\begin{thm}\label{thm:main} Let $F$ be a totally real field,
and $F'$ be a real quadratic extension of $F$. Assume that $p$ is
unramified in both $F$ and $F'$. Let $f$ be a Hilbert newform of
parallel even weight $k$ over $F$ with trivial central character and
level $\mathfrak{n}_f$. Let $f'$ be the base change of $f$ to $F'$.
Let $\mathfrak{n}_{f'}$ be the level of $f'$.

Let $p$ be a prime number satisfying $p\geq\mathrm{max}(k+2,7)$. We
assume that $\mathfrak{n}_f$ is prime to $p$ and $f$ is ordinary at
each prime above $p$. When $[F:\mathbb{Q}]$ is odd, we assume that
there exists at least one prime $\mathfrak{q}$ such that
$\mathfrak{q}|| \mathfrak{n}_f$, and $\mathfrak{q}$ is split in
$F'$.

Suppose the following conditions hold.
\begin{enumerate}
\item The restrictions of $\bar{\rho}_f$ to $G_{F'(\xi_p)}$ and $G_{F'(
\sqrt{ p^*})}$ are absolutely irreducible, where
$p^*=(-1)^{\frac{p-1}{2}}p$.
\item In the case of $k=2$, for each place $v$ of $F$ above $p$ we
have  $a_v^2(f)\ {\backslash\hskip -10pt \equiv } 1 \ (\mathrm{mod}
\ p)$; similarly, for each place $v'$ of $F'$ above $p$ we have
$a_{v'}^2(f')\ {\backslash\hskip -10pt \equiv } 1 \ (\mathrm{mod} \
p)$.
\item For each $\mathfrak{l}|\mathfrak{n}$, if
$\bar{\rho}_f|_{I_{F_\mathfrak{l}}}$ is absolutely irreducible, then
$\mathrm{N}(\mathfrak{l})\equiv  {\hskip -10pt /}-1 \ (\mathrm{mod
}\ p)$. Similarly, for each $\mathfrak{l}'|\mathfrak{n}'$, if
$\bar{\rho}_{f'}|_{I_{F'_{\mathfrak{l}'}}}$ is absolutely
irreducible, then $\mathrm{N}(\mathfrak{l}')\equiv  {\hskip -10pt
/}-1 \ (\mathrm{mod }\ p)$.
\item $\rho_f$ is a minimal modular lifting of $\bar{\rho}_f$.
\end{enumerate} Then $$
\Omega_{f'}^\mathrm{can}=(\Omega_f^\mathrm{can})^2 $$ up to a
$p$-adic unit.
\end{thm}

\subsection{Special value formula, Gross period and Hida canonical period}

In \cite{Gro87} Gross provided a formula expressing the special
values of $L$-functions via Heegner points. To describe his result,
let $f$ be a newform on $\Gamma_0(N)$  of even weight $k$, and let $K$ be an imaginary
quadratic field of discriminant $D$ such that $D$ is prime to $N$.

Assume $k=2$ and $N$ is a prime inert in $K$. Let $B$ be the definite quaternion algebra that is ramified exactly at $N$. Fix a prime number $p$. Let $H_n$ be the ring class field of $K$ of conductor $p^n$. When $\chi$ is a character of the Galois group $\mathrm{Gal}(H_n/K)$, one forms the Rankin-Selberg $L$-function $L(f,\chi,s)$.
Gross \cite{Gro87} showed that there exists a period $\Omega_{f,K}$
depending on $f$ and $K$ but independent of $n$ and $\chi$, called Gross
period, such that $\frac{L(f,\chi,1)}{\Omega_{f,K}}$ are algebraic
and they satisfy
$$\frac{L(f,\chi,1)}{\Omega_{f,K}}\cdot\sqrt{D}p^{n}=|\sum\limits_{\sigma\in\mathrm{Gal}(H_n/K)}\chi(\sigma)\psi(P^{\sigma})|^{2},$$
where $P$ is the Heegner point on the Shimura set attached to $B$.

Gross formula is generalized by Shouwu Zhang \cite[Theorem 7.1]{Zha04} for $k=2$ and general $N$, and by Haiping Yuan \cite{Yua05} for $k>2$.

In \cite{BD07}  Bertolini and Darmon used Gross' special value
formula to construct anticyclotomic $p$-adic $L$-functions, i.e.
$p$-adic $L$-functions for anticyclotomic $\BZ_p$-extension instead
of the cyclotomic $\BZ_p$-extension. In the same paper they proved
one divisibility for Iwasawa main conjecture for elliptic curves in
the setting of anticyclotomic $\BZ_p$-extension (\cite[Theorem
1]{BD07}), and also gave new evidence for Birch and Swinnerton-Dyer
conjecture (\cite[Corollary 4]{BD07}).

Gross period serves as a bridge between the complex $L$-function and
the $p$-adic $L$-function. But it depends on the choice of an imaginary
quadratic extension $K$.

Hida \cite{Hid81, Hid81.2} introduced a period that is independent
of $K$ and called the {\it canonical period}. Attached to $f$ there
exists a homomorphism $\lambda_f:T\longrightarrow \mathcal{O}_f$
from the Hecke algebra $T$ of level $N$ to a discrete valuation ring
$\mathcal{O}_f$ that is finite over $\BZ_p$. Let $\eta $ be the
congruence number for $\lambda_f$ defined by
$$\eta =\lambda_f(\mathrm{Ann}(\mathrm{Ker}(\lambda_f))).$$ Then the
canonical period is defined by
$$\Omega_{f}^{\mathrm{can}}=\frac{\langle f, f
\rangle_{\mathrm{Pet}}}{\eta }.$$ The reader may consult
\cite[Section 2]{Vat03} about more knowledge on Gross period and the
canonical period.

Vatsal \cite{Vat03} showed that
$\frac{\Omega_{f,K}}{\Omega_{f}^{\mathrm{can}}}$ lies in
$\mathcal{O}_f$. Pollack and Weston \cite{PW11} pointed out that
$\frac{\Omega_{f,K}}{\Omega_{f}^{\mathrm{can}}}\in\mathcal{O}_f^{\times}$
is equivalent to the freeness of spaces of modular forms on $B$ over
the associated Hecke algebra and the vanishing of certain local
Tamagawa components. In \cite{CH18} Chida and Hsieh actually proved
that $\frac{\Omega_{f,K}}{\Omega_{f}^{\mathrm{can}}}$ is a $p$-adic
unit, i.e.
$\frac{\Omega_{f,K}}{\Omega_{f}^{\mathrm{can}}}\in\mathcal{O}_f^{\times}$
under a condition ($\mathrm{CR^{+}}$). Wang \cite{Wang} generalized Chida and Hsieh's result to the setting of Hilbert
modular forms under similar hypothesis.
There are two normalizations of anticyclotomic $p$-adic
$L$-functions depending on which period one uses. However, in the
present paper we assume those hypothesis in \cite{Wang} in order
that there is essentially no difference between these two
normalizations.

\subsection{Comparison of periods under base change}

The goal of the present paper is to compare the canonical period of
a Hilbert modular form and that of its base change to a (totally)
real quadratic extension.

To make it precise, let $F$ be a totally real number field, and  $f$
be a Hilbert newform of parallel even weight over $F$. Let $F'$ be a
totally real quadratic extension of $F$, and $f'$ be the base change
of $f$ to $F'$. One may expect the following holds.

\begin{conj} We have
$$ \Omega_{f'}^\mathrm{can}=(\Omega_f^\mathrm{can})^2 $$ up to a
$p$-adic unit.
\end{conj}

The main result of this paper is the following theorem. Let $\rho_f$ be $p$-adic Galois representation attached to $f$ (see \cite{Wil88,Tay}), and let $\bar{\rho}_f$ denote the residue representation of $\rho_f$.

Assume $k=2$ and $N$ is a prime inert in $K$. Let $B$ be the definite quaternion algebra that is ramified exactly at $N$. Fix a prime number $p$. Let $H_n$ be the ring class field of $K$ of conductor $p^n$. When $\chi$ is a character of the Galois group $\mathrm{Gal}(H_n/K)$, one forms the Rankin-Selberg $L$-function $L(f,\chi,s)$.
Gross \cite{Gro87} showed that there exists a period $\Omega_{f,K}$
depending on $f$ and $K$ but independent of $n$ and $\chi$, called Gross
period, such that $\frac{L(f,\chi,1)}{\Omega_{f,K}}$ are algebraic
and they satisfy
$$\frac{L(f,\chi,1)}{\Omega_{f,K}}\cdot\sqrt{D}p^{n}=|\sum\limits_{\sigma\in\mathrm{Gal}(H_n/K)}\chi(\sigma)\psi(P^{\sigma})|^{2},$$
where $P$ is the Heegner point on the Shimura set attached to $B$.

Hida \cite{Hid81, Hid81.2} introduced a period that is independent
of $K$ and called the {\it canonical period}. Attached to $f$ there
exists a homomorphism $\lambda_f:T\longrightarrow \mathcal{O}_f$
from the Hecke algebra $T$ of level $N$ to a discrete valuation ring
$\mathcal{O}_f$ that is finite over $\BZ_p$. Let $\eta $ be the
congruence number for $\lambda_f$ defined by
$$\eta =\lambda_f(\mathrm{Ann}(\mathrm{Ker}(\lambda_f))).$$ Then the
canonical period is defined by
$$\Omega_{f}^{\mathrm{can}}=\frac{\langle f, f
\rangle_{\mathrm{Pet}}}{\eta }.$$ The reader may consult
\cite[Section 2]{Vat03} about more knowledge on Gross period and the
canonical period.

Vatsal \cite{Vat03} showed that
$\frac{\Omega_{f,K}}{\Omega_{f}^{\mathrm{can}}}$ lies in
$\mathcal{O}_f$. Pollack and Weston \cite{PW11} pointed out that
$\frac{\Omega_{f,K}}{\Omega_{f}^{\mathrm{can}}}\in\mathcal{O}_f^{\times}$
is equivalent to the freeness of spaces of modular forms on $B$ over
the associated Hecke algebra and the vanishing of certain local
Tamagawa components. In \cite{CH18} Chida and Hsieh actually proved
that $\frac{\Omega_{f,K}}{\Omega_{f}^{\mathrm{can}}}$ is a $p$-adic
unit, i.e.
$\frac{\Omega_{f,K}}{\Omega_{f}^{\mathrm{can}}}\in\mathcal{O}_f^{\times}$
under a condition ($\mathrm{CR^{+}}$). Wang \cite{Wang} generalized Chida and Hsieh's result to the setting of Hilbert
modular forms under similar hypothesis.
There are two normalizations of anticyclotomic $p$-adic
$L$-functions depending on which period one uses. However, in the
present paper we assume those hypothesis in \cite{Wang} in order
that there is essentially no difference between these two
normalizations.

To make it precise, let $F$ be a totally real number field, and $f$
be a Hilbert newform of parallel even weight over $F$. Let $F'$ be a
totally real quadratic extension of $F$, and $f'$ be the base change
of $f$ to $F'$. One may expect the following holds.

\begin{conj} We have
$$ \Omega_{f'}^\mathrm{can}=(\Omega_f^\mathrm{can})^2 $$ up to a
$p$-adic unit.
\end{conj}

The main result of this paper is the following theorem. Let $\rho_f$ be $p$-adic Galois representation attached to $f$ (see \cite{Wil88,Tay}), and let $\bar{\rho}_f$ denote the residue representation of $\rho_f$.

The main result of this paper is the following theorem. Let $\rho_f$ be $p$-adic Galois representation attached to $f$ (see \cite{Wil88,Tay}), and let $\bar{\rho}_f$ denote the residue representation of $\rho_f$.

We choose an imaginary quadratic extension $K$ of $F$ that is
inertia at $\mathfrak{q}$ when $[F:\mathbb{Q}]$ is odd. Denote $KF'$
by $K'$. Fix a place $\mathfrak{p}$ of $F$ above $p$. Let $K_\infty$ be the $\mathfrak{p}$-anticyclotomic
$\mathbb{Z}_p$-extension of $K$.

\begin{lem}\label{lem:fitt} Let $S$ be the set of primes dividing $\mathfrak{n}^-$. Then
$$ \mathrm{Fitt}(\prod_{\mathfrak{l}|
\mathfrak{n}^- }H^1((\mathcal{K}_\infty)_\mathfrak{l},
F^-_\mathfrak{l}A_\rho)^\vee) =\left(\frac{L^S_p (\mathcal{K}_\infty,
f)}{L_p (\mathcal{K}_\infty, f)}\right).
$$
\end{lem} This should be well known.
However, the authors do not have a reference about it.
\begin{proof}
By definition (see Section \ref{ss:selmer}) $H^{1}((\mathcal{K}_{\infty})_{\mathfrak{l}},F_{\mathfrak{l}}^{-}A_{\rho})$ is the inductive limit $\lim\limits_{\overrightarrow{\;\;L\;\;}} H^{1}(L_{\mathfrak{l}},F_{\mathfrak{l}}^{-}A_{\rho}),$ where $L$ runs through all finite extensions of $K$ contained in $\mathcal{K}_\infty$. For each $L$, $H^{1}(L_{\mathfrak{l}},F_{\mathfrak{l}}^{-}A_{\rho})$ is isomorphic to
$\bigoplus_{\mathfrak{L}|\mathfrak{l}} H^1(G_{L_\mathfrak{L}}, F_{\mathfrak{l}}^{-}A_{\rho})$ where $\mathfrak{L}$ runs over all places of $L$ above $\mathfrak{l}$,
and sits in the following exact sequence
{\small $$ \bigoplus_{\mathfrak{L}|\mathfrak{l}} H^1(G_{L_\mathfrak{L}}/I_{L_\mathfrak{L}}, (F_{\mathfrak{l}}^{-}A_{\rho})^{I_{L_\mathfrak{L}}})\rightarrowtail \bigoplus_{\mathfrak{L}|\mathfrak{l}} H^1(G_{L_\mathfrak{L}}, F_{\mathfrak{l}}^{-}A_{\rho})\twoheadrightarrow \bigoplus_{\mathfrak{L}|\mathfrak{l}} H^1(I_{L_\mathfrak{L}}, F_{\mathfrak{l}}^{-}A_{\rho})^{G_{L_\mathfrak{L}}/I_{L_\mathfrak{L}}} . $$}
As $G_{L_{\mathfrak{L}}}$ acts trivially on $F_\mathfrak{l}^-A_\rho$, we have
$$ H^1(G_{L_\mathfrak{L}}/I_{L_\mathfrak{L}}, (F_{\mathfrak{l}}^{-}A_{\rho})^{I_{L_\mathfrak{L}}}) \cong F_{\mathfrak{l}}^{-}A_{\rho}. $$ If $L'$ is a finite extension of $L$ contained in $\mathcal{K}_\infty$, and $\mathfrak{L}'$ is a place of $L'$ above $\mathfrak{L}$, we have the following commutative diagram
\[ \xymatrix{ H^1(G_{L_\mathfrak{L}}/I_{L_\mathfrak{L}}, (F_{\mathfrak{l}}^{-}A_{\rho})^{I_{L_\mathfrak{L}}})\ar[rr]^{\hskip45pt \simeq}\ar[d] && F_{\mathfrak{l}}^{-}A_{\rho} \ar[d]^{\times[G_{L_\mathfrak{L}}/I_{L_\mathfrak{L}}:G_{L'_{\mathfrak{L}'}}/I_{L'_{\mathfrak{L}'}}]} \\ H^1(G_{L'_{\mathfrak{L}'}}/I_{L'_{\mathfrak{L}'}}, (F_{\mathfrak{l}}^{-}A_{\rho})^{I_{L'_{\mathfrak{L}'}}})\ar[rr]^{\hskip45pt\simeq} && F_{\mathfrak{l}}^{-}A_{\rho}, } \] where the left vertical arrow is the restriction map. Since both $L_{\mathfrak{L}}$ and $L'_{\mathfrak{L}'}$ are unramified over $K_\mathfrak{l}$, $$[G_{L_\mathfrak{L}}/I_{L_\mathfrak{L}}:G_{L'_{\mathfrak{L}'}}/I_{L'_{\mathfrak{L}'}}]
=|\mathrm{Gal}(L'_{\mathfrak{L}'}/L_{\mathfrak{L}})|=[L'_{\mathfrak{L}'}:L_{\mathfrak{L}}]$$ is a power of $p$. Thus taking limit we have $$\lim\limits_{\overrightarrow{\;\;L\;\;}}\bigoplus_{\mathfrak{L}: \text{places of } L \text{ above }\mathfrak{l}} H^1(G_{L_\mathfrak{L}}/I_{L_\mathfrak{L}}, (F_{\mathfrak{l}}^{-}A_{\rho})^{I_{L_\mathfrak{L}}})=0.$$

