\label{main result}
Let $k$ be a positive integer such that $k\in \{7, 8, 10,$ $11,12,13\}.$ If $\{a, b, c, d\}$ is a $D(4)$-quadruple with $b=ka$, then it is regular. In other words, we have $d=d_{\pm}.$

\label{intr}

\begin{definition}Let $n\neq0$ be an integer. We call a set of $m$ distinct positive integers a $D(n)$-$m$-tuple, or $m$-tuple with the property $D(n)$, if the product of any two of its distinct elements increased by $n$ is a perfect square.
\end{definition}
We research the $n=4$ case, which has many similarities to the classical $n=1$ case. First author and Filipin proved in \cite{blizfil} the nonexistence of $D(4)$-quintuples.

For a $D(4)$-triple $\{a,b,c\}, a<b<c$, we define
 \[ d_{\pm}(a,b,c)=a+b+c+\frac{1}{2}(abc \pm \sqrt{(ab+4)(ac+4)(bc+4)}).\]

$D(4)$-quadruple  $\{a,b,c,d_{+}\}$ is called a \textbf{regular quadruple}
 if $d_{-} \neq 0$, then $\{a,b,c,d_{-}\}$ is a regular $D(4)$-quadruple with $d_{-}<c$. It is easy to verify that $c=d_+(a,b,d_-)$.

In both the classical case, $n=1$, and $n=4$, conjectures about the uniqueness of an extension of a triple to a quadruple with a larger element are still open.

\begin{conjecture}\label{conj}
Any $D(4)$-quadruple is regular.
\end{conjecture}

In this paper, we study the extensibility of $D(4)$-pairs $\{a,b\}$, where $b = ka$ and $k \in \{7,8,10,11,12,13\}$ are defined by following the methods in \cite{glavni}. So, the main theorem we prove is:

\label{intr}

\begin{definition}Let $n\neq0$ be an integer. We call a set of $m$ distinct positive integers a $D(n)$-$m$-tuple, or $m$-tuple with the property $D(n)$, if the product of any two of its distinct elements increased by $n$ is a perfect square.
\end{definition}
We research the $n=4$ case, which has many similarities to the classical $n=1$ case. First author and Filipin proved in \cite{blizfil} the nonexistence of $D(4)$-quintuples.

$D(4)$-quadruple $\{a,b,c,d_{+}\}$ is called a \textbf{regular quadruple}
 if $d_{-} \neq 0$, then $\{a,b,c,d_{-}\}$ is a regular $D(4)$-quadruple with $d_{-}<c$. It is easy to verify that $c=d_+(a,b,d_-)$.

In both the classical case, $n=1$, and $n=4$, conjectures about the uniqueness of an extension of a triple to a quadruple with a larger element are still open.

\begin{conjecture}\label{conj}
Any $D(4)$-quadruple is regular.
\end{conjecture}

In this paper, we study the extensibility of $D(4)$-pairs $\{a,b\}$, where $b = ka$ and $k \in \{7,8,10,11,12,13\}$ are defined by following the methods in \cite{glavni}. So, the main theorem we prove is:

\begin{conjecture}\label{conj}
Any $D(4)$-quadruple is regular.
\end{conjecture}

Firstly, in Section $2$ we show that $D(4)$-pairs $\{a,ka\}$, $k\in \{7, 8, 10,$ $11,12,13\}$ can be extended to a $D(4)$-triple only with an element $c$, which is a member of a family of positive integers depending on $a$. Then in Sections $3-5$, we prove that these triples can only be extended to regular $D(4)$-quadruples $\{a, ka, c, d\}$.\ These examples support the Conjecture \ref{conj}.
\section{Extensions of pairs to triples}\label{section_2}

\begin{cor}\label{nova lema}
Let $\{a,ka,c\}$ be a $D(4)$-triple, $k \in \{10,11,12,13\}$. Then $c=c_{\nu}^{\pm}$.
\end{cor}
\begin{proof}
Let's show that $\{1,5,a,ka \}$ is not a $D(4)$-quadruple for $k \in \{10,11,12,13\}$. If $\{1,5,a,ka \}$ is a $D(4)$-quadruple, by using \cite[Lemma 6.1]{mbt}, Lemma \ref{lower_on_b}, and Lemma \ref{pomocna lema za lemu ispod}, we obtain $a=a_{\nu}^{\pm}$, $ka=d_+(1,5,a)$, and $d_+(1,5,a)=a_{\nu+1}^{\pm}$ for the same choice of $\pm$. We divide both sides of \begin{equation}\label{pomocna}
 ka = d_+(1,5,a)=1+5+a+\frac{1}{2}(5a+3\sqrt{5a+4}\sqrt{a+4})
\end{equation} by $a$ and, using the fact that $a \geq 12$, we obtain $k \leq 8$.
\end{proof}

