methods for harmonic and caloric functions seem to have analogs for biharmonic and bicaloric functions, which we will see as we prove Theorem \ref{dimension bound theorem}. We are still limited to some extent, however, because when performing integrations by parts we are forced to use the Bochner formula
\begin{equation*}
    \frac{1}{2}\Delta|\nabla u|^2 = |\nabla^2u| + \langle \nabla \Delta u, \nabla u\rangle + \Ric(\nabla u, \nabla u)
\end{equation*}
to control the factor \(\langle \nabla \Delta u, \nabla u\rangle\). It is the appearance of the Ricci term here that makes the decay on Ricci curvature crucial for our result.

Our methodology is inspired by Colding and Minicozzi's in \cite{colding2021optimal}. We will show a reverse Poincaré inequality for bicaloric functions on ``heat balls'' \(B_R(p) \times [-R^4, 0]\). Because we are considering ancient bicaloric functions, we will be able to apply the inequality as \(R \rightarrow \infty\) to get strong, global control of their behavior. In particular we will see that high order time derivatives \(\partial_t^ku\) must vanish identically, allowing us to write for some finite \(d\):
\[u(x, t) = p_d(x)t^d + \cdots + p_1(x)t + p_0(x)\]
with \(\Delta \Delta p_d = 0\) and \(\Delta \Delta p_j = -(j + 1)p_{j + 1}\). This will allow us to directly compare the spaces \(\mathcal{H}_{4k, 4\ell}(M)\) with \(\mathcal{P}_{4k, 4\ell}(M)\).

To show the dimension estimates are sharp in \(\mathbb{R}^n\), we will consider biharmonic and bicaloric polynomials (analogs of the harmonic polynomials), enabling us to explicitly compute the dimensions of the spaces \(\mathcal{H}_{4k, 4\ell}(\mathbb{R}^n)\) and \(\mathcal{P}_{4k, 4\ell}(\mathbb{R}^n)\).

The introduction indicates the proof strategy for Theorem~\ref{dimension bound theorem} as follows.

1. **Use energy methods and Bochner to handle fourth order terms.** It says that while working "as we prove Theorem \ref{dimension bound theorem}" one uses analogs of energy methods for harmonic/caloric functions, but when integrating by parts one is "forced to use the Bochner formula"
\[
\frac{1}{2}\Delta|\nabla u|^2 = |\nabla^2u| + \langle \nabla \Delta u, \nabla u\rangle + \Ric(\nabla u, \nabla u)
\]
"to control the factor \(\langle \nabla \Delta u, \nabla u\rangle\)," and "the appearance of the Ricci term here" is why the quadratic lower bound/decay assumption on Ricci is crucial.

2. **Establish a reverse Poincar\'e inequality on heat balls for bicaloric functions.** Inspired by Colding--Minicozzi, the paper will "show a reverse Poincar\'e inequality for bicaloric functions on `heat balls' \(B_R(p)\times[-R^4,0]\)."

3. **Send \(R\to\infty\) using the ancient assumption to get global control.** Because the functions are ancient bicaloric, "we will be able to apply the inequality as \(R\to\infty\) to get strong, global control of their behavior."

4. **Deduce vanishing of high time derivatives and a finite time-polynomial expansion.** From this global control, "high order time derivatives \(\partial_t^k u\) must vanish identically," so for some finite \(d\)
\[
u(x,t)=p_d(x)t^d+\cdots+p_1(x)t+p_0(x),
\]
with relations "\(\Delta\Delta p_d=0\)" and "\(\Delta\Delta p_j=-(j+1)p_{j+1}\)."

5. **Compare \(\mathcal{H}_{4k,4\ell}(M)\) and \(\mathcal{P}_{4k,4\ell}(M)\) via the coefficient relations.** The stated expansion and recursions "will allow us to directly compare the spaces \(\mathcal{H}_{4k,4\ell}(M)\) with \(\mathcal{P}_{4k,4\ell}(M)\)," yielding the dimension bound.

6. **Sharpness in \(\mathbb{R}^n\) by explicit polynomial models.** To show sharpness, it says they will consider "biharmonic and bicaloric polynomials (analogs of the harmonic polynomials), enabling us to explicitly compute the dimensions" of \(\mathcal{H}_{4k,4\ell}(\mathbb{R}^n)\) and \(\mathcal{P}_{4k,4\ell}(\mathbb{R}^n)\).