\begin{proof} The representation $\bar{\rho}_f$ factors through
$\mathrm{Gal}(F_1/F)$ for a finite Galois extension $F_1$ of $F$.
Take a finite place $\mathfrak{q}\nmid p\mathfrak{n}$. For each
$\mathfrak{l}|p\mathfrak{n}$ take a unit $u_\mathfrak{l}$ of
$\mathcal{O}_{F_{\mathfrak{l}}}$ such that the image
$u_\mathfrak{l}$ in $k_{\mathfrak{l}}$ by the natural homomorphism
$$ \mathcal{O}_{F_\mathfrak{l}}\rightarrow
\mathcal{O}_{F_\mathfrak{l}}/\mathfrak{l}\cong k_\mathfrak{l} $$
generates the multiplicative group $k_\mathfrak{l}^\times$. Take a
uniformizing element $\omega_\mathfrak{q}$ of
$\mathcal{O}_{F_{\mathfrak{q}}}$. By the weak approximation theorem
there exists a totally positive element $a$ of $F$ such that
$a_{\mathfrak{l}}-u_{\mathfrak{l}} \in
\mathfrak{l}\mathcal{O}_{F_\mathfrak{l}}$ for each
$\mathfrak{l}|p\mathfrak{n}$, and
$a_\mathfrak{q}-\omega_\mathfrak{q}\in
\mathfrak{q}^2\mathcal{O}_{F_\mathfrak{q}}$. Put $F'=F(\sqrt{a})$.
Then each place dividing $p\mathfrak{n}$ splits in $F'$. As $a$ is
totally positive, $F'$ is totally real. Let $f'$ be the base change
of $f$ to $F'$.

\begin{thm}\label{thm:Iw-main} Assume $f$ satisfies
$(\mathrm{CR}^+)$, $(\mathfrak{n}^+\text{-}\mathrm{DT})$,
$(\mathrm{PO})$ and $(\mathrm{Fuji}1$-$4)$.
\begin{enumerate}
\item\label{it:Iw-main-a}
We have \begin{equation}\label{eq:char} \mathrm{char}\ \mathrm{Sel}
(K_\infty, A_{\rho})^\vee = (L_p(K_\infty, f)). \end{equation}
\item\label{it:Iw-main-b} Let $\mathcal{L}$ be $\mathcal{K}_\infty,
\mathcal{K}^-_\infty$ or $K^-_J$. If $[F:\BQ]$ is even, then
$$ \mathrm{char}\ \mathrm{Sel}(\mathcal{L}, A_\rho)^\vee = (L_p(\mathcal{L}, f))
$$ in $\mathcal{O}_\mathfrak{P}[[\mathrm{Gal}(\mathcal{L}/K)]]_E$.
\end{enumerate}
\end{thm}
\begin{proof} Let $F'$ be a totally real quadratic extension of $F$, as in Lemma \ref{lem:quad-ext}. Then $[F':\BQ]$ is even.
Let $\chi_{F'/F}$ be the quadratic character corresponding to the
extension $F'/F$. Put $K'=F'K$. Let $\mathcal{K}'^-_\infty$ be the
maximal abelian anticyclotomic $\BZ_p$-extension of $K'$ unramified
outside $p$. Put $\mathcal{K}'^+_\infty=F_\infty K'$,
$\mathcal{K}'_\infty=\mathcal{K}'^+_\infty\mathcal{K}'^-_\infty$ and
$\Gamma_{K'}=\mathrm{Gal}(\mathcal{K}'_\infty/K')$. Then
$K'\mathcal{K}_\infty=F'\mathcal{K}_\infty$ is contained in
$\mathcal{K}'_\infty$.

\begin{thm}\label{thm:main-stronger} Let $F$ be a totally real field,
and $F'$ be a real quadratic extension of $F$. Assume that $p$ is
unramified in both $F$ and $F'$. Let $f$ be a Hilbert newform of
parallel even weight $k$ over $F$ with trivial central character and
level $\mathfrak{n}_f$. Let $f'$ be the base change of $f$ to $F'$.
Let $\mathfrak{n}_{f'}$ be the level of $f'$.

We assume that $\mathfrak{n}_f$ is prime to $p$ and $f$ is ordinary
at each place above $p$. When $[F:\mathbb{Q}]$ is odd, we assume
that there exists at least one prime $\mathfrak{q}$ such that
$\mathfrak{q}|| \mathfrak{n}_f$, $\mathfrak{q}$ is split in $F'$,
and $\bar{\rho}_{f}$ is ramified at $\mathfrak{q}$. Suppose that, if
$\mathfrak{l}'||\mathfrak{n}_{f'}$ and
$\mathrm{N}(\mathfrak{l}')\equiv 1 \ (\mathrm{mod} \ p)$, then
$\bar{\rho}_{f'}$ is ramified at $\mathfrak{l}'$.

\begin{conj}\label{conj-mu} Let $F$ be a totally real number field, $K$ a quadratic imaginary
extension of $F$. Let $F'$ be a finite totally real extension of
$F$, and $f_{F'}$ be a Hilbert modular form over $F'$ that is
ordinary at each place of $F'$ above $p$. Then
$$\mu(L_p(F'K_\infty, f_{F'}))=0.$$
\end{conj}
Corollary \ref{thm:mu} says that Conjecture \ref{conj-mu} holds when
$F'$ is a quadratic real extension of $F$ and $f_{F'}$ is the base
change of a Hilbert modular form over $F$ (that satisfies some
conditions). It seems impossible to use the method in \cite{CH18,
Hung} to prove Conjecture \ref{conj-mu}, though it works when
$F'=F$.

We choose an imaginary quadratic extension $K$ of $F$ that is
inertia at $\mathfrak{q}$ when $[F:\mathbb{Q}]$ is odd. Denote $KF'$
by $K'$. Fix a place $\mathfrak{p}$ of $F$  above $p$. Let $K_\infty$ be the $\mathfrak{p}$-anticyclotomic
$\mathbb{Z}_p$-extension of $K$.

Using the canonical periods one attaches to $f$,
$f\otimes\chi_{F'/F}$ and $f'$ the $p$-adic $L$-functions
$$L_{p}(K_{\infty} ,f), \
L_{p}(K_{\infty} ,f\otimes\chi_{F'/F})  \ \text{ and }
L_{p}(K'K_{\infty} ,f').$$ All of them can be considered as elements
of $\mathcal{O}_{\mathfrak{P}}[[\mathrm{Gal}(K_{\infty} /K)]]$. The
ring $\mathcal{O}_{\mathfrak{P}}\supset \mathcal{O}_f$ is some
coefficient ring that is clearly defined in our context. To prove
Theorem \ref{thm:main} we only need to show
\begin{equation}\label{eq:main}
(L_{p}(K'K_{\infty} ,f'))=(L_{p}(K_{\infty} ,f))\cdot
(L_{p}(K_{\infty} ,f\otimes\chi_{F'/F}))  \end{equation} in
$\mathcal{O}_{\mathfrak{P}}[[\mathrm{Gal}(K_{\infty} /K)]].$

Our strategy is to use Iwasawa main conjecture that provides an
equality between a quantity measuring Selmer groups and $p$-adic
$L$-functions. Actually what we need is Iwasawa main conjecture for
Hilbert modular forms in the anticyclotomic setting.

Its proof is divided into two parts, one part proving one
divisibility by Ribet's method, and the other proving the converse
divisibility by Euler systems. The former divisibility was proved by
Skinner and Urban \cite{S-U} for elliptic modular forms, and was
proved by Wan \cite{Wan} for Hilbert modular forms. When
$[F:\mathbb{Q}]$ is odd, Wan needs the condition that Ihara Lemma
for Shimura curves holds. The latter divisibility was proved by
Bertolini and Darmon \cite{BD07} for elliptic curves, and by Chida
and Hsieh \cite{CH15} for elliptic modular forms. It was proved by
Longo \cite{Longo} and Wang \cite{Wang} for Hilbert modular forms
assuming Ihara Lemma for Shimura curves. This condition was removed
by the second author \cite{Xie}.

Combining results in \cite{Wan} and \cite{Xie} we obtain the
following theorem. Remark that we do not need Ihara Lemma for
Shimura curves even when $[F:\mathbb{Q}]$ is odd, though such a
condition is needed in \cite{Wan}.

\begin{thm}\label{thm:Iw-main-int}
Assume that $f$ satisfies conditions $(\mathrm{CR}^+)$,
$(\mathfrak{n}^+\text{-}\mathrm{DT})$, $(\mathrm{PO})$ and $(\mathrm{Fuji}1$-$4)$.
Then we have $$(L_p(K_\infty, f))=\mathrm{char}\ \mathrm{Sel}
(K_\infty, A_{\rho})^\vee .$$
\end{thm} See our context for Conditions $(\mathrm{CR}^+)$,
$(\mathfrak{n}^+\text{-}\mathrm{DT})$, $(\mathrm{PO})$ and
$(\mathrm{Fuji}1$-$4)$.

By Theorem \ref{thm:Iw-main-int} we have $$
(L_{p}(K_{\infty},f))=\mathrm{char}\
\mathrm{Sel}(K_{\infty},A_{\rho})^{\vee} $$ and
    $$  (L_{p}(K_{\infty},f\otimes\chi_{F'/F}))=\mathrm{char}\ \mathrm{Sel}(K_{\infty},A_{\rho\otimes\chi_{F'/F}})^{\vee}    .$$
Using the relation
$$\mathrm{Sel}(K'K_{\infty},A_{\rho})=\mathrm{Sel}(K_{\infty},A_{\rho})\bigoplus\mathrm{Sel}(K_{\infty},A_{\rho\otimes\chi_{F'/F}})$$
we obtain \begin{equation}\label{eq:Lp-eq} (L_{p}(K_{\infty},f)\cdot
L_{p}(K_{\infty},f\otimes\chi_{F'/F}))= \mathrm{char}\
\mathrm{Sel}(K'K_{\infty},A_{\rho})^{\vee}. \end{equation}

Let $\mathfrak{p}'$ be a place of $F'$ above $\mathfrak{p}$. Let
$K'_\infty$ be the $\mathfrak{p}'$-anticyclotomic
$\mathbb{Z}_p$-extension of $K'$.    With $K'_\infty$ instead of
$K_\infty$ we also have $$ (L_{p}(K_{\infty}',f'))=\mathrm{char}\
\mathrm{Sel}(K_{\infty}',A_{\rho})^{\vee},$$ from which we deduce
that \begin{equation}\label{eq:con}(L_{p}(K'K_{\infty},f'))\supseteq
\mathrm{char}\ \mathrm{Sel}(K'K_{\infty},A_{\rho})^{\vee}.
\end{equation} Combining (\ref{eq:Lp-eq}) and (\ref{eq:con}) we
obtain
\begin{equation}\label{eq:contain} (L_{p}(K'K_{\infty},f'))\supseteq (L_{p}(K_{\infty},f)\cdot
L_{p}(K_{\infty},f\otimes\chi_{F'/F})) . \end{equation} As
(\ref{eq:main}) holds in
$\mathcal{O}_{\mathfrak{P}}[[\mathrm{Gal}(K_{\infty}
/K)]][\frac{1}{p}]$, and as the $\mu$-invariants of
$L_{p}(K_{\infty} ,f)$ and $L_{p}(K_{\infty} ,f\otimes\chi_{F'/F})$
are zero, (\ref{eq:contain}) implies (\ref{eq:main}).

The paper is organized as follows. In Section \ref{sec:a} we collect
some basic facts about automorphic forms on definite quaternion
algebras. In Section \ref{sec:b} we recall the construction of
anticyclotomic $p$-adic $L$-functions via Theta elements. In Section
\ref{sec:d}, we combine Wan's result in \cite{Wan} with the result
in \cite{Xie} to prove Iwasawa main conjecture for Hilbert modular
form in the anticyclotomic setting (under certain conditions).
Finally we prove Theorem \ref{thm:main} in Section \ref{sec:e}.

In Sections \ref{sec:a}-\ref{sec:d}, the imaginary quadratic
extension $K$ of $F$ is fixed. In Section \ref{sec:e}, we will
choose a suitable $K$.

The authors thank B.H. Gross for his helpful advice.

\subsection*{Notations}

We fix a totally real number field \(F\) over \(\mathbb{Q}\). Let
\(\Sigma_{F}\) be the set of all real embeddings of \(F\) and let
\(\Sigma_{p}\) be the set of all places of \(F\) above \(p\).

For a number field \(L\), and each place \(v\) of \(L\) above a
prime number \(l\), let \(|\cdot|_{v}\) or \(|\cdot|_{L_{v}}\) be
the absolute value on \(L_{v}\) defined by
\(|x|_{v}=|N_{L_{v}/\mathbb{Q}_{l}}(x)|_{l}\), and  let
\(\varpi_{v}\) be a uniformizer of \(\mathcal{O}_{L_{v}}\).

Let $E$ be a finite extension of $\mathbb{Q}$, $\mathcal{O}$ be the ring of
integers in $E$ and $\mathfrak{P}$ be a prime  of $\mathcal{O}$
above $p$ such that $\mathcal{O}_\mathfrak{P}$ contains
$\mathcal{O}_f$. Let $\omega$ be a uniformizer of
$\mathcal{O}_\mathfrak{P}$. For each positive integer $n$ we put
$\mathcal{O}_{n}=\mathcal{O}_\mathfrak{P}/\omega^n$. We consider
$E_\mathfrak{P}$, $\mathcal{O}_\mathfrak{P}$ and $\mathcal{O}_n$ as
coefficient rings, and let $G_F=\mathrm{Gal}(\overline{F}/F)$ act trivially on them.

Let $\epsilon$ be the $p$-adic cyclotomic character of $G_F$.