Secondly, we observe the solutions of the system of equations (\ref{eqn:sol_pell_bc}) and (\ref{eqn:sol_pell_ab}) and determine the intersections $y=A_n=B_l$ of sequences $(A_n)_n$ and $(B_l)_l$ defined by
\begin{align}
 &A_0=y_1,\ A_1=\frac{1}{2}(ty_1+bz_1),\ A_{n+2}=tA_{n+1}-A_n,\label{seq:An}\\
 &B_0=y_2,\ B_1=\frac{1}{2}(ry_2+bx_2),\ B_{l+2}=rB_{l+1}-B_l, \ n,l\geq 0.\label{seq:Bl}
\end{align}
The initial terms of these sequences are described in the next lemma, whose proof follows \cite[Lemma 2.5]{glavni}.
\begin{lemma}\label{lem:lem_4}
Assume that $\{a, b, c', c\}$ is not a $D(4)$-quadruple for any $c'$ with $0<c'<c^{\pm}_{\nu-1}$ and $b\geq 556881$. Then, $A_{2n}=B_{2l+1}$ has no solution. Moreover, if $A_{2n} = B_{2l}$, then $y_2 = 2$. In other cases, we have $y_2 =\pm 2.$
\end{lemma}

For any nonzero algebraic number $\alpha$ of degree $d$ over $\QQ$ whose minimal polynomial over $\ZZ$ is $a_0 \prod_{j=1}^{d}(X-\alpha^{(j)}),$ we denote by
$$
 h(\alpha)= \frac{1}{d}\left(\log |a_0|+\sum_{j=1}^{d}\log\max\left(1,\left|\alpha^{(j)} \right| \right) \right)
$$
its absolute logarithmic height. We recall the following result due to Matveev \cite{Matveev:2000}.
\begin{lemma}\label{Matveev}
Denote by $\alpha_1,\ldots,\alpha_j$ algebraic numbers, not $0$ or $1,$ by $\log\alpha_1,\ldots,$ $\log\alpha_j$ determinations of their logarithms, by $D$ the degree over $\QQ$ of the number field $\KK=\QQ(\alpha_1,\ldots,\alpha_j),$ and by $b_1,\ldots,b_j$ integers. Define $B=\max\{|b_1|,\ldots,|b_j|\}$ and
$$A_i=\max\{Dh(\alpha_i),|\log\alpha_i|,0.16\}\;\; (1\leq i\leq j),$$
where $h(\alpha)$ denotes the absolute logarithmic Weil height of $\alpha.$
Assume that the number
$$
\Lambda= b_1\log\alpha_1+\cdots+b_n\log\alpha_j
$$
does not vanish. Then
$$
|\Lambda|\geq \exp\{-C(j,\chi)D^2A_1\cdots A_j\log(eD)\log(eB)\},
$$
where $\chi=1$ if $\KK \subset\RR$ and $\chi=2$ otherwise and
$$
C(j,\chi)=\min\left\{\frac{1}{\chi}\left( \frac{1}{2}ej\right)^{\chi}30^{j+3}j^{3.5},2^{6j+20}\right\}.
$$
\end{lemma}
By applying this result we obtain upper bound for $l$ and $m$ in terms of $a$.
\begin{proposition}\label{proposition-1}