The introduction indicates the proof strategy for the following theorem.

\begin{theorem}\label{dimension bound theorem}
    Let \(M\) be a complete Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Let \(k\), \(\ell\) be nonnegative integers. Then
    \begin{equation}\label{equation in the first statement of the main theorem}
        \dim \mathcal{P}_{4k, 4\ell}(M) \le \begin{cases}
             \displaystyle \sum_{i = 0}^{k} \dim \mathcal{H}_{4(k - i), 4(\ell - i)}(M) & \quad k \le \ell + 1, \\ \displaystyle
            1 + \sum_{i = 0}^{\ell} \dim \mathcal{H}_{4(k - i), 4(\ell - i)}(M) & \quad k > \ell + 1
        \end{cases}
    \end{equation}
    Moreover, these inequalities are sharp in \(\mathbb{R}^n\).
\end{theorem}

1. **Use energy methods and Bochner to handle fourth order terms.** In establishing the main theorem, one uses analogs of energy methods for harmonic/caloric functions, but when integrating by parts one is forced to use the Bochner formula
\[
\frac{1}{2}\Delta|\nabla u|^2 = |\nabla^2u| + \langle \nabla \Delta u, \nabla u\rangle + \Ric(\nabla u, \nabla u)
\]
so as to control the factor \(\langle \nabla \Delta u, \nabla u\rangle\); the appearance of the Ricci term here is why the quadratic lower bound/decay assumption on Ricci is crucial.

2. **Establish a reverse Poincar\'e inequality on heat balls for bicaloric functions.** Inspired by Colding--Minicozzi, the paper will show a reverse Poincar\'e inequality for bicaloric functions on “heat balls” \(B_R(p)\times[-R^4,0]\).

3. **Send \(R\to\infty\) using the ancient assumption to get global control.** Because the functions are ancient bicaloric, one will be able to apply the inequality as \(R\to\infty\) to get strong, global control of their behavior.

4. **Deduce vanishing of high time derivatives and a finite time-polynomial expansion.** From this global control, high order time derivatives \(\partial_t^k u\) must vanish identically, so for some finite \(d\)
\[
u(x,t)=p_d(x)t^d+\cdots+p_1(x)t+p_0(x),
\]
with relations “\(\Delta\Delta p_d=0\)” and “\(\Delta\Delta p_j=-(j+1)p_{j+1}\).”

5. **Compare \(\mathcal{H}_{4k,4\ell}(M)\) and \(\mathcal{P}_{4k,4\ell}(M)\) via the coefficient relations.** The stated expansion and recursions allow one to directly compare the spaces \(\mathcal{H}_{4k,4\ell}(M)\) with \(\mathcal{P}_{4k,4\ell}(M)\), yielding the dimension bound \eqref{equation in the first statement of the main theorem}.

6. **Sharpness in \(\mathbb{R}^n\) by explicit polynomial models.** To show sharpness, one considers biharmonic and bicaloric polynomials (analogs of the harmonic polynomials), enabling one to explicitly compute the dimensions of \(\mathcal{H}_{4k,4\ell}(\mathbb{R}^n)\) and \(\mathcal{P}_{4k,4\ell}(\mathbb{R}^n)\).

Question: Let \(M\) be a complete Riemannian manifold. Say that \(M\) has polynomial volume growth if there exist \(p\in M\) and constants \(C,d_V>0\) such that \(\operatorname{Vol}(B_R(p))\le C(1+R)^{d_V}\) for all \(R>0\). Say that the Ricci curvature of \(M\) is bounded below quadratically if there exist \(p\in M\) and \(K>0\) such that \(\sup_{v\in TB_R(p)}\frac{\operatorname{Ric}(v,v)}{|v|^2}\ge -\frac{K}{R^2}\) for all \(R>0\). For \(d,d'\), let \(\mathcal{P}_{d,d'}(M)\) denote the space of ancient functions \(u:M\times(-\infty,0]\to\mathbb{R}\) satisfying the biharmonic heat equation \(\partial_t u+\Delta\Delta u=0\) and such that for some constants \(C,C'>0\), \(\sup_{B_R(q)\times[-R^4,0]}|u(x,t)|\le C(1+R)^d\) and \(\sup_{B_R(q)\times[-R^4,0]}|\nabla u(x,t)|\le C'(1+R)^{d'}\) for every \(q\in M\) and \(R>0\). Let \(\mathcal{H}_{d,d'}(M)\) denote the space of functions \(u:M\to\mathbb{R}\) satisfying \(\Delta\Delta u=0\) and the analogous bounds over \(B_R(q)\). For nonnegative integers \(k\) and \(\ell\), which statement holds for every such manifold \(M\)?