Choose an imaginary quadratic extension $K/F$ (inert at $\mathfrak{q}$ when $[F:\mathbb{Q}]$ is odd) and put $K'=KF'$. Fix $\mathfrak{p}\mid p$ of $F$ and let $K_\infty$ be the $\mathfrak{p}$-anticyclotomic $\mathbb{Z}_p$-extension of $K$. Using the canonical periods, attach $p$-adic $L$-functions $L_p(K_\infty,f)$, $L_p(K_\infty,f\otimes\chi_{F'/F})$ and $L_p(K'K_\infty,f')$ in $\mathcal{O}_{\mathfrak{P}}[[\mathrm{Gal}(K_\infty/K)]]$. To prove Theorem~\ref{thm:main} it suffices to show the ideal identity
\[\tag{\ref{eq:main}} (L_{p}(K'K_{\infty} ,f'))=(L_{p}(K_{\infty} ,f))\cdot (L_{p}(K_{\infty} ,f\otimes\chi_{F'/F}))\]
inside $\mathcal{O}_{\mathfrak{P}}[[\mathrm{Gal}(K_\infty/K)]]$.

The strategy is to use the anticyclotomic Iwasawa main conjecture, i.e. an equality between $p$-adic $L$-functions and characteristic ideals of Selmer groups. By Theorem~\ref{thm:Iw-main-int},
\[(L_p(K_\infty,f))=\mathrm{char}\,\mathrm{Sel}(K_\infty,A_{\rho})^\vee,\qquad (L_p(K_\infty,f\otimes\chi_{F'/F}))=\mathrm{char}\,\mathrm{Sel}(K_\infty,A_{\rho\otimes\chi_{F'/F}})^\vee.
\]
Using the Selmer decomposition
\[\mathrm{Sel}(K'K_{\infty},A_{\rho})=\mathrm{Sel}(K_{\infty},A_{\rho})\bigoplus\mathrm{Sel}(K_{\infty},A_{\rho\otimes\chi_{F'/F}}),\]
one gets
\[\tag{\ref{eq:Lp-eq}} (L_p(K_\infty,f)\cdot L_p(K_\infty,f\otimes\chi_{F'/F}))=\mathrm{char}\,\mathrm{Sel}(K'K_\infty,A_{\rho})^\vee.
\]
Then, letting $\mathfrak{p}'\mid\mathfrak{p}$ in $F'$ and $K'_\infty$ be the $\mathfrak{p}'$-anticyclotomic $\mathbb{Z}_p$-extension of $K'$, one also has $(L_p(K'_\infty,f'))=\mathrm{char}\,\mathrm{Sel}(K'_\infty,A_{\rho})^\vee$, which implies the containment
\[\tag{\ref{eq:con}} (L_p(K'K_\infty,f'))\supseteq \mathrm{char}\,\mathrm{Sel}(K'K_\infty,A_{\rho})^\vee.
\]
Combining \eqref{eq:Lp-eq} and \eqref{eq:con} yields
\[\tag{\ref{eq:contain}} (L_{p}(K'K_{\infty},f'))\supseteq (L_{p}(K_{\infty},f)\cdot L_{p}(K_{\infty},f\otimes\chi_{F'/F})).
\]
Finally, since \eqref{eq:main} holds after inverting $p$ and the $\mu$-invariants of $L_p(K_\infty,f)$ and $L_p(K_\infty,f\otimes\chi_{F'/F})$ are zero, the containment \eqref{eq:contain} implies the equality \eqref{eq:main}, hence Theorem~\ref{thm:main}.

Choose an imaginary quadratic extension $K/F$ (inert at $\mathfrak{q}$ when $[F:\mathbb{Q}]$ is odd) and put $K'=KF'$. Fix $\mathfrak{p}\mid p$ of $F$ and let $K_\infty$ be the $\mathfrak{p}$-anticyclotomic $\mathbb{Z}_p$-extension of $K$. Using the canonical periods, attach $p$-adic $L$-functions $L_p(K_\infty,f)$, $L_p(K_\infty,f\otimes\chi_{F'/F})$ and $L_p(K'K_\infty,f')$ in $\mathcal{O}_{\mathfrak{P}}[[\mathrm{Gal}(K_\infty/K)]]$. To prove the main theorem, it suffices to show the ideal identity
\begin{equation}\label{eq:main}
(L_{p}(K'K_{\infty} ,f'))=(L_{p}(K_{\infty} ,f))\cdot
(L_{p}(K_{\infty} ,f\otimes\chi_{F'/F}))  \end{equation}
inside $\mathcal{O}_{\mathfrak{P}}[[\mathrm{Gal}(K_\infty/K)]]$.

The strategy is to use the anticyclotomic Iwasawa main conjecture, i.e. an equality between $p$-adic $L$-functions and characteristic ideals of Selmer groups. We first use the following theorem.
\begin{thm}\label{thm:Iw-main-int}
Assume that $f$ satisfies conditions $(\mathrm{CR}^+)$,
$(\mathfrak{n}^+\text{-}\mathrm{DT})$, $(\mathrm{PO})$ and $(\mathrm{Fuji}1$-$4)$.
Then we have $$(L_p(K_\infty, f))=\mathrm{char}\ \mathrm{Sel}
(K_\infty, A_{\rho})^\vee .$$
\end{thm}
Thus
\[(L_p(K_\infty,f))=\mathrm{char}\,\mathrm{Sel}(K_\infty,A_{\rho})^\vee,\qquad (L_p(K_\infty,f\otimes\chi_{F'/F}))=\mathrm{char}\,\mathrm{Sel}(K_\infty,A_{\rho\otimes\chi_{F'/F}})^\vee.
\]
Using the Selmer decomposition
\[\mathrm{Sel}(K'K_{\infty},A_{\rho})=\mathrm{Sel}(K_{\infty},A_{\rho})\bigoplus\mathrm{Sel}(K_{\infty},A_{\rho\otimes\chi_{F'/F}}),\]
one gets
\begin{equation}\label{eq:Lp-eq} (L_{p}(K_{\infty},f)\cdot
L_{p}(K_{\infty},f\otimes\chi_{F'/F}))= \mathrm{char}\
\mathrm{Sel}(K'K_{\infty},A_{\rho})^{\vee}. \end{equation}
Then, letting $\mathfrak{p}'\mid\mathfrak{p}$ in $F'$ and $K'_\infty$ be the $\mathfrak{p}'$-anticyclotomic $\mathbb{Z}_p$-extension of $K'$, one also has $(L_p(K'_\infty,f'))=\mathrm{char}\,\mathrm{Sel}(K'_\infty,A_{\rho})^\vee$, which implies the containment
\begin{equation}\label{eq:con}(L_{p}(K'K_{\infty},f'))\supseteq
\mathrm{char}\ \mathrm{Sel}(K'K_{\infty},A_{\rho})^{\vee}.
\end{equation}
Combining the equation above and the containment above yields
\begin{equation}\label{eq:contain} (L_{p}(K'K_{\infty},f'))\supseteq (L_{p}(K_{\infty},f)\cdot
L_{p}(K_{\infty},f\otimes\chi_{F'/F})) . \end{equation}
Finally, since the displayed ideal identity above holds after inverting $p$ and the $\mu$-invariants of $L_p(K_\infty,f)$ and $L_p(K_\infty,f\otimes\chi_{F'/F})$ are zero, the containment above implies the ideal identity
\begin{equation}\label{eq:main}
(L_{p}(K'K_{\infty} ,f'))=(L_{p}(K_{\infty} ,f))\cdot
(L_{p}(K_{\infty} ,f\otimes\chi_{F'/F}))  \end{equation}
and this completes the proof of the main theorem.

Question:
Let $F$ be a totally real field and let $F'/F$ be a real quadratic extension. Let $f$ be a Hilbert newform over $F$ of parallel even weight $k$, trivial central character, and level $\mathfrak{n}_f$, and let $f'$ be the base change of $f$ to $F'$ with level $\mathfrak{n}_{f'}$. Let $p$ be a prime such that $p\ge \max(k+2,7)$ and $p$ is unramified in both $F$ and $F'$. Assume that $\mathfrak{n}_f$ is prime to $p$ and that $f$ is ordinary at every prime of $F$ above $p$. If $[F:\mathbb{Q}]$ is odd, assume there exists a prime $\mathfrak{q}$ such that $\mathfrak{q}\parallel \mathfrak{n}_f$ (that is, $\mathfrak{q}$ divides $\mathfrak{n}_f$ exactly once) and $\mathfrak{q}$ is split in $F'$. Let $\rho_f$ be the $p$-adic Galois representation attached to $f$, and let $\bar\rho_f$ and $\bar\rho_{f'}$ denote the residual Galois representations attached to $f$ and $f'$, respectively. Suppose moreover that:

1. the restrictions of $\bar\rho_f$ to $G_{F'(\xi_p)}$ and to $G_{F'(\sqrt{p^*})}$ are absolutely irreducible, where $p^*=(-1)^{\frac{p-1}{2}}p$;
2. if $k=2$, then for every place $v$ of $F$ above $p$, $a_v(f)^2\not\equiv 1\pmod p$, and for every place $v'$ of $F'$ above $p$, $a_{v'}(f')^2\not\equiv 1\pmod p$;
3. for every prime $\mathfrak{l}\mid \mathfrak{n}_f$, if $\bar\rho_f|_{I_{F_{\mathfrak{l}}}}$ is absolutely irreducible, then $\mathrm N(\mathfrak{l})\not\equiv -1\pmod p$, and for every prime $\mathfrak{l}'\mid \mathfrak{n}_{f'}$, if $\bar\rho_{f'}|_{I_{F'_{\mathfrak{l}'}}}$ is absolutely irreducible, then $\mathrm N(\mathfrak{l}')\not\equiv -1\pmod p$;
4. $\rho_f$ is a minimal modular lifting of $\bar\rho_f$.

For a modular form $g$, write its canonical period as $\Omega_g^{\mathrm{can}}=\langle g,g\rangle_{\mathrm{Pet}}/\eta_g$, where $\langle g,g\rangle_{\mathrm{Pet}}$ is the Petersson inner product and $\eta_g$ is the associated congruence number. Under these hypotheses, which statement about the canonical periods of $f$ and $f'$ is valid?

The canonical periods satisfy
$$\Omega_{f'}^{\mathrm{can}}=(\Omega_f^{\mathrm{can}})^2$$
up to a $p$-adic unit; equivalently, the ratio $\Omega_{f'}^{\mathrm{can}}/(\Omega_f^{\mathrm{can}})^2$ is a $p$-adic unit.

Choices:
B.
The canonical periods satisfy
$$\Omega_{f'}^{\mathrm{can}}=(\Omega_f^{\mathrm{can}})^2$$
exactly in the coefficient ring, i.e. the ratio $\Omega_{f'}^{\mathrm{can}}/(\Omega_f^{\mathrm{can}})^2$ is equal to $1$.
C.
The canonical periods satisfy
$$\Omega_{f'}^{\mathrm{can}}=u\, (\Omega_f^{\mathrm{can}})^2$$
for some nonzero element $u$ in the coefficient field; in particular, $\Omega_{f'}^{\mathrm{can}}$ and $(\Omega_f^{\mathrm{can}})^2$ agree up to multiplication by a scalar.
D.
The canonical periods satisfy
$$\Omega_{f'}^{\mathrm{can}}=(\Omega_f^{\mathrm{can}})^2$$
up to a $p$-adic unit whenever the restrictions of $\bar\rho_f$ to $G_{F(\xi_p)}$ and to $G_{F(\sqrt{p^*})}$ are absolutely irreducible; no hypothesis involving the corresponding restrictions over $F'$ is needed.
E.
The canonical periods satisfy
$$\Omega_{f'}^{\mathrm{can}}=(\Omega_f^{\mathrm{can}})^2$$
up to multiplication by an arbitrary $p$-power; equivalently, the ratio $\Omega_{f'}^{\mathrm{can}}/(\Omega_f^{\mathrm{can}})^2$ is merely $p$-adically integral.

unit-ambiguity from ideal identity and period normalization

dropped the requirement that the scalar be a p-adic unit

base-change irreducibility hypotheses over F'

mu=0 used to upgrade containment to equality up to unit

The stem gives hypotheses but does not reveal the conclusion. The correct choice is not explicitly stated, and the reader must distinguish it from closely related alternatives.

This is essentially a theorem-recall question: under a long list of hypotheses, the task is to identify the exact conclusion. It is not a pure verbatim restatement because the options vary in strength and precision, but it remains close to theorem matching.

Some reasoning is required to separate 'up to a p-adic unit' from stronger or weaker formulations, but the main task is recognizing the precise theorem statement rather than generating a new argument or synthesis.

The distractors are mathematically plausible and target common failure modes: overstating to exact equality, weakening to mere scalar commensurability, allowing extra p-power ambiguity, or adding unnecessary hypotheses. They are distinct and well-designed.

A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it mainly tests precise recall of a known conclusion rather than deeper generative reasoning.


Classical Ramsey theory typically asks which structured subsets must appear in any finite colouring of the natural numbers. Raimi proposed a complementary point of view: he asked which partitions of $\mathbb{N}$ cannot be avoided by any finite colouring of $\mathbb{N}$, even after allowing a shift.

More precisely, given a partition $\mathbb{N}=E_1\cup E_2$, we say that $(E_1,E_2)$ is \emph{unavoidable} if for every finite partition $\mathbb{N}=\bigcup\limits_{j=1}^t F_j$ there exist $j\in\{1,\dots,t\}$ and $k\in\mathbb{N}$ such that both $(F_j+k)\cap E_1$ and $(F_j+k)\cap E_2$ are infinite. This formalises Raimi’s viewpoint on unavoidable partitions under shifts.

A classical theorem of Raimi \cite{Raimi} shows that such partitions do exist: there is a partition $\mathbb{N}=E_1\cup E_2$ with the property that for every finite partition $\mathbb{N}=\bigcup\limits_{j=1}^t F_j$ with $t\in\mathbb{N}$, one can find $j\in\{1,\dots,t\}$ and $k\in\mathbb{N}$ such that both $(F_j+k)\cap E_1$ and $(F_j+k)\cap E_2$ are infinite. Raimi’s original proof used topological methods. Hindman later gave an elementary proof \cite[p.~180, Theorem~11.15]{HM79} and showed that one may take $E_1$ to be the set of natural numbers whose last non-zero digit in the ternary expansion is $1$, and $E_2=\mathbb{N}\setminus E_1$.

Strengthenings of this phenomenon, imposing density conditions on the partition sets or guaranteeing positive densities in the conclusion, were obtained by Hegyv\'ari \cite{NH} and by Bergelson and Weiss \cite{Bergelson2}. More recently, Hegyv\'ari, Pach, and Pham \cite{HPP25} introduced a powerful and flexible framework, combining tools from harmonic analysis, additive combinatorics, and group theory, which yields polynomial and finite-group extensions of Raimi’s theorem and makes its connection to Ramsey theory explicit. Their beautiful construction in the finite-group setting has since been extended successfully to the continuous setting for circles by Kang, Koh, and the author in \cite{KKT25}, which is stated as follows.

A classical theorem of Raimi \cite{Raimi} shows that such partitions do exist: there is a partition $\mathbb{N}=E_1\cup E_2$ with the property that for every finite partition $\mathbb{N}=\bigcup\limits_{j=1}^t F_j$ with $t\in\mathbb{N}$, one can find $j\in\{1,\dots,t\}$ and $k\in\mathbb{N}$ such that both $(F_j+k)\cap E_1$ and $(F_j+k)\cap E_2$ are infinite. Raimi’s original proof used topological methods. Hindman later gave an elementary proof \cite[p.~180, Theorem~11.15]{HM79} and showed that one may take $E_1$ to be the set of natural numbers whose last non-zero digit in the ternary expansion is $1$, and $E_2=\mathbb{N}\setminus E_1$.

Before stating the main results, we need to introduce some notation.