Assume that $c\in \{c_1^+, c_2^\pm\}.$ If $P_l=Q_m,$ with $m>1$ then
\begin{align*}
\frac{l}{\log(el)}&< 3.34\cdot 10^{13}\cdot\log^2(8.09c^2),\quad \text{with solutions of Type $1$},\; \\
\frac{l}{\log(el)}&< 6.63\cdot 10^{13}\cdot\log^2(8.09c^2), \quad \text{with solutions of Type $2$}.
\end{align*}
If $Q_m=P_l,\ l\geq1$ with $c=c_1^-,$ then we get
\begin{align*}
\frac{m}{\log(em)}< 13.36\cdot 10^{13}\cdot\log^2(21.3a),\quad \text{for}\; k=7,8,10,11,12,13.
\end{align*}
\end{proposition}
\begin{proof}
 We will demonstrate the proof for the case $c=c_1^-$ since the proof for the other cases closely follows \cite[Proposition 2.9]{glavni}. We apply Lemma \ref{proposition-1}
 with $j=3$ and $\chi=1$ to the linear form \ref{Lambda'} and take \[D=4,\ b_1=m,\ b_2=-l,\ b_3=1,\ \alpha_1=\beta,\ \alpha_2=\alpha,\ \alpha_3=\gamma'.\]
 Since $l\leq m$, we can take $B=m$. Also, we have
 \[h(\alpha_1)=\frac{1}{2}\log\beta,\ h(\alpha_2)=\frac{1}{2}\log\alpha.\]
 Since $\gamma'=\gamma^{-1}$, then $h(\gamma)=h(\gamma')$ and from \cite[Proposition 2.9]{glavni} we have
 \[h(\gamma)<\frac{1}{4}\log\left[\frac{2^4r^4c^4(1+\sqrt{k})^4}{(c-a)^2}\right].\]
 We have \begin{align*}(c_1^--a)^2&=a^2(2\sqrt{k+\frac{4}{a^2}}-k)^2>a^2(2\sqrt{k}-k)^2,\\
 r&<a\sqrt{1.1k},\\ c=c_1^-&=a(k+1-2\sqrt{k+\frac{4}{a^2}})<a(\sqrt{k}-1)^2,\end{align*} and now it follows that
 \[h(\gamma')=h(\gamma)<\frac{1}{4}\log(93309063a^6)<\frac{3}{2}\log(21.3a).\]
 By applying Lemma \ref{proposition-1} with
 \[A_1=2\log\beta,\ A_2=2\log\alpha,\ A_3=6\log(21.3a)\]
 we get
 \begin{align}\label{Matveev_0}
\log|\Lambda'|>-1.3901&\cdot 10^{11}\cdot16\cdot2 \cdot\log\beta\\&\cdot2\log\alpha\cdot6\log(21.3a)\cdot\log(4e)\cdot\log(em).\nonumber
\end{align}
From $1)$ of Lemma \ref{bound-Lambda} and using $l\geq1$, it is easy to conclude that $$m\log\beta<l\log\alpha+2.6\alpha^{-2l}-\log\gamma'<2l\log\alpha.$$
Also,
$$\log|\Lambda'|<-1.9069l\log\alpha$$
and
$$\log\alpha=\log\left[a\left(\sqrt{k+\frac{4}{a^2}}+\sqrt{k}\right)\right]<\log(7.212a)<\log(21.3a).$$
Now, by combining everything, we get
\[\frac{m}{\log(em)}<1.336\cdot10^{14}\cdot\log^2(21.3a),\ \text{for}\ k=7,8,10,11,12,13. \qedhere\]
\end{proof}