\[\dim \mathcal{P}_{4k,4\ell}(M)\le \begin{cases}\displaystyle \sum_{i=0}^{k}\dim \mathcal{H}_{4(k-i),4(\ell-i)}(M), & k\le \ell+1,\\[4pt]\displaystyle 1+\sum_{i=0}^{\ell}\dim \mathcal{H}_{4(k-i),4(\ell-i)}(M), & k>\ell+1,\end{cases}\] and moreover these inequalities are sharp in \(\mathbb{R}^n\).

Choices:
B. \[\dim \mathcal{P}_{4k,4\ell}(M)\le \sum_{i=0}^{\min\{k,\ell\}}\dim \mathcal{H}_{4(k-i),4(\ell-i)}(M),\] and moreover this inequality is sharp in \(\mathbb{R}^n\).
C. \[\dim \mathcal{P}_{4k,4\ell}(M)<\infty\] for all nonnegative integers \(k\) and \(\ell\).
D. \[\dim \mathcal{P}_{4k,4\ell}(M)\le \begin{cases}\displaystyle \sum_{i=0}^{k}\dim \mathcal{H}_{4(k-i),4(\ell-i)}(M), & k\le \ell,\\[4pt]\displaystyle 1+\sum_{i=0}^{\ell}\dim \mathcal{H}_{4(k-i),4(\ell-i)}(M), & k>\ell,\end{cases}\] and moreover these inequalities are sharp in \(\mathbb{R}^n\).
E. \[\dim \mathcal{P}_{4k,4\ell}(M)= \begin{cases}\displaystyle \sum_{i=0}^{k}\dim \mathcal{H}_{4(k-i),4(\ell-i)}(M), & k\le \ell+1,\\[4pt]\displaystyle 1+\sum_{i=0}^{\ell}\dim \mathcal{H}_{4(k-i),4(\ell-i)}(M), & k>\ell+1,\end{cases}\] and moreover these equalities are sharp in \(\mathbb{R}^n\).

The stem gives definitions and hypotheses but does not reveal the dimension bound itself. There is no explicit or obvious hint that singles out choice A.

The item is essentially asking for the exact theorem statement under the stated hypotheses. The multiple close alternatives prevent it from being a pure verbatim restatement, but it still mainly tests theorem recall rather than an independently derived conclusion.

Some reasoning is needed to distinguish subtle variants: off-by-one thresholding, inequality versus equality, and the extra constant term in one regime. However, the task is mostly recognition of the correct theorem statement rather than substantial mathematical generation.

The distractors are strong: one is a weaker true statement, others are plausible near-miss versions with boundary errors or overstrong equality claims. They are distinct and reflect realistic failure modes in reading or recalling the result.

A solid theorem-recall MCQ with strong distractors and no answer leakage, but it mainly tests precise recall/discrimination among near-identical statements rather than deep generative reasoning.


The study of \hk \ manifolds is a central topic in the theory of compact Kähler manifolds: in virtue of the Beauville--Bogomolov decomposition theorem \cite[Thm.\ 1]{beauville}, they represent one of three ``fundamental" types of compact Kähler manifolds with \(c_1(K_X)=0\), together with abelian varieties and strict Calabi-Yau manifolds.
Their topology is extremely interesting: on the one hand it is severely restricted compared to that of a general compact \Kah \ manifold, for example by the Fujiki relations, \cite{Fujiki}. Moreover, endowed with the usual Hodge structure, the cohomology of \hkm s largely determines their geometry by the Torelli Theorem, see for example \cite{Markman_torelli}. On the other hand, the topology of \hkm s remains very mysterious, with a lot of basic open questions. One of these questions is the following:

\begin{question}\label{que:hk_torsion}
Let $X$ be a \hk \ manifold. Is $H^*(X, \mathbb Z)$ torsion-free?
\end{question}
For \hk \ manifolds of $K3^{[n]}$-type, the answer is positive by \cite{Markman2007_integral}, which has then been further generalized by \cite{Totaro_2020_integral} to the Hilbert scheme of points on any smooth projective surface with torsion-free cohomology. Already for the next most understood \hk \ deformation type, that of \textit{generalized Kummer varieties} (or \textit{Kummer manifolds}), denoted as \(\Kum_n(A)\), this question is open in its full generality. In \cite{kapfermenet}, however, Kapfer and Menet give a positive answer in the case of Kummer fourfolds.