For every finite measurable cover
\[
\mathbb{S}^{n-1} \subset F_1\cup\cdots\cup F_t,
\]
there exist an index $m_0\in\{1,\dots,t\}$ and $\theta_0\in C$ such that
\[
\sigma_{n-1}\big(R_{\theta_0}(F_{m_0})
 \cap E_i^{\mathbb{S}^{n-1}}\big)>0
\qquad\text{for all }1\le i\le r.
\]
where $R_\theta$ denotes the rotation defined in \eqref{definition-rotation} as restricted to $\mathbb{S}^{n-1}$.
\end{theorem}

\begin{theorem}[Rotational power surfaces]\label{thm:power-surface-Raimi}
Let $n\ge 3$ and $k>0$. Define
\[
\mathcal{S}_{k,R}
 := \big\{(x',x_n)\in\mathbb{R}^{n-1}\times\mathbb{R}
 : x_n = |x'|^{\,k},\ 0<|x'|\le R\big\}.
\]
Let
$\sigma_{k,R}$ be the normalized surface measure on $\mathcal{S}_{k,R}$.
Then there exists a measurable partition
$\{E_i^{\mathcal{S}_{k,R}}\}_{i=1}^r$ of $\mathcal{S}_{k,R}$ with the following property:
For every finite measurable cover
\[
\mathcal{S}_{k,R} \subset F_1\cup\cdots\cup F_t
\]
there exist an index $m_0\in\{1,\dots,t\}$ and $\theta_0\in C$ such that
\[
\sigma_{k,R}\big(R_{\theta_0}(F_{m_0})
 \cap E_i^{\mathcal{S}_{k,R}}\big)>0
\qquad\text{for all }1\le i\le r.
\]
In particular, $k=1$ yields the cone, and $k=2$ yields the paraboloid.
\end{theorem}

For every finite measurable cover
\[
\mathcal{C}_{R,\Omega} \subset F_1\cup\cdots\cup F_t
\]
there exist an index $m_0\in\{1,\dots,t\}$ and $\theta_0\in C$ such that
\[
\mu_{\text{\tiny $R,\Omega$}}\big(R_{\theta_0}(F_{m_0})\cap E_i^{\mathcal{C}_{R,\Omega}}\big)>0
\qquad\text{for all }1\le i\le r,
\]
where $R_\theta$ denotes the rotation defined in \eqref{definition-rotation} as restricted to $\mathcal{C}_{R,\Omega}$.
\end{theorem}

We now present the following general theorem, of which the spherical, cylindrical, and rotational power surface cases are specific instances.
\begin{theorem}\label{thm:general-circle-bundle}
Let $(X,\mu)$ be a probability space equipped with a measurable, measure-preserving action
\[
\{R_\theta\}_{\theta\in C},\qquad C:=\mathbb{R}/\mathbb{Z},
\]
of the circle group $C$. Assume there exist
\begin{itemize}
 \item a probability space $(Y,\nu)$,
 \item a measurable set $N\subset X$ with $\mu(N)=0$,
 \item a measurable bijection
 \[
 \Phi : C\times Y \longrightarrow X\setminus N,
 \]
\end{itemize}
satisfying:
\begin{enumerate}
 \item[(i)] (\emph{Equivariance}) For all $\theta,\alpha\in C$ and $y\in Y$,
 \[
 \Phi(\theta+\alpha,y) = R_\alpha(\Phi(\theta,y)).
 \]
 \item[(ii)] (\emph{Product disintegration}) For every bounded measurable function $f:X\to\mathbb{R}$,
 \[
 \int_X f(x)\,d\mu(x)
 = \int_C \int_Y f(\Phi(\theta,y))\,d\nu(y)\,d\mu_1(\theta),
 \]
 where $\mu_1$ is the normalized Lebesgue measure on $C$.
\end{enumerate}
Let $\{E_i^C\}_{i=1}^r$ be a measurable partition of $C$ with the Raimi property from Theorem~\ref{thm:KKT-circle}. Define
\begin{equation*}
 \begin{cases}
 E_1^X := \Phi(E_1^C\times Y) \cup N,\\
 E_i^X:=\Phi(E_i^C\times Y), ~~~~~~~~2 \leq i\leq r.
 \end{cases}
\end{equation*}
Then $\{E_i^X\}_{i=1}^r$ is a measurable partition of $X$ with the following property:

Now define a measurable partition of $C$ by
\[
 C_m := \{\theta\in C : m(\theta)=m\},\qquad 1\le m\le t.
\]
By construction, for every $\theta\in C_m$ we have
\begin{equation}\label{eq:Am-lower-bound}
 \nu(A_m(\theta))\ge \frac1t.
\end{equation}
By applying Theorem~ \ref{thm:KKT-circle} to the partition $\{C_m\}_{m=1}^t$ of $C$, there exist an index $m_0\in\{1,\dots,t\}$ and a rotation
\[
 R_{\theta_0}: C\to C,\qquad R_{\theta_0}(\theta)=\theta+\theta_0,
\]
such that
\begin{equation}\label{eq:KKT-circle-hit}
 \mu_1\big(R_{\theta_0}(C_{m_0})\cap E_i^C\big)>0,
 \qquad\, \forall\, 1\le i\le r.
\end{equation}

To complete the proof of the theorem, it remains to use \eqref{eq:KKT-circle-hit} to show that
\begin{align}\label{ineq:intersection-rotation-E_i^X}
 \mu\bigl(R_{\theta_0}(F_{m_0}) \cap E_i^X\bigr) > 0
 \qquad \text{for all } 1 \le i \le r.
\end{align}
Let
\[
 R_{\theta_0}(\theta,y):=(\theta+\theta_0,y)
\]
be the induced rotation on $C\times Y$. By equivariance, we have
\[
 R_{\theta_0}(F_{m_0})\cap (X\setminus N)
 = R_{\theta_0}\big(\Phi(F_{m_0}^{\mathrm e})\big)
 = \Phi\big(R_{\theta_0}(F_{m_0}^{\mathrm e})\big),
\]
which, together with the definition of $E_i^X$, gives
\[
 R_{\theta_0}(F_{m_0})\cap E_i^X
 = \Phi\Big(R_{\theta_0}(F_{m_0}^{\mathrm e})\cap (E_i^C\times Y)\Big)
 \cup \Big(R_{\theta_0}(F_{m_0})\cap N\cap E_i^X\Big).
\]
By the assumption on $N$, the definition of definition of $F_{m_0}^{\mathrm e}$, and the rotation on $C\times Y$, we have
\begin{align*}
 \mu\big(R_{\theta_0}(F_{m_0})\cap E_i^X\big)
 =& (\mu_1\times\nu)\big(R_{\theta_0}(F_{m_0}^{\mathrm e})\cap (E_i^C\times Y)\big) \\
 = &\int_{C}\int_Y
 \mathbf{1}_{R_{\theta_0}(F_{m_0}^{\mathrm e})}(\theta,y)
 \mathbf{1}_{E_i^C}(\theta)\,d\nu(y)\,d\mu_1(\theta) \\
 =& \int_{C}\int_Y
 \mathbf{1}_{A_{m_0}(\theta-\theta_0)}(y)\mathbf{1}_{E_i^C}(\theta)
 \,d\nu(y)\,d\mu_1(\theta) \\
 =& \int_{C} \nu\big(A_{m_0}(\theta-\theta_0)\big)\,\mathbf{1}_{E_i^C}(\theta)\,d\mu_1(\theta).
\end{align*}
By restricting the outer integral to those $\theta$ satisfying
$\theta-\theta_0\in C_{m_0}$, i.e.\ $\theta\in R_{\theta_0}(C_{m_0})$, and using \eqref{eq:Am-lower-bound} together with \eqref{eq:KKT-circle-hit}, we obtain
\begin{align*}
 \mu\big(R_{\theta_0}(F_{m_0})\cap E_i^X\big)
 \geq & \int_{R_{\theta_0}(C_{m_0})\cap E_i^C} \nu\big(A_{m_0}(\theta-\theta_0)\big)\,d\mu_1(\theta) \\
 &\ge \frac1t\,\mu_1\big(R_{\theta_0}(C_{m_0})\cap E_i^C\big)>0
\end{align*}
This is precisely \eqref{ineq:intersection-rotation-E_i^X}. This completes the proof.
\end{proof}

\begin{align}\label{definition-rotation}
R_\theta(x_1,x_2,x'',x_n)
 :=\bigl(\,
        x_1\cos(2\pi\theta) - x_2\sin(2\pi\theta),\
        x_1\sin(2\pi\theta) + x_2\cos(2\pi\theta),\
        x'',\ x_n
      \,\bigr),
\end{align}

\begin{theorem}[Kang--Koh--Tran {\cite{KKT25}}]\label{thm:KKT-circle}
Let $r,t\in\mathbb{N}$ with $r,t\ge2$. There exists a measurable partition
\[
C = \bigcup_{i=1}^r E_i
\]
such that for every finite measurable cover
\[
C \subset F_1 \cup \cdots \cup F_t,
\]
there exist an index $m \in \{1,\dots,t\}$ and a rotation $R_{\theta}$ satisfying
\begin{align*}
    \mu_1\big(R_{\theta}(F_m) \cap E_i\big) > 0
\quad\text{for all } 1\le i\le r.
\end{align*}
\end{theorem}

The theorem provides a partition of the circle with the property that every finite measurable cover admits a translate meeting each partition element in positive measure.  This paper shows that this phenomenon extends beyond compact abelian groups to a wide class of non-group geometric surfaces that still exhibit \textit{a hidden one-dimensional symmetry}. This answers a question raised in \cite{KKT25} concerning the extension
from the circle $C$ to the unit sphere $\mathbb{S}^{n-1}\subset \mathbb{R}^n$.

Before stating the main results, we need to introduce some notation.

For notational convenience, given a point
$x=(x_1,\dots,x_n)\in\mathbb{R}^n$, we write
\[
x' := (x_1,\dots,x_{n-1}) \in \mathbb{R}^{\,n-1},
\qquad
x'' := (x_3,\dots,x_{n-1}) \in \mathbb{R}^{\,n-3}.
\]
Throughout the paper, we denote by $C=\mathbb{R}/\mathbb{Z}$ the circle group. Let $\{E_i^C\}_{i=1}^r$ be the measurable partition of $C$ provided by Theorem~\ref{thm:KKT-circle}.

We also introduce the rotation
\begin{align}\label{definition-rotation}
R_\theta(x_1,x_2,x'',x_n)
 :=\bigl(\,
        x_1\cos(2\pi\theta) - x_2\sin(2\pi\theta),\
        x_1\sin(2\pi\theta) + x_2\cos(2\pi\theta),\
        x'',\ x_n
      \,\bigr),
\end{align}
that is, rotation by angle $2\pi\theta$ in the $(x_1,x_2)$–plane.

\begin{theorem}[Spheres]\label{thm:sphere-Raimi}
Let $n\ge 3$,
\[
\mathbb{S}^{n-1} := \{x\in\mathbb{R}^n : |x|=1\}
\]
be the unit sphere equipped with the normalized surface measure $\sigma_{n-1}$.
Then there exists a measurable partition $\{E_i^{\mathbb{S}^{n-1}}\}_{i=1}^r$ of $\mathbb{S}^{n-1}$ such that:

For every finite measurable cover
\[
\mathbb{S}^{n-1} \subset F_1\cup\cdots\cup F_t,
\]
there exist an index $m_0\in\{1,\dots,t\}$ and $\theta_0\in C$ such that
\[
\sigma_{n-1}\big(R_{\theta_0}(F_{m_0})
    \cap E_i^{\mathbb{S}^{n-1}}\big)>0
\qquad\text{for all }1\le i\le r.
\]
where $R_\theta$ denotes the rotation defined in \eqref{definition-rotation} as restricted to $\mathbb{S}^{n-1}$.
\end{theorem}

\begin{theorem}[Rotational power surfaces]\label{thm:power-surface-Raimi}
Let $n\ge 3$ and $k>0$. Define
\[
\mathcal{S}_{k,R}
 := \big\{(x',x_n)\in\mathbb{R}^{n-1}\times\mathbb{R}
        : x_n = |x'|^{\,k},\ 0<|x'|\le R\big\}.
\]
Let
$\sigma_{k,R}$ be the normalized surface measure on $\mathcal{S}_{k,R}$.
Then there exists a measurable partition
$\{E_i^{\mathcal{S}_{k,R}}\}_{i=1}^r$ of $\mathcal{S}_{k,R}$ with the following property:
For every finite measurable cover
\[
\mathcal{S}_{k,R} \subset F_1\cup\cdots\cup F_t
\]
there exist an index $m_0\in\{1,\dots,t\}$ and $\theta_0\in C$ such that
\[
\sigma_{k,R}\big(R_{\theta_0}(F_{m_0})
       \cap E_i^{\mathcal{S}_{k,R}}\big)>0
\qquad\text{for all }1\le i\le r.
\]
In particular, $k=1$ yields the cone, and $k=2$ yields the paraboloid.
\end{theorem}

\begin{theorem}[Cylindrical surface]\label{thm:cylinder-Raimi}
Let $n\ge 3$, $R>0$, and $\Omega\subset\mathbb{R}^{n-2}$ be a
bounded Borel set. Define the cylindrical surface
\[
\mathcal{C}_{R,\Omega}
:= \big\{(x_1,x_2, x'', x_n)\in\mathbb{R}^2\times\mathbb{R}^{n-2}
 : x_1^2+x_2^2=R^2,\ (x'', x_n)\in\Omega\big\}.
\]
Let $\mu_{\text{\tiny $R,\Omega$}}$ be the normalized surface measure on $\mathcal{C}_{R,\Omega}$.
Then there exists a measurable partition $\{E_i^{\mathcal{C}_{R,\Omega}}\}_{i=1}^r$ of $\mathcal{C}_{R,\Omega}$ with the following property:

For every finite measurable cover
\[
\mathcal{C}_{R,\Omega} \subset F_1\cup\cdots\cup F_t
\]
there exist an index $m_0\in\{1,\dots,t\}$ and $\theta_0\in C$ such that
\[
\mu_{\text{\tiny $R,\Omega$}}\big(R_{\theta_0}(F_{m_0})\cap E_i^{\mathcal{C}_{R,\Omega}}\big)>0
\qquad\text{for all }1\le i\le r,
\]
where $R_\theta$ denotes the rotation defined in \eqref{definition-rotation} as restricted to $\mathcal{C}_{R,\Omega}$.
\end{theorem}

\medskip
\paragraph{Sketch of proof.}
The essential observation is that spheres, rotational power surfaces, and cylinders all carry a natural measure-preserving action of the circle $C$ given by rotation in the $(x_1,x_2)$-plane.  Moreover, each of these surfaces admits a measurable, measure-preserving trivialization
\[
\Phi : C\times Y \longrightarrow X\setminus N,
\]
where $Y$ is a suitable parameter space and $N$ is a lower-dimensional set of measure zero.  This allows the surface measure on $X$ to disintegrate as
\[
d\mu_X(x)
 = d\mu_1(\theta)\, d\nu_Y(y),
\]
mirroring the product structure of $C\times Y$.
The equivariance identity
\[
\Phi(\theta+\alpha,y)=R_\alpha(\Phi(\theta,y))
\]
then enables the measurable Raimi partition on the base circle to be lifted directly to a partition of $X$.

The proofs of our three geometric theorems reduce to verifying the structural hypotheses of the general circle-bundle theorem introduced in Section~\ref{section-main-result}.  Once this framework is in place, the partition constructed in the previous work of Kang, Koh, and the author on the circle $C$ automatically induces the desired Raimi-type partitions on $\mathbb{S}^{n-1}$, on rotational power surfaces, and on circular cylindrical surfaces.