The abelian group $H^3(X, \mathbb Z)_{\tors}$ is of particular interest as it sits in the short exact sequence
$$0 \to H^2(X, \mathbb Z) / \NS(X, \mathbb Z) \otimes_\Z \Q/\Z \to \Br(X) \to H^3(X, \mathbb Z)_{\tors} \to 0.$$
For this reason it is often called the \textit{topological Brauer group}, see also \cite[Sec.\ 1]{grothendieck}, and it is the lowest degree cohomology group whose torsion part is not known for any \hk \ manifold other than those of K3\(^{[n]}\)-type and Kummer fourfolds.

The aim of this note is to prove the following result:

\begin{question}\label{que:hk_torsion}
Let $X$ be a \hk \ manifold. Is $H^*(X, \mathbb Z)$ torsion-free?
\end{question}
For \hk \ manifolds of $K3^{[n]}$-type, the answer is positive by \cite{Markman2007_integral}, which has then been further generalized by \cite{Totaro_2020_integral} to the Hilbert scheme of points on any smooth projective surface with torsion-free cohomology. Already for the next most understood \hk \ deformation type, that of \textit{generalized Kummer varieties} (or \textit{Kummer manifolds}), denoted as \(\Kum_n(A)\), this question is open in its full generality. In \cite{kapfermenet}, however, Kapfer and Menet give a positive answer in the case of Kummer fourfolds.

The abelian group $H^3(X, \mathbb Z)_{\tors}$ is of particular interest as it sits in the short exact sequence
$$0 \to H^2(X, \mathbb Z) / \NS(X, \mathbb Z) \otimes_\Z \Q/\Z \to \Br(X) \to H^3(X, \mathbb Z)_{\tors} \to 0.$$
For this reason it is often called the \textit{topological Brauer group}, see also \cite[Sec.\ 1]{grothendieck}, and it is the lowest degree cohomology group whose torsion part is not known for any \hk \ manifold other than those of K3\(^{[n]}\)-type and Kummer fourfolds.

The strategy of the proof is inspired by and generalizes the approach of \cite{kapfermenet}:
we first consider a rational cover of degree \(n\) by a smooth projective variety with no torsion in degree three cohomology, which forces \(H^3(\Kum_{n-1}(A),\Z)_{\tors}\) to be entirely of \(n\)-torsion. By considering a double cover of a suitable open of \(\Kum_{n-1}(A)\), we then show that \(H^3(\Kum_{n-1}(A),\Z)_{\tors}\) is a $2$-group, via an analysis of the cohomology of the alternating group and the Cartan--Leray spectral sequence. When \(n\) is odd, we obtain Corollary \ref{cor:nice_statement_even}.
We are then left with asking ourselves the following

\end{lemma}
\begin{proof}
Since \(H^3(-,\Z)_{\tors}\) is a birational invariant, by passing to a resolution of indeterminacies we may assume that \(f\) is everywhere defined. The result then follows from the projection formula.
\end{proof}
\begin{corollary}\label{cor:n-tors_of_Kum}
 The abelian group $H^3(\Kum_{n-1}(A), \mathbb Z)_{\tors}$ is $n$-torsion.
\end{corollary}
\begin{proof}
 The map \(K_{n-1}(A) \dashrightarrow \Kum_{n-1}(A)\) factors through the quotient by \(\mathfrak{S}_{n-1}\) acting on the first \(n-1\) entries of \(A^n\), yielding a map of degree \(n\).
 The quotient \(K_{n-1}(A)/\mathfrak{S}_{n-1}\) is birational to \(A^{[n-1]}\), hence we have a generically finite rational map of degree \(n\)
 \[A^{[n-1]} \dashrightarrow \Kum_{n-1}(A).\]
By \cite{Markman2007_integral} the group $H^3(A^{[n-1]},\Z)$ is torsion-free. The result then follows from Lemma \ref{lem:dn_tors} with \(N=1\).
\end{proof}
\section{Quotient by the alternating group}
Ideally, one would like to apply the argument of the previous section to the rational double cover $$A^{n-1} / \mathfrak A_n \dashrightarrow \Kum_{n-1}(A).$$ However, there is no smooth projective model \(Z\) of $A^{n-1} / \alt_n$ for which \(H^3(Z, \Z)_{\tors}\) is known.

Let $U \coloneqq K_{n-1}(A) \cap (A^n\setminus \Delta_2) \subset A^n$ and $V \coloneqq \Kum_{n-1}(A) \cap (A^{[n]} \setminus D_2) \subset A^{[n]}$.
For codimension reasons, avoiding the smaller diagonals does not change $H^3(-, \mathbb Z)_{\tors}$:
\begin{lemma}\label{lem:torsvkum}
 We have
 $H^3(V, \mathbb Z)_{\tors} \simeq H^3(\Kum_{n-1}(A), \mathbb Z)_{\tors}.$
\end{lemma}
\begin{proof}
We consider the stratification
\[D_{2}= Z_0 \supset Z_1\supset \cdots \]
where \(Z_{i+1}\) is the singular locus of \(Z_i\). By the long exact sequence of relative cohomology and the Thom isomorphism, we have
\[H^3(\Kum_{n-1}\setminus Z_{1},\Z)\simeq H^3(\Kum_{n-1}\setminus Z_{2},\Z) \simeq \dots \simeq H^3(\Kum_{n-1}(A), \mathbb Z),\]
see \cite[Lemma 11.13]{Voisin_book_I}. By the same argument we also have the exact sequence
$$ 0\to H^3(\Kum_{n-1} \setminus Z_1,\mathbb Z) \to H^3( \Kum_{n-1}(A) \setminus Z_0, \mathbb Z) \to \mathbb Z^{\pi_0(Z_0 \setminus Z_1)}.$$
Thus
\[H^3(\Kum_{n-1}(A),\Z)_{\tors}\simeq H^3(\Kum_{n-1}(A)\setminus Z_0,\Z)_{\tors}.\]
Since $V = \Kum_{n-1}(A) \setminus Z_0$, we conclude.
\end{proof}
Since the alternating group acts freely on $U$, the relation between the quotient $U / \alt_n$ and the open subset $V \subset \Kum_{n-1}(A)$ can be made precise as follows:
\begin{lemma}\label{lem:doublecover_V}
There exists a ramified double cover
 $$\pi \colon \Bl_{\overline{\Delta}} (U/\alt_n) \to V,$$
 where \(\overline{\Delta}\) is the image in \(U/\alt_n\) of \(\Delta\subset U\).
\end{lemma}
\begin{proof}
By \cite[Sec.\ 8, p.\ 770]{beauville}, we have
\[(\Bl_{\Delta} U )/ \mathfrak S_n \simeq V\]
Moreover, since $\mathfrak A_n$ acts freely on $U$ and stabilizes $\Delta$, we have
$$\Bl_{\Delta}(U) / \mathfrak A_n \simeq \Bl_{\overline{\Delta}}(U / \mathfrak A_n).$$
The action of \(\Z/2\Z\simeq\mathfrak{S}_n/\alt_n\) induced onto the quotient \(\Bl_{\Delta}(U) / \mathfrak A_n \) produces a double cover
 $$\Bl_{\overline\Delta}(U / \mathfrak A_n) \simeq (\Bl_{\Delta} U ) / \mathfrak A_n \to (\Bl_{\Delta} U) / \mathfrak S_n \simeq V,$$
 which is what we wanted.
 \end{proof}