Our approach naturally leads to the following open question:
Let $M$ be a compact hyperbolic surface with its normalized area measure.
Does $M$ admit a measurable Raimi--type partition (in the sense of this paper), even though it has no circle action and therefore lies outside our circle--bundle framework?

{\bf The paper is organized as follows.}
In Section~\ref{section-main-result}, we establish the general circle-bundle theorem, which serves as the main structural tool of the paper.  Sections~\ref{section-sphere}, \ref{section-power-surface}, and \ref{section-cylinder} are devoted to the proofs of the three principal applications: the sphere, the rotational power surfaces, and the circular cylindrical surfaces, respectively.  Each result follows by verifying the hypotheses of the general theorem and applying the measurable Raimi partition on the base circle.

The post-theorem text does not sketch a proof of Theorem~\ref{thm:KKT-circle}; it only states that Theorem~\ref{thm:KKT-circle} provides a partition of the circle with a translate/rotation intersection property and then uses it as an input for later results. The only explicit \paragraph{Sketch of proof.} concerns the later geometric theorems (sphere/power surfaces/cylinders): it observes these surfaces have a measure-preserving circle action by rotation in the $(x_1,x_2)$-plane, admits a measurable measure-preserving trivialization \(\Phi: C\times Y\to X\setminus N\) leading to disintegration \(d\mu_X=d\mu_1(\theta)\,d\nu_Y(y)\), uses equivariance \(\Phi(\theta+\alpha,y)=R_\alpha(\Phi(\theta,y))\) to lift the partition from the base circle, and reduces the proofs to verifying hypotheses of a general circle-bundle theorem and then applying the partition from Kang--Koh--Tran on \(C\).

The post-theorem text does not sketch a proof of the main theorem; it only states that the main theorem provides a partition of the circle with a translate/rotation intersection property and then uses it as an input for later results. The only explicit \paragraph{Sketch of proof.} concerns the later geometric theorems (sphere/power surfaces/cylinders): it observes these surfaces have a measure-preserving circle action by rotation in the $(x_1,x_2)$-plane, admits a measurable measure-preserving trivialization \(\Phi: C\times Y\to X\setminus N\) leading to disintegration \(d\mu_X=d\mu_1(\theta)\,d\nu_Y(y)\), uses equivariance \(\Phi(\theta+\alpha,y)=R_\alpha(\Phi(\theta,y))\) to lift the partition from the base circle, and reduces the proofs to verifying hypotheses of a general circle-bundle theorem and then applying the partition from the main theorem on \(C\).

Question: Let $C:=\mathbb{R}/\mathbb{Z}$ be the circle group with normalized Haar measure $\mu_1$, and for $\theta\in C$ let the rotation $R_\theta:C\to C$ be given by $R_\theta(x)=x+\theta \pmod 1$. Fix integers $r,t\in\mathbb{N}$ with $r,t\ge 2$. Which statement holds about measurable partitions of $C$ and measurable covers of $C$?

There exists a measurable partition $C=\bigcup_{i=1}^r E_i$ such that for every measurable cover $C\subset F_1\cup\cdots\cup F_t$, there exist an index $m\in\{1,\dots,t\}$ and a rotation $R_\theta$ with \[\mu_1\bigl(R_\theta(F_m)\cap E_i\bigr)>0\qquad\text{for all }1\le i\le r.\]

Choices:
B. There exists a measurable partition $C=\bigcup_{i=1}^r E_i$ such that for every measurable cover $C\subset F_1\cup\cdots\cup F_t$, there exist an index $m\in\{1,\dots,t\}$ and a rotation $R_\theta$ with \[\mu_1\bigl(R_\theta(F_m)\cap E_i\bigr)=1/r\qquad\text{for all }1\le i\le r.\]
C. There exists a measurable partition $C=\bigcup_{i=1}^r E_i$ such that for every measurable cover $C\subset F_1\cup\cdots\cup F_t$, there exist an index $m\in\{1,\dots,t\}$, a rotation $R_\theta$, and an index $i_0\in\{1,\dots,r\}$ with \[\mu_1\bigl(R_\theta(F_m)\cap E_{i_0}\bigr)>0.\]
D. For every measurable partition $C=\bigcup_{i=1}^r E_i$ and every measurable cover $C\subset F_1\cup\cdots\cup F_t$, there exist an index $m\in\{1,\dots,t\}$ and a rotation $R_\theta$ with \[\mu_1\bigl(R_\theta(F_m)\cap E_i\bigr)>0\qquad\text{for all }1\le i\le r.\]
E. There exists a measurable partition $C=\bigcup_{i=1}^r E_i$ and a rotation $R_\theta$ such that for every measurable cover $C\subset F_1\cup\cdots\cup F_t$, there exists an index $m\in\{1,\dots,t\}$ with \[\mu_1\bigl(R_\theta(F_m)\cap E_i\bigr)>0\qquad\text{for all }1\le i\le r.\]

The stem does not explicitly reveal the correct statement or uniquely signal choice A. It asks which statement holds, and the alternatives differ by subtle quantifier and strength changes rather than by obvious cues.

The item is not tautological from the stem alone, but it does function largely as a theorem-identification question: the correct option appears to be the exact intended theorem statement, while the others are nearby strengthenings or weakenings.

Answering correctly requires genuine reasoning about quantifier order, existence versus universality, and the difference between positivity, exact measure constraints, and cover-dependent rotations. The strongest valid conclusion is not obvious.

The distractors are mathematically meaningful and plausible. They reflect common failure modes: overstrengthening to exact densities, weakening to a single partition cell, replacing existential with universal quantification, and mishandling dependence on the cover.

A strong MCQ with little answer leakage and high-quality distractors that test careful reasoning about quantifiers and statement strength, though it is somewhat theorem-recognition based rather than fully non-tautological.


\begin{align}\label{eq+1}
\sum_{\substack{bm\equiv \pm an\ppmod d\\(mn,q)=1}}\lambda(m) \tau(n)W\left(\frac mM\right)W\left(\frac {n}{N}\right),
\end{align}

\begin{align}
&E_{M,N}\ll q^{-1+\ve}(MN)^{\frac12}+q^\ve(M/N)^{\frac12}, \label{eqtbA}\\
&E_{M,N}\ll M^{\theta_f} \left(q^{-1+\ve}(MN)^{\frac12}+q^\ve(N/M)^{\frac12}\r). \label{eqtbB}
\end{align}

\begin{remark}\label{rem1}
Observe that $\ve(f,\chi)$ is independent of $\chi$ when $f$ is a Maa\ss~form.
When $\ve(f,\chi)=\ve(f)=-1$, summing both sides of equation \eqref{eqfe} over all primitive characters leads to the vanishing of $M_{f,E}(q;a,b)$ by symmetry. For holomorphic $f$, the root number $\ve(f,\chi)$ depends on $\chi$ at most though its parity $\chi(-1)$. The symmetry also implies that the total contribution of such characters $\chi$ satisfying $\ve(f,\chi)=-1$ to $M_{f,E}(q;a,b)$ is zero. Therefore, we restrict our consideration to the following cases:
\begin{itemize}
  \item for a Maa\ss~form $f$, we take $\ve(f)=1$,
  \item for a holomorphic $f$, we consider only characters with $\chi(-1)=\ve(f)$.
\end{itemize}
In both cases, it holds that
\[
\ve(f,\chi)=1,
\]
which we assume henceforth.
\end{remark}

\begin{theorem}\label{themain}
Let $f$ be a cuspidal Hecke eigenform of level $1$ with Hecke eigenvalues $\lambda(n)\ll \tau(n)n^{\theta_f}$. If $f$ is Maa\ss~form we further assume  that its root number satisfies $\ve(f)=1$. For $q\nequiv 2 \pmod4$ and $1\le a,b\le q$ satisfying $(a,b)=(ab,q)=1$, we have
\begin{align*}
M_{f,E}(q;a,b)=&c_f\prod_{p\mid qab}\left(1-\frac{\lambda(p)}{p}+\frac1{p^2}\r)\left(1-\frac1{p^2}\r)^{-2}\\
&\times\frac{c_{a,b}+c_{b,a}}{(ab)^{\frac12}}\frac{L(1,f)^2}{\zeta(2)} +O\left(q^{-\frac{1}{20}+\ve}a^{\frac{3}{10}}+q^{-\frac{1-2\theta_f}{22+16\theta_f}+\ve}b^{\frac{3+2\theta_f }{11+8\theta_f}}\r),
\end{align*}
where the coefficients $c_f$, $c_{a,b}$ are defined by
\begin{align}
c_f=
\begin{cases}
1/2,  &\text{if} \ f\ \text{is holomorphic},\\
1, &\text{if}\ f\ \text{is a Maa\ss\ form},
\end{cases}
\quad c_{a,b}=\sum_{a_1\mid a^\infty}\sum_{b_1\mid b^\infty} \frac{\lambda(aa_1b_1)\tau(ba_1b_1)}{a_1b_1}. \label{eqc1c2}
\end{align}
\end{theorem}

\subsection{Moments of twisted $L$-functions}
The study of moments in families of $L$-functions constitutes a central problem in modern number theory. As emphasized in the elegant work of Young \cite{You11}, these moments are not only pivotal for their wide-ranging applications but also serve as fundamental objects that unveil deep structural properties and inherent symmetries within the family.

Asymptotic formulae for moments of $L$-function with a power-saving error term are particularly crucial, as they play vital roles in amplification techniques, mollification methods, and resonance techniques. The complexity of such moment calculations can be quantified by the ratio $r=\log \mathcal{C}/\log |\mathcal{F}|$, where $|\mathcal{F}|$ denotes the size of the family and $\mathcal{C}$ its analytic conductor. Notably, computational difficulty increases with growing complexity. The threshold $r=4$ is the precise boundary, where most current analytic techniques fall just short of producing an asymptotic formula. In the few successful cases, some deep input is typically indispensable; see \cite{CLMR24} \cite{CIS12} \cite{IS00} \cite{Kha12} \cite{KMV00} \cite{Li24} for example.

Let $f$ and $g$ denote two fixed (holomorphic or non-holomorphic) Hecke eigenforms of level $1$, not necessarily cuspidal. The following second moment is defined in the context $r=4$: for $q\nequiv 2\pmod4$,
\begin{align}\label{eqgmoment}
M_{f,g}(q)=\frac1{\vp^*(q)}\ssum_{\chi\!\!\!\!\!\pmod q}L\left(\tfrac12,f\otimes\chi\r)\ol{L\left(\tfrac12,g\otimes\chi\r)},
\end{align}
where the sum runs over all primitive characters modulo $q$, and $\vp^*(q)$ denotes the number of such characters. Asymptotics for the moment \eqref{eqgmoment} with a power-saving error term are crucial to amplification and related analytic techniques in problems such as subconvexity, nonvanishing, and extreme values of $L$-functions; see \cite{BFKMMS23} for a rich sample of applications.

When both $f$ and $g$ correspond to non-cuspidal Eisenstein series, the moment \eqref{eqgmoment} reduces to
\[
M_{E,E}(q)=\frac1{\vp^*(q)}\ssum_{\chi\!\!\!\!\!\pmod q}\left|L\left(\tfrac12,\chi\r)\right|^4,
\]
the fourth moment of Dirichlet $L$-functions, which has a very long history and has been extensively studied; see \cite{BFK+17a} \cite{HB81} \cite{Sou07} \cite{Wu23} \cite{You11} for example. The first asymptotic formula with a power-savings error term was established by Young \cite{You11} for prime moduli. Wu \cite{Wu23} proved the asymptotic formula for all admissible moduli, and the best-known error term to date is $O\left(q^{-\frac1{20}+\ve}\r)$; see \cite{BFK+17b} \cite{GWZ25}.

When both $f$ and $g$ are cuspidal (holomorphic or Maa\ss) in \eqref{eqgmoment}, some special cases (ie. $f=g$) were studied earlier by Stefanicki \cite{Ste96} and Gao, Khan, and  Ricotta \cite{GKR09}. Blomer and Mili\'cevi\'c \cite{BM15} established a power-saving asymptotic for most moduli, specifically when $q$ is not close to a prime or to a product of two primes of comparable size. The main obstacle in these cases has been the lack of power-saving estimates for bilinear forms involving Kloosterman sums in the P\'olya-Vinogradov range.
The case of prime moduli was later addressed by Kowalski, Michel, and Sawin \cite{KMS17}, who proved the asymptotic with a power-saving error term $O\left(p^{-\frac1{144}+\ve}\r)$. The remaining case was recently resolved by  Mili\'cevi\'c, Qin, and Wu \cite{MQW25} and independently by  Pascadi~\cite{Pascadi2025}. Both works \cite{MQW25} and \cite{Pascadi2025} established a power-saving asymptotic for all admissible moduli $q$, with the former also removing the dependence on the Ramanujan--Petersson conjecture and achieving a sharper error term $O\left(q^{-\frac1{216}+\ve}\r)$.

These are termed mixed moments when one eigenform is cuspidal while the other corresponds to a non-cuspidal Eisenstein series. The first power-saving asymptotic formula for this case was established by Blomer, Fouvry, Kowalski, Michel, and Mili\'cevi\'c \cite{BFK+17a}, who proved that for prime moduli $p$
\[
M_{f,E}(p)=\frac1{\vp^*(p)}\ssum_{\chi\!\!\!\!\!\pmod p}L\left(\tfrac12,f\otimes\chi\r)\ol{L\left(\tfrac12,\chi\r)}^2=\frac{L(1,f)^2}{\zeta(2)}+O\left(p^{-\frac1{68}+\ve}\r).
\]
The error term was subsequently improved to $O\left(p^{-\frac1{64}+\ve}\r)$ by Shparlinski \cite{Shp19}, and more recently to $O\left(p^{-\frac1{22}+\ve}\r)$ for holomorphic $f$ and to $O\left(p^{-\frac5{152}+\ve}\r)$ for Maa\ss\ forms $f$ by Khan and Zhang \cite{KZ23}.

In this paper, we investigate the mixed moment for arbitrary moduli. More precisely, we study the following twisted mixed moment
\[
M_{f,E}(q;a,b)=\frac1{\varphi^*(q)}\ssum_{\chi\!\!\!\!\!\pmod q} L\left(\tfrac12,f\otimes\chi\r)L\left(\tfrac12,\ol\chi\r)^2\chi(\ol{a}b),
\]
for any integers $a,b\le q$ satisfying $(a,b)=(ab,q)=1$.

Asymptotic formulae for moments of $L$-function with a power-saving error term are particularly crucial, as they play vital roles in amplification techniques, mollification methods, and resonance techniques. The complexity of such moment calculations can be quantified by the ratio $r=\log \mathcal{C}/\log |\mathcal{F}|$, where $|\mathcal{F}|$ denotes the size of the family and $\mathcal{C}$ its analytic conductor. Notably, computational difficulty increases with growing complexity. The threshold $r=4$ is the precise boundary, where most current analytic techniques fall just short of producing an asymptotic formula. In the few successful cases, some deep input is typically indispensable; see \cite{CLMR24} \cite{CIS12} \cite{IS00} \cite{Kha12} \cite{KMV00} \cite{Li24} for example.

Let $f$ and $g$ denote two fixed (holomorphic or non-holomorphic) Hecke eigenforms of level $1$, not necessarily cuspidal. The following second moment is defined in the context $r=4$: for $q\nequiv 2\pmod4$,
\begin{align}\label{eqgmoment}
M_{f,g}(q)=\frac1{\vp^*(q)}\ssum_{\chi\!\!\!\!\!\pmod q}L\left(\tfrac12,f\otimes\chi\r)\ol{L\left(\tfrac12,g\otimes\chi\r)},
\end{align}
where the sum runs over all primitive characters modulo $q$, and $\vp^*(q)$ denotes the number of such characters. Asymptotics for the moment \eqref{eqgmoment} with a power-saving error term are crucial to amplification and related analytic techniques in problems such as subconvexity, nonvanishing, and extreme values of $L$-functions; see \cite{BFKMMS23} for a rich sample of applications.

When both $f$ and $g$ correspond to non-cuspidal Eisenstein series, the moment \eqref{eqgmoment} reduces to
\[
M_{E,E}(q)=\frac1{\vp^*(q)}\ssum_{\chi\!\!\!\!\!\pmod q}\left|L\left(\tfrac12,\chi\r)\right|^4,
\]
the fourth moment of Dirichlet $L$-functions, which has a very long history and has been extensively studied; see \cite{BFK+17a} \cite{HB81} \cite{Sou07} \cite{Wu23} \cite{You11} for example. The first asymptotic formula with a power-savings error term was established by Young \cite{You11} for prime moduli. Wu \cite{Wu23} proved the asymptotic formula for all admissible moduli, and the best-known error term to date is $O\left(q^{-\frac1{20}+\ve}\r)$; see \cite{BFK+17b} \cite{GWZ25}.

When both $f$ and $g$ are cuspidal (holomorphic or Maa\ss) in \eqref{eqgmoment}, some special cases (ie. $f=g$) were studied earlier by Stefanicki \cite{Ste96} and Gao, Khan, and Ricotta \cite{GKR09}. Blomer and Mili\'cevi\'c \cite{BM15} established a power-saving asymptotic for most moduli, specifically when $q$ is not close to a prime or to a product of two primes of comparable size. The main obstacle in these cases has been the lack of power-saving estimates for bilinear forms involving Kloosterman sums in the P\'olya-Vinogradov range.
The case of prime moduli was later addressed by Kowalski, Michel, and Sawin \cite{KMS17}, who proved the asymptotic with a power-saving error term $O\left(p^{-\frac1{144}+\ve}\r)$. The remaining case was recently resolved by Mili\'cevi\'c, Qin, and Wu \cite{MQW25} and independently by Pascadi~\cite{Pascadi2025}. Both works \cite{MQW25} and \cite{Pascadi2025} established a power-saving asymptotic for all admissible moduli $q$, with the former also removing the dependence on the Ramanujan--Petersson conjecture and achieving a sharper error term $O\left(q^{-\frac1{216}+\ve}\r)$.

These are termed mixed moments when one eigenform is cuspidal while the other corresponds to a non-cuspidal Eisenstein series. The first power-saving asymptotic formula for this case was established by Blomer, Fouvry, Kowalski, Michel, and Mili\'cevi\'c \cite{BFK+17a}, who proved that for prime moduli $p$
\[
M_{f,E}(p)=\frac1{\vp^*(p)}\ssum_{\chi\!\!\!\!\!\pmod p}L\left(\tfrac12,f\otimes\chi\r)\ol{L\left(\tfrac12,\chi\r)}^2=\frac{L(1,f)^2}{\zeta(2)}+O\left(p^{-\frac1{68}+\ve}\r).
\]
The error term was subsequently improved to $O\left(p^{-\frac1{64}+\ve}\r)$ by Shparlinski \cite{Shp19}, and more recently to $O\left(p^{-\frac1{22}+\ve}\r)$ for holomorphic $f$ and to $O\left(p^{-\frac5{152}+\ve}\r)$ for Maa\ss\ forms $f$ by Khan and Zhang \cite{KZ23}.

In this paper, we investigate the mixed moment for arbitrary moduli. More precisely, we study the following twisted mixed moment
\[
M_{f,E}(q;a,b)=\frac1{\varphi^*(q)}\ssum_{\chi\!\!\!\!\!\pmod q} L\left(\tfrac12,f\otimes\chi\r)L\left(\tfrac12,\ol\chi\r)^2\chi(\ol{a}b),
\]
for any integers $a,b\le q$ satisfying $(a,b)=(ab,q)=1$.

\begin{align}\label{eqgmoment}
M_{f,g}(q)=\frac1{\vp^*(q)}\ssum_{\chi\ppmod q}L\left(\tfrac12,f\otimes\chi\r)\ol{L\left(\tfrac12,g\otimes\chi\r)},
\end{align}

\subsection{Twisted $L$-functions}
We review some basic facts on twisted $L$-functions in this section; for further details, see \cite[Section 2.3]{BFK+17a}.
Let $\chi$ be a primitive Dirichlet character modulo $q$ and let $f$ be a cuspidal Hecke eigenform with Hecke eigenvalues $\lambda(n)$. The twisted form $f\otimes\chi$ remains cuspidal for the congruence subgroup $\Gamma_0(q^2)$ with nebentypus $\chi^2$ (see e.g \cite[Propositions 14.19 \& 14.20]{IK04}). Its $L$-function is given by
\[
L(s,f\otimes\chi)=\sum_{n\ge1}\frac{\lambda(n)\chi(n)}{n^{s}}=\prod_p\left(1-\frac{\lambda(p)\chi(p)}{p^s}+\frac{\chi^2(p)}{p^{2s}}\r)^{-1},\ \ \text{for}\ \ \re(s)>1,
\]
where the product is over primes $p$.
The completed $L$-function is defined by
\begin{align*}
\Lambda(s,f\otimes\chi)=q^s L_\infty(s,f\otimes\chi)L(s,f\otimes\chi),
\end{align*}
with
\begin{align*}
L_\infty(s,f\otimes\chi)=\begin{cases}
(2\pi)^{-\frac{k-1}2-s}\Gamma\left(\frac{k-1}2+s\r), &\text{if} \ f\ \text{is holomorphic of weight}\ k,\\
\pi^{-s-\ma}\Gamma\left(\frac{s+i\kappa+\ma}2\r)\Gamma\left(\frac{s-i\kappa+\ma}2\r), &\text{if}\ f\ \text{is a Maa\ss\ form with eigenvalue}\ \tfrac14+\kappa^2.
\end{cases}
\end{align*}
This completed $L$-function admits analytic continuation to $\mathbb{C}$ and satisfies the functional equation (see e.g \cite[Theorem 14.17, Proposition 14.20]{IK04}):
\[
\Lambda(s,f\otimes\chi)=\ve(f\otimes\chi)\Lambda(1-s,f\otimes\ol{\chi}),
\]
where the root number is given by
\begin{align*}
\ve(f\otimes\chi)=\begin{cases}
\ve(f)\ve_\chi^2, &\text{if} \ f\ \text{is holomorphic},\\
\chi(-1)\ve(f)\ve_\chi^2, &\text{if}\ f\ \text{is a Maa\ss\ form},
\end{cases}
\end{align*}
and $\ve(f)=\pm 1$ denotes the root number of $L(s,f)$.

Using the identity
\[
S(hq\ol{b}, m;a_1k)=e\left(-m\frac{\ol{a}_1}{b}\right)S_{\infty,1/a_1}(hq,m;\gammaup),
\]
where $\gammaup=a_1\ssqrt{b}k$ and $Q=a_1b$, we derive from \eqref{eqdefmS} that
\begin{align*}
\mS(a_1,r)=&\frac{\sqrt{b}}{aN_1}\sum_{h\sim H}\sum_{m\sim M^*} \lambda(m)e\left(-m\frac{\ol{a}_1}{b}\right)\\
&\times\sum_{\gammaup}\frac 1{\gammaup}S_{\infty,1/a_1}(hq, m;\gammaup)W\left(\frac {r\gammaup}{a_1\ssqrt{b}N_1}\r)\mathring{V}_{+}\left(\frac{bm}{\gammaup^2},h\right).
\end{align*}
Performing the variable substitution $(bx-hq)/N\rightarrow x$ in \eqref{eqV0} yields
\[
\mathring{V}_{+}\left(\frac{bm}{\gammaup^2},h\right)=\frac{N}{b}\Omega\Bigg(\frac{4\pi\ssqrt{hqm}}{\gammaup},\frac{4\pi\ssqrt{mN}}{\gammaup},h,m\Bigg),
\]
where
\[
\Omega(y,z,h,m)=\int_{0}^\infty W(x)W\left(\frac {hq+xN}M\r) \mJ_{+}\left(\ssqrt{y^2+xz^2}\right)\d x
\]
with
\begin{equation}\label{eqyz}
y\asymp Y:=\frac{r\ssqrt{MM^*}}{a_1\ssqrt{b}N_1}\gg q^\ve,\quad z\asymp \frac{r\ssqrt{M^*N}}{a_1\ssqrt{b}N_1},\quad h\asymp H,\quad m\asymp M^*.
\end{equation}
For $M\gg q^\ve N$, it is easy to see from \eqref{eqlargeM} and \eqref{eqyz} that
\[
y\gg z\quad \text{and}\quad y\gg z^2q^{-\ve}.
\]
Then Lemma \ref{lemdecomJ} yields the decomposition
\[
\Omega(y,z,h,m)=\Omega_+(y,z,h,m)e^{iy}+\Omega_-(y,z,h,m)e^{-iy}+O_A\left(q^{-A}\r),
\]
where the functions $\Omega_{\pm}$ satisfy
\begin{align}\label{eqWpmd}
y^{j_1}z^{j_2}h^{j_3}m^{j_4}\frac{\partial^{j_1+j_2+j_3+j_4}}{\partial y^{j_1}\partial z^{j_2}\partial h^{j_3}\partial m^{j_4}}\Omega_{\pm}(y,z,h,m)\ll_{j_1,j_2,j_3,j_4}q^{(j_1+j_2+j_3+j_4)\ve} Y^{-\frac12}.
\end{align}
After separating variables in $\Omega_\pm$ via the Mellin transform, we bound $\mS(a_1,r)$ by sums of the form
\begin{equation*}
\mS(a_1,r)\ll q^\ve\frac{N Y^{-\frac12}}{a\ssqrt{b}N_1}\B|\sum_{H<h\le 2H}\alpha_h\sum_{M^*<m\le 2M^*} \beta_m\lambda(m)\sum_{\gammaup}\frac1{\gammaup}S_{\infty,1/a_1}(hq,m;\gammaup)\Theta \B(\frac{4\pi\ssqrt{hqm}}{\gammaup}\B)\B|
\end{equation*}
for some coefficients satisfying $\alpha_h\ll h^\ve$, $\beta_m\ll m^\ve$ and a test function
\[
\Theta (y)=W\B(\frac yY\B)\exp(\pm iy)y^{-u}
\]
with $\re(u)=\ve$.

An application of the Kuznetsov formula (Lemma \ref{lemKf}) followed by spectral truncation at parameter $Y^{\frac12+\ve}$ ( Lemma \ref{lemBe}) produces
\begin{equation*}
\mS(a_1,r)\ll q^\ve\frac{N Y^{-\frac12}}{a\ssqrt{b}N_1}\B|\sum_{H<h\le 2H}\alpha_h\sum_{M^*<m\le 2M^*} \beta_m\lambda(m)\left(\mH+\mM+\mE\r)\B|,
\end{equation*}
where the spectral contributions decompose as
\[
\mH=\sum_{\substack{2\le k\le Y^{\frac12+\ve}\\ k ~\text{even}}}\sum_{f\in \mathcal{B}_k(Q)}\tilde{\Theta} (k)\ssqrt{hqm}~ \bar\rho_f(hq)~\rho_f(a_1^{-1},m),
\]
\[
\mM=\sum_{\substack{f\in \mathcal{B}(Q)\\ \kappa_f\ll Y^{\frac12+\ve}}}\hat{\Theta} (\kappa_f)\frac{\ssqrt{hqm}~ \bar\rho_f(hq)~\rho_f(a_1^{-1},m)}{\cosh(\pi \kappa_f)},
\]
\[
\mE=\frac1{4\pi}\sum_{\mc}\int_{|\kappa|\ll Y^{\frac12+\ve}}\hat{\Theta} (\kappa)\ssqrt{hqm} ~\frac{\bar\rho_{\mc}(hq,\kappa)~\rho_{1/{a_1},\mc}(m,\kappa)}{\cosh(\pi\kappa)}\d \kappa,
\]
representing the holomorphic cusp forms, Maa\ss\ cusp forms, and the Eisenstein series respectively.
We foucus our detailed analysis on the Maa\ss\ case (the holomorphic and Eisenstein cases are identical). Since $\hat{\Theta} (\kappa_f)\ll Y^{-\frac12+\ve}$ (see \eqref{eqphi1}), the contribution of the Maa\ss\ case is bounded by
\[
\ll q^\ve\frac{NY^{-1}}{a\ssqrt{b}N_1}\sum_{\substack{f\in \mathcal{B}(Q)\\ \kappa_f\ll Y^{\frac12+\ve}}}\B|\sum_{H<h\le 2H}\alpha_h\sum_{M^*<m\le 2M^*} \beta_m\lambda(m)\frac{\ssqrt{hqm} ~\bar\rho_f(hq)~\rho_f(a_1^{-1},m)}{\cosh(\pi \kappa_f)}\B|,
\]
which, after applying the Cauchy-Schwarz inequality, can be bounded by
\begin{align*}
\ll q^\ve\frac{NY^{-1}}{a\ssqrt{b}N_1}&\B(\sum_{\substack{f\in \mathcal{B}(Q)\\ \kappa_f\ll Y^{\frac12+\ve}}}\frac{1}{\cosh(\pi \kappa_f)}\B|\sum_{H<h\le 2H}\alpha_h \ssqrt{hq} ~\rho_f(hq)\B|^2\B)^{\frac12}\\
&\times \B(\sum_{\substack{f\in \mathcal{B}(Q)\\ \kappa_f\ll Y^{\frac12+\ve}}}\frac{1}{\cosh(\pi \kappa_f)}\B|\sum_{M^*<m\le 2M^*}\beta_m\lambda(m)\ssqrt{m} ~\rho_f(a_1^{-1},m)\B|^2\B)^{\frac12}.
\end{align*}
We may handle the second spectral sum via Lemma \ref{lemsls}, but for the first spectral sum, an application of Lemma \ref{lemsls} leads to an extra factor of $q$ to the final estimate. To address this issue, we apply the following estimate, established in \cite[Theorems 11.3-11.6]{GWZ25}.
\begin{lemma}\label{thmsls}
Let $Q,s\in\mathbb{N}$, $K, N\ge 1$, and let $(a_n)_{n\in[N,2N]}$ be complex coefficients. Then we have
\[
\sum_{\substack{2\le k\le K\\ k~ \text{even}}}\Gamma(k)\sum_{f\in\mathcal{B}_k(Q)}\B|\sum_{n}a_n\ssqrt{sn}~\rho_f(sn)\B|^2, \ \ \ \sum_{\mc}\int_{-K}^K\frac1{\cosh(\pi\kappa)}\B|\sum_n a_n\ssqrt{sn}~\rho_{\mc}(\pm sn,\kappa)\B|^2\d\kappa,
\]
\[
\sum_{\substack{f\in\mathcal{B}(Q)\\ |\kappa_f|\le K}}\frac1{\cosh(\pi\kappa_f)}\B|\sum_{n}a_n\ssqrt{sn}~\rho_f(sn)\B|^2
\]
are bounded by
\[
\ll (sQKN)^\ve\B(K+\frac{s^{\frac12}}{Q^{\frac12}}\B)\B(K+\frac{(s,Q)^{\frac12}N}{Q^{\frac12}}\B)N\|a_n\|_\infty.
\]
\end{lemma}
Applying Lemma \ref{thmsls} to the first spectral sum and Lemma \ref{lemsls} to the second one reduces the Maa\ss\ contribution to
\begin{equation*}
\ll q^\ve\frac{NY^{-1}}{a\ssqrt{b}N_1}\B(H^{\frac12}{M^*}^{\frac12}\left(Y+\frac{q}{a_1b}\r)^{\frac14} \left(Y+\frac{(a_1b,q)H^2}{a_1b}\r)^{\frac14}\left(Y+\frac{M^*}{a_1b}\r)^{\frac12}\B).
\end{equation*}
Taking
\[
 H=\frac{M}q,\ \ \ M^*= q^\ve\frac{a_1^2N_1^2}{r^2}\frac{bM}{N^2}, \ \ \ Y=\frac{r\ssqrt{MM^*}}{a_1\ssqrt{b}N_1}=\frac{M}{N},
\]
and observing that
$\frac{M^*}{a_1b}\ll Y$ ( since $N_1^2\ll N/a$), we obtain
for the Maa\ss\ contribution
\begin{align*}
&\ll q^\ve\frac{N}{r\ssqrt{q}}\left(Y+\frac{q}{ab}\r)^{\frac14} \left(Y+\frac{(ab,q)H^2}{ab}\r)^{\frac14}Y^{\frac12}\\
&\ll q^\ve r^{-1}\B(\frac{M}{q^{\frac12}}+\frac{(ab,q)^{\frac14}M^{\frac 54}N^{\frac14}}{(ab)^{\frac14}q} +\frac{M^{\frac 34}N^{\frac14}}{(ab)^{\frac14}q^{\frac14}}+\frac{(ab,q)^{\frac14}MN^{\frac12}}{(ab)^{\frac12}q^{\frac 34}}\B).
\end{align*}