\begin{proposition}\label{prop:spectral_sequence_argument}
 If there are integers $N_p$ for $1 \leq p \leq 3$ such that $$N_p \cdot H^p(\mathfrak A_n, H^{3-p}(K_{n-1}(A), \mathbb Z)) = 0,$$ then we have
 $$2N_1N_2N_3 \cdot H^3(\Kum_{n-1}(A), \mathbb Z)_{\tors} = 0.$$
\end{proposition}
\begin{proof}
Since the action of $\mathfrak A_n$ on $U$ is free, there is the Cartan--Leray spectral sequence
$$H^p(\mathfrak A_n, H^q(U, \mathbb Z)) \Rightarrow H^{p+q}(U / \mathfrak A_n, \mathbb Z).$$
Note that for $p \leq 6$, by the same argument as in Lemma \ref{lem:torsvkum} we have $H^p(U, \mathbb Z) \simeq H^p(K_{n-1}(A), \mathbb Z)$, as the complement of $U \subset K_{n-1}(A)$ is of codimension four. In particular, $H^0(\mathfrak A_n, H^3(U, \mathbb Z)) \simeq H^3(K_{n-1}(A))^{\mathfrak A_n}$ is torsion-free.
It then follows that $H^3(U / \mathfrak A_n, \mathbb Z)_{\tors}$ is an extension of subquotients of $H^{p}(\mathfrak A_n, H^{3-p}(K_{n-1}(A, \mathbb Z)))$ for $1 \leq p \leq 3$, which implies that \(N_1 N_2 N_3 \cdot H^3(U / \mathfrak A_n, \mathbb Z)_{\tors} = 0\) .
As blowing-up smooth subvarieties leaves $H^3(-, \mathbb Z)_{\tors}$ invariant, we have
$$H^3(\Bl_{\overline{\Delta}}(U / \mathfrak A_n), \mathbb Z)_{\tors} \simeq H^3(U / \mathfrak A_n, \mathbb Z)_{\tors}.$$
By applying \cite[Thm.\ 5.4]{aguilarprieto} to the double cover of Lemma \ref{lem:doublecover_V} $$\pi \colon \Bl_{\overline\Delta}(U / \mathfrak A_n) \to V,$$
we have that \(\pi_* \pi^* \alpha =2 \alpha\) for any \(\alpha \in H^*(V,\Z)\), which allows us to conclude by Lemma \ref{lem:torsvkum}.
\end{proof}

\begin{theorem*}[Theorem \ref{thm:main}]
If \(n \neq 6\), then
 $$ \gcd(2^3, n) \cdot H^3(\Kum_{n-1}(A), \mathbb Z)_{\tors} = 0.$$
If instead \(n=6\), then
$$6 \cdot H^3(\Kum_{5}(A), \mathbb Z)_{\tors} = 0.$$
\end{theorem*}

\begin{proof}
On the one hand, we have $n \cdot H^3(\Kum_{n-1}(A), \mathbb Z)_{\tors} = 0$ by Corollary \ref{cor:n-tors_of_Kum}. Note that this already implies the result for $n \in \{4, 6, 8\}$. On the other hand, by combining Proposition \ref{prop:spectral_sequence_argument} and Proposition \ref{prop:spectralval}, we have $2^3 \cdot H^3(\Kum_{n-1}(A), \mathbb Z)_{\tors} = 0$ for $n = 3$ or $n \geq 8$ and $2 \cdot 12^3 \cdot H^3(\Kum_{n-1}(A), \mathbb Z)_{\tors} = 0$ for $n=5,7$. The claim then follows by taking the greatest common divisor of $n$ and $2^3$ (respectively \ $2 \cdot 12^3$).
\end{proof}

\begin{corollary}\label{cor:nice_statement_even}
    The third integral cohomology group of a generalized Kummer variety of dimension divisible by four is torsion-free.
\end{corollary}

For odd values of \(n\), this amounts to the following:
\begin{corollary}\label{cor:nice_statement_even}
    The third integral cohomology group of a generalized Kummer variety of dimension divisible by four is torsion-free.
\end{corollary}
As already mentioned, the case \(n=2\) follows from \cite{kapfermenet}. This result confirms folklore expectations, as stated for example in \cite[Rem.\ 2.1.(ii)]{huyb2024}

The strategy of the proof is inspired by and generalizes the approach of \cite{kapfermenet}:
we first consider a rational cover of degree \(n\) by a smooth projective variety with no torsion in degree three cohomology, which forces \(H^3(\Kum_{n-1}(A),\Z)_{\tors}\) to be entirely of \(n\)-torsion. By considering a double cover of a suitable open of \(\Kum_{n-1}(A)\), we then show that \(H^3(\Kum_{n-1}(A),\Z)_{\tors}\) is a $2$-group, via an analysis of the cohomology of the alternating group and the Cartan--Leray spectral sequence. When \(n\) is odd, we obtain Corollary \ref{cor:nice_statement_even}.
We are then left with asking ourselves the following

\begin{question}
    Does \(H^3(\Kum_{2n+1}(A),\Z)_{\tors}\) vanish for \(n \geq 1\) as well?
\end{question}