\begin{remark}
The dependence on  $\theta_f$ in the error term of Theorem \ref{themain} arises from the trivial estimate \eqref{eqtbB}, which constitutes the sole component requiring an upper bound for the Hecke eigenvalues of $f$. The assumption $\ve(f)=1$ for a Maa\ss~form is natural, as otherwise $M_{f,E}(q;a,b)=0$ for parity reasons (see Remark \ref{rem1}). In the Maa\ss~case, the coefficient $c_f$ addresses a literature discrepancy originating from a typographical oversight in \cite[Theorem 1.2]{BFK+17a} and was subsequently replicated in \cite[Thoerem 1.1]{KZ23}.
\end{remark}

Setting $a=b=1$ in Theorem \ref{themain} results in a power-saving asymptotic for $M_{f,E}(q)$ across all admissible moduli $q$.

\begin{corollary}\label{cor}
Let $f$ be a cuspidal Hecke eigenform of level $1$ with Hecke eigenvalues $\lambda(n)\ll \tau(n)n^{\theta_f}$. If $f$ is Maa\ss~form we further assume that its root number satisfies $\ve(f)=1$. For $q\nequiv 2 \pmod4$, we have
\begin{align*}
M_{f,E}(q)=c_f\prod_{p\mid q}\left(1-\frac{\lambda(p)}{p}+\frac1{p^2}\r)\left(1-\frac1{p^2}\r)^{-2}\frac{L(1,f)^2}{\zeta(2)}+O\left(q^{-\frac{1-2\theta_f }{22+16\theta_f }+\ve}\r),
\end{align*}
where $c_f$ is the same constant as previously defined.
\end{corollary}
For the holomorphic case, Deligne's well-known work \cite{Del74} gives $\theta_f=0$, whereas for the Maa\ss~form we may take $\theta_f= 7/{64}$ due to Kim--Sarnak \cite{Kim03}. This yields an error term as sharp as the best bound \cite{KZ23} known for prime moduli.

Through the functional equations of $L$-functions and the orthogonality of characters, the evaluation of the moment $M_{f,E}(q,a,b)$ reduces to analyzing the shifted convolution sum
\begin{align}\label{eq+1}
\sum_{\substack{bm\equiv \pm an\!\!\!\!\!\pmod d\\(mn,q)=1}}\lambda(m) \tau(n)W\left(\frac mM\right)W\left(\frac {n}{N}\right),
\end{align}
where $a, b,M,N\ge1$ and $W$ is smooth test functions. The classical approach to such shifted convolution sum involves either the $\delta$-method or Jutila's circle method to eliminate the congruence condition (see \cite{BM15}). For \eqref{eq+1}, a more efficient approach is to expand the divisor function using the multiplicative decomposition $\tau(n)=\sum_{n_1n_2=n}1$, which decomposes the $n$-sum into two sums with smooth coefficients, making them amenable to Fourier-analytic techniques.

The diagonal terms (where $bm=an$) in \eqref{eq+1} contribute to the main term, which can be evaluated explicitly using standard techniques. The primary challenge arise from the off-diagonal terms, which we classify into two distinct cases based on the relative magnitudes of $bM$ and $aN$: the balanced case and the unbalanced case. These two cases demand fundamentally different approaches.

In the balanced case where $bM$ and $aN$ are of comparable size, we first remove the coprimality condition and reduce the problem to estimating the sum
\begin{align}\label{eqAdef}
A_q(a,b,M,N)=\sum_{\substack{bm\equiv \pm an \!\!\!\!\!\pmod q\\ bm\neq an}}\lambda(m) \tau(n) W\left(\frac {bm}M\r) W\left(\frac {an}N\r),
\end{align}
where $W$ are some smooth functions.
For $M\ge N$, we establish the bound
\[
A_q(a,b,M,N)\ll  q^\ve\Bigg(\frac{M}{q^{\frac12}}+\frac{(ab,q)^{\frac14}M^{\frac54}N^{\frac14}}{(ab)^{\frac14}q} +\frac{M^{\frac34}N^{\frac14}}{(ab)^{\frac14}q^{\frac14}} +\frac{(ab,q)^{\frac14}MN^{\frac12}}{(ab)^{\frac12}q^{\frac34}}\Bigg)
\]
with an analogous result holding for $N\ge M$. Compared to the bound in \cite[(8.13)]{BM15}, our result achieves two key improvements: reducing the influence of the factor $(ab,q)^{\frac12}$ and recovering a saving $(ab)^{-\frac14}$ in the third term. This improvements stem from two innovation: the elimination of dependence on the Ramanujan--Petersson conjecture and the minimization of the impact of $a, b$ through new upper bounds for sums involving Kloosterman sums, as established in the recent work of Gao, Wu, and Zhao \cite{GWZ25}.

For the unbalanced case where $BM$ and $AN$ differ significantly in size, the theory of algebraic trace functions becomes inapplicable due to its restriction to prime moduli. To address this, we employ the multiplicative decomposition of the divisor function $\tau(n)$, which allows us to reformulate the problem as analyzing a bilinear sum involving incomplete Kloosterman sum. The estimate of this bilinear form relies primarily on a recent result by Kerr, Shparlinski, Wu, and Xi \cite{KSWX23} concerning arbitrary moduli.
The approach also requires the Weil bound for Kloosterman sums and several Fourier-analytic tools, including the Poisson summation formula and the Voronoi summation formula.

\subsection{Notation}
We adopt the standard convention that $\ve$ denotes an arbitrarily small positive constant whose values may vary from one occurrence to another. All implied constants in our estimates may depend on $\ve$ as well as on the fixed cusp form $f$.
Throughout this paper, $W(x)$ denotes a smooth complex-valued function with compact support in $[1/2,3]$, satisfying
\begin{align}\label{eqW}
W^{(j)}(x)\ll_{j}1,
\end{align}
for all integers $j\ge0$. The function $W(x)$ may also vary at each occurrence. We call a modulus $q$ admissible if $q\nequiv 2\pmod4$, which is a natural assumption since no primitive characters exist modulo $q$ otherwise.

Through the functional equations of $L$-functions and the orthogonality of characters, the evaluation of the moment $M_{f,E}(q,a,b)$ (hence Theorem~\ref{themain}) “reduces to analyzing the shifted convolution sum”
\[
\sum_{\substack{bm\equiv \pm an\!\!\!\!\!\pmod d\\(mn,q)=1}}\lambda(m) \, \tau(n)\,W\!\left(\frac mM\right)W\!\left(\frac {n}{N}\right).
\]
Instead of using “the $\delta$-method or Jutila's circle method,” the argument uses “a more efficient approach” by expanding the divisor function via “the multiplicative decomposition $\tau(n)=\sum_{n_1n_2=n}1$,” which splits the $n$-sum into two smooth sums “amenable to Fourier-analytic techniques.”

The “diagonal terms (where $bm=an$)” give the main term and “can be evaluated explicitly using standard techniques.” The remaining “off-diagonal terms” are treated by splitting into two regimes “based on the relative magnitudes of $bM$ and $aN$”:

1) **Balanced case** ($bM$ comparable to $aN$): “we first remove the coprimality condition and reduce the problem” to bounding
\[
A_q(a,b,M,N)=\sum_{\substack{bm\equiv \pm an \!\!\!\!\!\pmod q\\ bm\neq an}}\lambda(m)\tau(n) W\!\left(\frac {bm}M\right) W\!\left(\frac {an}N\right),
\]
and establish the stated upper bound for $A_q(a,b,M,N)$ (and its analogue when $N\ge M$). The text attributes the improvement to “two innovation: the elimination of dependence on the Ramanujan--Petersson conjecture and the minimization of the impact of $a, b$ through new upper bounds for sums involving Kloosterman sums” from \cite{GWZ25}.

2) **Unbalanced case** ($BM$ and $AN$ very different): since “the theory of algebraic trace functions becomes inapplicable due to its restriction to prime moduli,” one again uses the decomposition of $\tau(n)$ to “reformulate the problem as analyzing a bilinear sum involving incomplete Kloosterman sum.” This bilinear form is estimated using \cite{KSWX23} (for “arbitrary moduli”), together with “the Weil bound for Kloosterman sums” and Fourier-analytic tools including “the Poisson summation formula and the Voronoi summation formula.”

Finally, the remark notes that the $\theta_f$-dependence in the error term “arises from the trivial estimate \eqref{eqtbB},” which is “the sole component requiring an upper bound for the Hecke eigenvalues of $f$.”

Through the functional equations of $L$-functions and the orthogonality of characters, the evaluation of the moment $M_{f,E}(q,a,b)$ (hence, in establishing the main theorem) “reduces to analyzing the shifted convolution sum”
\[
\sum_{\substack{bm\equiv \pm an\!\!\!\!\pmod d\\(mn,q)=1}}\lambda(m) \, \tau(n)\,W\!\left(\frac mM\right)W\!\left(\frac {n}{N}\right).
\]
Instead of using “the $\delta$-method or Jutila's circle method,” the argument uses “a more efficient approach” by expanding the divisor function via “the multiplicative decomposition $\tau(n)=\sum_{n_1n_2=n}1$,” which splits the $n$-sum into two smooth sums “amenable to Fourier-analytic techniques.”

The “diagonal terms (where $bm=an$)” give the main term and “can be evaluated explicitly using standard techniques.” The remaining “off-diagonal terms” are treated by splitting into two regimes “based on the relative magnitudes of $bM$ and $aN$”:

1) **Balanced case** ($bM$ comparable to $aN$): “we first remove the coprimality condition and reduce the problem” to bounding
\[
A_q(a,b,M,N)=\sum_{\substack{bm\equiv \pm an \!\!\!\!\pmod q\\ bm\neq an}}\lambda(m)\tau(n) W\!\left(\frac {bm}M\right) W\!\left(\frac {an}N\right),
\]
and establish the stated upper bound for $A_q(a,b,M,N)$ (and its analogue when $N\ge M$). The text attributes the improvement to “two innovation: the elimination of dependence on the Ramanujan--Petersson conjecture and the minimization of the impact of $a, b$ through new upper bounds for sums involving Kloosterman sums” from \cite{GWZ25}.

2) **Unbalanced case** ($BM$ and $AN$ very different): since “the theory of algebraic trace functions becomes inapplicable due to its restriction to prime moduli,” one again uses the decomposition of $\tau(n)$ to “reformulate the problem as analyzing a bilinear sum involving incomplete Kloosterman sum.” This bilinear form is estimated using \cite{KSWX23} (for “arbitrary moduli”), together with “the Weil bound for Kloosterman sums” and Fourier-analytic tools including “the Poisson summation formula and the Voronoi summation formula.”

Finally, the remark notes that the $\theta_f$-dependence in the error term “arises from the trivial estimate
\begin{align}
&E_{M,N}\\ll q^{-1+\ve}(MN)^{\frac12}+q^\ve(M/N)^{\frac12}, \label{eqtbA}\\
&E_{M,N}\\ll M^{\theta_f} \left(q^{-1+\ve}(MN)^{\frac12}+q^\ve(N/M)^{\frac12}\r). \label{eqtbB}
\end{align}
,” which is “the sole component requiring an upper bound for the Hecke eigenvalues of $f$.”

Question:
Let f be a cuspidal Hecke eigenform of level 1 with Hecke eigenvalues λ(n) satisfying λ(n) ≪ τ(n)n^{θ_f}, where τ(n) is the divisor function. If f is a Maaß form, also assume that its root number satisfies ε(f)=1. For an integer q with q ≢ 2 (mod 4), and integers 1 ≤ a,b ≤ q satisfying (a,b)=(ab,q)=1, define the twisted mixed moment
M_{f,E}(q;a,b) = (1/φ*(q)) ∑_{χ primitive mod q} L(1/2,f⊗χ) L(1/2,χ̄)^2 χ(āb),
where φ*(q) is the number of primitive Dirichlet characters modulo q and ā denotes the inverse of a modulo q. Also define
c_f = 1/2 if f is holomorphic, and c_f = 1 if f is a Maaß form,
and
c_{a,b} = ∑_{a_1|a^∞} ∑_{b_1|b^∞} λ(aa_1b_1)τ(ba_1b_1)/(a_1b_1),
with c_{b,a} defined by swapping a and b. Which statement holds for every such f, q, a, and b?

One has
M_{f,E}(q;a,b) = c_f · ∏_{p∣qab} (1 − λ(p)/p + 1/p^2)(1 − 1/p^2)^(−2) · ((c_{a,b}+c_{b,a})/(ab)^(1/2)) · (L(1,f)^2/ζ(2)) + O(q^(−1/20+ε)a^(3/10) + q^(−(1−2θ_f)/(22+16θ_f)+ε) b^((3+2θ_f)/(11+8θ_f))).