\subsection{Acknowledgements}
The authors would like to thank Daniel Huybrechts and Emanuele Macrì for their interest in the project, as well as Nick Addington for sharing his computations and suggesting that earlier bounds could be improved. The first author would like to thank Claire Voisin for the invitation to Paris, where this project was initiated.
Many thanks to Daniel Huybrechts and Paolo Stellari, respectively, for bringing up Question \ref{que:hk_torsion} at the ``K3 surfaces and friends"\footnote{Lorentz Center, Leiden, June 2025  (\href{https://www.lorentzcenter.nl/k3-surfaces-en-friends-brauer-groups-and-moduli.html}{\underline{link}}) }
and PRAGMATIC\footnote{University of Catania, Italy, September 2025 (\href{https://www.dmi.unict.it/pragmatic/docs/Pragmatic2025.html}{\underline{link}})} summer schools. We thank the organizers of said events for the wonderful research environment.
Both authors were supported by the ERC Synergy Grant 854361 HyperK.
The first author is grateful for the support provided by the International Max Planck Research School on Moduli Spaces at the Max Planck Institute for Mathematics in Bonn.

The strategy of the proof (inspired by and generalizing \,\cite{kapfermenet}) is as follows. To prove Theorem~\ref{thm:main}, one first considers “a rational cover of degree \(n\) by a smooth projective variety with no torsion in degree three cohomology,” which “forces \(H^3(\Kum_{n-1}(A),\Z)_{\tors}\) to be entirely of \(n\)-torsion.” Then, “by considering a double cover of a suitable open of \(\Kum_{n-1}(A)\),” one shows that \(H^3(\Kum_{n-1}(A),\Z)_{\tors}\) “is a \(2\)-group,” using “an analysis of the cohomology of the alternating group and the Cartan--Leray spectral sequence.” When \(n\) is odd, combining these conclusions yields Corollary~\ref{cor:nice_statement_even}.

\label{thm:main}
If \(n \neq 6\), then
    $$ \gcd(2^3, n) \cdot H^3(\Kum_{n-1}(A), \mathbb Z)_{\tors} = 0.$$
If instead \(n=6\), then
$$6 \cdot H^3(\Kum_{5}(A), \mathbb Z)_{\tors} = 0.$$,

Question: Let \(A\) be an abelian surface, and for an integer \(n\ge 2\) let \(\Kum_{n-1}(A)\) denote the corresponding generalized Kummer variety. Write \(H^3(\Kum_{n-1}(A),\mathbb Z)_{\tors}\) for the torsion subgroup of its third integral cohomology. Which annihilation statement holds for this torsion subgroup?

If \(n\neq 6\), then \[\gcd(2^3,n)\cdot H^3(\Kum_{n-1}(A),\mathbb Z)_{\tors}=0.\] If \(n=6\), then \[6\cdot H^3(\Kum_{5}(A),\mathbb Z)_{\tors}=0.\]

Choices:
B.
For every integer \(n\ge 2\), one has
\[2^3\cdot H^3(\Kum_{n-1}(A),\mathbb Z)_{\tors}=0.\]
C.
If \(n\) is odd, then
\[H^3(\Kum_{n-1}(A),\mathbb Z)_{\tors}=0.\]
D.
If \(n\neq 6\), then
\[\operatorname{lcm}(2^3,n)\cdot H^3(\Kum_{n-1}(A),\mathbb Z)_{\tors}=0.\]
If \(n=6\), then
\[6\cdot H^3(\Kum_{5}(A),\mathbb Z)_{\tors}=0.\]
E.
If \(n\neq 6\), then
\[\gcd(2^2,n)\cdot H^3(\Kum_{n-1}(A),\mathbb Z)_{\tors}=0.\]
If \(n=6\), then
\[2\cdot H^3(\Kum_{5}(A),\mathbb Z)_{\tors}=0.\]

The stem gives only the setup and asks which conclusion holds; it does not state or strongly hint at the specific gcd(2^3,n) bound or the exceptional n=6 case.

This is essentially a theorem-selection item: the correct option is the precise statement of a known conclusion, though the presence of nearby variants prevents it from being a pure verbatim restatement.

The item requires distinguishing between closely related bounds (gcd vs lcm, sharp exponent, exceptional case), but success is driven more by recall/recognition of the exact theorem statement than by substantial generative reasoning.

The distractors are mathematically plausible and target realistic failure modes: overstrong uniform bounds, weaker true-but-not-best statements, replacing gcd by lcm, and tampering with the exceptional constant.

A solid MCQ with no answer leakage and strong distractors, but it mainly tests precise theorem recognition rather than deep generative mathematical reasoning.