Choices:
B.
One has
M_{f,E}(q;a,b) = c_f \cdot \prod_{p\mid qab} (1 - \lambda(p)/p + 1/p^2)(1 - 1/p^2)^{(-2)} \cdot \frac{c_{a,b}+c_{b,a}}{(ab)^{1/2}} \cdot \frac{L(1,f)^2}{\zeta(2)} + O\!\left(q^{-1/20+\varepsilon}(ab)^{3/10} + q^{-(1-2\theta_f)/(22+16\theta_f)+\varepsilon}(ab)^{(3+2\theta_f)/(11+8\theta_f)}\right).
C.
One has
M_{f,E}(q;a,b) = c_f \cdot \prod_{p\mid qab} (1 - \lambda(p)/p + 1/p^2)(1 - 1/p^2)^{(-2)} \cdot \frac{c_{a,b}+c_{b,a}}{(ab)^{1/2}} \cdot \frac{L(1,f)^2}{\zeta(2)} + O\!\left(q^{-1/20+\varepsilon}a^{3/10} + q^{\varepsilon} b^{(3+2\theta_f)/(11+8\theta_f)}\right).
D.
One has
M_{f,E}(q;a,b) = c_f \cdot \prod_{p\mid qab} (1 - \lambda(p)/p + 1/p^2)(1 - 1/p^2)^{(-2)} \cdot \frac{c_{a,b}+c_{b,a}}{(ab)^{1/2}} \cdot \frac{L(1,f)^2}{\zeta(2)} + O\!\left(q^{-1/20+\varepsilon}a^{3/10} + q^{-(1-2\theta_f)/(22+16\theta_f)+\varepsilon} a^{(3+2\theta_f)/(11+8\theta_f)}\right).
E.
One has
M_{f,E}(q;a,b) = c_f \cdot \prod_{p\mid qab} (1 - \lambda(p)/p + 1/p^2)(1 - 1/p^2)^{(-2)} \cdot \frac{c_{a,b}}{(ab)^{1/2}} \cdot \frac{L(1,f)^2}{\zeta(2)} + O\!\left(q^{-1/20+\varepsilon}a^{3/10} + q^{-(1-2\theta_f)/(22+16\theta_f)+\varepsilon} b^{(3+2\theta_f)/(11+8\theta_f)}\right).

separate balanced/unbalanced dependence on a and b in the error term

dropped the power-saving q-decay in the second error term

assignment of the second-regime parameter dependence to b rather than a

two-sided diagonal contribution c_{a,b}+c_{b,a}

The stem gives hypotheses and definitions but does not reveal the conclusion. The correct asymptotic formula is not stated or strongly hinted at in the prompt itself.

This is essentially a theorem-identification question: the setup is followed by 'Which statement holds?'. Although the options introduce nearby alternatives, the task is still largely selecting the exact theorem conclusion rather than resolving a broader mathematical issue.

Some reasoning or precise recall is needed because the choices differ in subtle but meaningful ways (symmetry in c_{a,b}+c_{b,a}, dependence on a versus b, and strength of q-decay). However, it does not strongly induce generative reasoning from the hypotheses alone.

The distractors are mathematically plausible and target realistic failure modes: overstrong combined error terms, weakened q-saving, swapped variable dependence, and omission of the symmetric c_{b,a} term.

A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it leans more toward precise recall/discrimination than genuine generative reasoning.


In this paper we are interested in the following well-known conjecture of Hindman.

\begin{conj}[\cite{hindman.conjecture}]\label{hindman conjecture}
    Any finite coloring of $\mathbb{N}$ contains monochromatic sets of the form $\{x,y,xy,x+y\}.$
\end{conj}

Despite its simplicity, Hindman's conjecture has remained recalcitrant for decades, and even special cases and relaxations of it have been the subject of much recent interest \cite{shkredov,cilleruelo2012combinatorial,green2016monochromatic,moreira2017monochromatic,bowen2025monochromatic,bowen.sabok,alweiss2022monochromatic,alweiss2023monochromatic,kousek2024revisiting,richter2025sums,green2025bounds}.

Most relevant to the present paper, Moreira \cite{moreira2017monochromatic} has shown that any finite coloring of $\mathbb{N}$ contains monochromatic sets $\{x,xy,x+y\}$, i.e., Hindman's conjecture is true if we do not require that the step size $y$ is also the desired color, and the first named author has recently given an alternative proof \cite{alweiss2022monochromatic}.  The second author \cite{bowen2025monochromatic} has shown that Hindman's conjecture is true for colorings of $\mathbb{N}$ into two colors, the second and the third authors  have shown \cite{bowen.sabok} that Hindman's conjecture holds for arbitrary finite colorings of $\mathbb{Q}$ and the first author has shown \cite{alweiss2023monochromatic} that the general version of the Hindman conjecture with more than two variables also holds for arbitrary finite colorings of $\mathbb{Q}$.

In the present paper we return to the two-color case analyzed in \cite{bowen2025monochromatic}. In particular, we give a simpler proof of the main result from \cite{bowen2025monochromatic} that generalizes to deal with more complicated configurations.  Our main result is the following, which was previously only known in the case $k=1.$

\begin{conj}[\cite{hindman.conjecture}]\label{hindman conjecture}
 Any finite coloring of $\mathbb{N}$ contains monochromatic sets of the form $\{x,y,xy,x+y\}.$
\end{conj}

Despite its simplicity, Hindman's conjecture has remained recalcitrant for decades, and even special cases and relaxations of it have been the subject of much recent interest \cite{shkredov,cilleruelo2012combinatorial,green2016monochromatic,moreira2017monochromatic,bowen2025monochromatic,bowen.sabok,alweiss2022monochromatic,alweiss2023monochromatic,kousek2024revisiting,richter2025sums,green2025bounds}.

Most relevant to the present paper, Moreira \cite{moreira2017monochromatic} has shown that any finite coloring of $\mathbb{N}$ contains monochromatic sets $\{x,xy,x+y\}$, i.e., Hindman's conjecture is true if we do not require that the step size $y$ is also the desired color, and the first named author has recently given an alternative proof \cite{alweiss2022monochromatic}. The second author \cite{bowen2025monochromatic} has shown that Hindman's conjecture is true for colorings of $\mathbb{N}$ into two colors, the second and the third authors have shown \cite{bowen.sabok} that Hindman's conjecture holds for arbitrary finite colorings of $\mathbb{Q}$ and the first author has shown \cite{alweiss2023monochromatic} that the general version of the Hindman conjecture with more than two variables also holds for arbitrary finite colorings of $\mathbb{Q}$.

In the present paper we return to the two-color case analyzed in \cite{bowen2025monochromatic}. In particular, we give a simpler proof of the main result from \cite{bowen2025monochromatic} that generalizes to deal with more complicated configurations. Our main result is the following, which was previously only known in the case $k=1.$

In the present paper we return to the two-color case analyzed in \cite{bowen2025monochromatic}. In particular, we give a simpler proof of the main result from \cite{bowen2025monochromatic} that generalizes to deal with more complicated configurations. Our main result is the following, which was previously only known in the case $k=1.$

In addition to proving Theorem \ref{t1}, a primary goal of this paper is to highlight that its basic proof strategy is fairly robust and allows for many alterations and adaptations. Indeed, the second author has recently used a similar strategy to give a new proof and generalizations of the non-commuting Schur theorem for finite colorings of amenable groups \cite{bowen.non.commuting}. The basic strategy used in the proof of Theorem \ref{t1} and the related results \cite{bowen2025monochromatic,bowen.sabok,bowen.non.commuting} is essentially as follows:

\begin{theorem}\label{t2}
 For any $k\in \mathbb{N}$ and any $2$-coloring of $\mathbb{N},$ there exist $x,y\in\mathbb{N}$ with $\{x,y,x^y, xy^i: i\leq k\}$ monochromatic.
\end{theorem}

\begin{theorem}\label{rational.moreira}
 Let $P\subseteq\pol$ be finite. For any finite coloring of $\mathbb{N}$ there exist $x,y\in\mathbb{N}$ and a color class $C$ such that $\{x,xy,x+p(y):p\in P\}\subseteq C$.
\end{theorem}

\begin{prop}\label{central*}
 For any finite set of polynomials $P\subseteq\pol$ and any finite coloring of $\mathbb{N}$ the set $$\{y: \exists x \textnormal{ with } \{x,xy,x+p(y):p\in P\} \textnormal{ monochromatic} \} $$

\begin{proof}[Second proof of Corollary \ref{all_syndetic}]
 Fix a finite set $P\subseteq \pol.$ By definition, for each $i$ there is a finite set $F_i\subset \mathbb{N}$ such that $S_i/F_i\supseteq \mathbb{N}.$ Let $k=\max_{i\leq r}|F_i|$ and consider the coloring $c:\mathbb{N}\rightarrow [k]^r$ where an $n\in\mathbb{N}$ is colored based on the $r$-tuple listing the minimal $f_i\in F_i$ such that $f_in\in S_i.$ Let $P'=\{p/f: p\in P, f\in \bigcup_{i\leq r}F_i\}$. By Theorem \ref{rational.moreira}, there are integers $x',y$ such that $S=\{x', x'y, x'+p'(y): p'\in P'\}$ is monochromatic according to the coloring $c.$ Now if $y\in S_i$, by the definition of $c$ there is an $f_i$ such that $f_iS\subseteq S_i.$ Consequently, setting $x=x'f_i$ is as desired.
\end{proof}

\begin{lemma}\label{all thick}
 Suppose $k\in\mathbb{N}$ and $\mathbb{N}=R\cup B$ with both $R$ and $B$ multiplicatively thick. Then one of the colors contains a set $\{x,y,xy,x+iy:i\leq k\}$.
\end{lemma}
\begin{proof}
Suppose for the sake of contradiction that the lemma is false.
 Without loss of generality assume that $R$ is additively piecewise syndetic. Choose $N\in\mathbb{N}$ large enough depending on $k$.

\begin{lemma}\label{log reduction}
 If any finite coloring of $\mathbb{N}$ contains a monochromatic set $\{x,y, x2^y, x+iy: i\leq k\}$ then any finite coloring contains a monochromatic set of the form $\{x,y,x^y,xy^i: i\leq k\}.$
\end{lemma}

\begin{theorem}\label{t1}
    For any $k\in \mathbb{N}$ and $2$-coloring of $\mathbb{N},$ there exist $x,y\in\mathbb{N}$ such that $\{x,y,xy, x+iy: i\leq k\}$ is monochromatic.
\end{theorem}

In addition to proving Theorem \ref{t1}, a primary goal of this paper is to highlight that its basic proof strategy is fairly robust and allows for many alterations and adaptations.  Indeed, the second author has recently used a similar strategy to give a new proof and generalizations of the non-commuting Schur theorem for finite colorings of amenable groups \cite{bowen.non.commuting}.  The basic strategy used in the proof of Theorem \ref{t1} and the related results \cite{bowen2025monochromatic,bowen.sabok,bowen.non.commuting} is essentially as follows:

\begin{enumerate}
    \item First, show that the result holds when the color of the step size $y$ is ignored.  In this case, we are interested in finding monochromatic sets $\{x,xy,x+iy: i\leq k\},$ which can be done either through Moreira's work \cite{moreira2017monochromatic} or the first author's \cite{alweiss2022monochromatic}.

    \item Define an appropriate structure vs randomness dichotomy, and argue that a $2$-coloring must either be structured or random.

    \item Argue that in either case the extra information we obtain can be used to upgrade the proof from step (1) to control the color of the $y$ term as well. \end{enumerate}

In order to illustrate this technique we give two proofs of Theorem \ref{t1} based on the two ways of handling step (1) and using two different structure vs randomness dichotomies.  These are presented in the two subsections of Section \ref{section +}.

We also show that the same basic approach can be adapted to prove the following exponential version of the result.

\begin{theorem}\label{t2}
    For any $k\in \mathbb{N}$ and any $2$-coloring of $\mathbb{N},$ there exist $x,y\in\mathbb{N}$ with $\{x,y,x^y, xy^i: i\leq k\}$ monochromatic.
\end{theorem}

The $k=1$ case of this result was proven by Sahasrabudhe \cite{sahasrabudhe2018monochromatic}, even for arbitrary finite colorings.  However, the result seems to be new for $k>1.$  Along the way, we also prove that any finite coloring of $\mathbb{N}$ contains monochromatic sets of the form $\{x,x^y,xy^i: i\leq k\},$ which gives a new exponential version of Moreira's theorem.

The authors state that “the basic strategy used in the proof of Theorem \ref{t1} … is essentially as follows:”
\begin{enumerate}
\item “First, show that the result holds when the color of the step size $y$ is ignored.” Then one finds monochromatic sets $\{x,xy,x+iy: i\leq k\}$, “which can be done either through Moreira's work … or the first author's …”.
\item “Define an appropriate structure vs randomness dichotomy, and argue that a $2$-coloring must either be structured or random.”
\item “Argue that in either case the extra information we obtain can be used to upgrade the proof from step (1) to control the color of the $y$ term as well.”
\end{enumerate}
They further note that they “give two proofs of Theorem \ref{t1} based on the two ways of handling step (1) and using two different structure vs randomness dichotomies.”

The authors state that “the basic strategy used in the proof of the main theorem … is essentially as follows:”
\begin{enumerate}
\item “First, show that the result holds when the color of the step size $y$ is ignored.” Then one finds monochromatic sets $\{x,xy,x+iy: i\leq k\}$, “which can be done either through Moreira's work … or the first author's …”.
\item “Define an appropriate structure vs randomness dichotomy, and argue that a $2$-coloring must either be structured or random.”
\item “Argue that in either case the extra information we obtain can be used to upgrade the proof from step (1) to control the color of the $y$ term as well.”
\end{enumerate}
They further note that they “give two proofs of the main theorem based on the two ways of handling step (1) and using two different structure vs randomness dichotomies.”

\label{t1}
    For any $k\in \mathbb{N}$ and $2$-coloring of $\mathbb{N},$ there exist $x,y\in\mathbb{N}$ such that $\{x,y,xy, x+iy: i\leq k\}$ is monochromatic.,

Question: Let $k\in\mathbb{N}$, and let $\mathbb{N}$ be colored with two colors (equivalently, let $c:\mathbb{N}\to\{1,2\}$ be any 2-coloring). Which statement holds?

There exist $x,y\in\mathbb{N}$ such that all elements of the set $\{x,\, y,\, xy,\, x+iy: i\le k\}$ have the same color; equivalently, $\{x,y,xy,x+y,x+2y,\dots,x+ky\}$ is monochromatic.

Choices:
B. There exist $x,y\in\mathbb{N}$ such that all elements of the set $\{x,\, xy,\, x+iy: i\le k\}$ have the same color; equivalently, $\{x,xy,x+y,x+2y,\dots,x+ky\}$ is monochromatic.
C. There exist $x,y\in\mathbb{N}$ such that all elements of the set $\{x,\, y,\, x+y,\, x+2y,\dots,x+ky\}$ have the same color.
D. For every color class $C$ of the 2-coloring, there exist $x,y\in\mathbb{N}$ such that $\{x,\, y,\, xy,\, x+iy: i\le k\}\subseteq C$.
E. There exist $x,y\in\mathbb{N}$ such that all elements of the set $\{x,\, y,\, x^y,\, xy^i: i\le k\}$ have the same color; equivalently, $\{x,y,x^y,xy,xy^2,\dots,xy^k\}$ is monochromatic.

The stem only gives the coloring setup and asks which conclusion is valid; it does not state or strongly hint at the specific monochromatic configuration in the correct option.

This is not a direct restatement in the stem. The item asks the learner to distinguish the actual theorem-strength conclusion from weaker, stronger, and altered alternatives.

Some reasoning is required to compare quantifiers and structural differences among the options, especially against the weaker true statement and the overly strong version. However, the item still mainly tests theorem recognition rather than substantial generative derivation.

The distractors are mathematically meaningful and varied: one is a weaker true statement, one improperly strengthens the quantifier, one omits a needed term, and one introduces an unrelated exponential pattern. These reflect realistic failure modes.

A strong MCQ overall: it avoids answer leakage, uses plausible and diagnostically useful distractors, and is not tautological, though it leans more toward theorem identification than deep generative reasoning.