In this section, we examine the equation \[x=Q_m=P_l.\] Firstly, we state a useful result from \cite{glavni} and then apply it to obtain lower bounds for the indices $m$ and $l$ in terms of $a$.
\begin{lemma}\cite[Lemma 3.1]{glavni}\label{lem_6} If $a$ is odd, then
\begin{align}
\label{cong_1} Q_{2m}&\equiv x_0+\dfrac{1}{2} a(cx_0m^2+sz_0m)\pmod{a^2},\\
P_{2l}&\equiv x_2+\dfrac{1}{2} a(bx_2l^2+ry_2l)\pmod{a^2} .
\end{align}
If $a$ is even, then
\begin{align}
 Q_{2m}&\equiv x_0+\dfrac{1}{2} a(cx_0m^2+sz_0m)\pmod{\dfrac{1}{2} a^2},\\
P_{2l}&\equiv x_2+\dfrac{1}{2} a(bx_2l^2+ry_2l)\pmod{\dfrac{1}{2} a^2} .
\end{align}
\end{lemma}
\begin{lemma}\label{lower-Type-a} \hfill
\begin{enumerate}[1)]
 \item Assume that $b=ka,\ k\in \{7,8,10,11,12,13\}$ and $c=c_1^-$. If the equation $P_l=Q_m$ has a solution $(l,m)$ (of Type $1$), then we have
$$
m \geq \frac{1}{4}\left(-2+\sqrt{4+\sqrt{a}}
\right).$$
 \item Assume that $b=ka,\ k\in \{7,8,10,11,12,13\}$ and $c \in \{c_1^+,c_2^{\pm}\}$. If the equation $P_l=Q_m$ has a solution $(l,m)$ (of Type $1$), then we have
$$
l \geq \frac{1}{12}\left(-2+\sqrt{4+3\sqrt{a}}
\right).$$
\end{enumerate}
\end{lemma}
\begin{proof}
 Proof follows \cite[Lemma 3.2]{glavni} and here we will demonstrate the proof of $1)$. From the recurrent sequence (\ref{sequence_s_nu}), we have $s \equiv 2,r\ (\bmod\ a)$. Also, $b=ka \equiv 0\ (\bmod\ a)$ and $c=c_{1}^{-}\equiv -2 r\ (\bmod\ a)$. Using the previous Lemma, with solutions in Type $1$, we get
 $$
- 4 r m^2\pm 4m\equiv \pm2rl\pmod{a},\quad \mbox{if}\quad s\equiv 2\pmod{a}
$$
and
$$
\ - 4m^2\pm 2m\mp2l\equiv 0\pmod{\dfrac{a}{\gcd (a,r)}},\quad \mbox{if}\quad s\equiv r\pmod{a}.
$$
In the first case, we multiply the congruence by $r$ and since $r^2 \equiv 4\ (\bmod\ a)$ we have $r \equiv \pm2\ (\bmod\ a')$ for some $a'$ that divides $a$ and $a' \geq \sqrt{a}$. So we get
$$-16m^2\pm8m\mp8l \equiv 0\ (\bmod\ a').$$
Using $l\leq m$ it follows that
$$16m^2+16m \geq |-16m^2\pm8m\mp8l|\geq a'\geq \sqrt{a},$$
which implies
\begin{align}\label{1}
 m \geq \frac{1}{4}\left(-2 +\sqrt{4+\sqrt{a}} \right).\end{align}
In the second case, since $\gcd(a,r)\leq2$ and $l \leq m$, we have
$$4m^2+4m \geq |- 4m^2\pm 2m\mp2l|\geq\frac{a}{\gcd(a,r)}\geq\frac{a}{2},$$
which implies
\begin{align}\label{2}
 m \geq \frac{1}{4}\left(-2+\sqrt{4+2a}\right).
\end{align}
Combining (\ref{1}) and (\ref{2}) we obtain the desired inequality.
\end{proof}

The cases $k \in \{2,3,5,6\}$ have already been observed in \cite{glavni} and \cite{Filipin}, and it is easy to verify that pairs $\{a,4a\}$ and $\{a,9a\}$ cannot satisfy the $D(4)$ property.

Firstly, in Section $2$ we show that $D(4)$-pairs $\{a,ka\}$, $k\in \{7, 8, 10,$ $11,12,13\}$ can be extended to a $D(4)$-triple only with an element $c$, which is a member of a family of positive integers depending on $a$. Then in Sections $3-5$, we prove that these triples can only be extended to regular $D(4)$-quadruples $\{a, ka, c, d\}$.\ These examples support the Conjecture \ref{conj}.

Firstly, in Section $2$ we show that $D(4)$-pairs $\{a,ka\}$, $k\in \{7, 8, 10, 11,12,13\}$ can be extended to a $D(4)$-triple only with an element $c$, which is a member of a family of positive integers depending on $a$. Then in Sections $3$--$5$, we prove that these triples can only be extended to regular $D(4)$-quadruples $\{a, ka, c, d\}$ (i.e., $d=d_{\pm}$).

Firstly, we show that $D(4)$-pairs $\{a,ka\}$, $k\in \{7, 8, 10, 11,12,13\}$ can be extended to a $D(4)$-triple only with an element $c$, which is a member of a family of positive integers depending on $a$. Then we prove that these triples can only be extended to regular $D(4)$-quadruples $\{a, ka, c, d\}$ (i.e., $d=d_{\pm}$).

Question: Let \(k\) be a positive integer with \(k\in\{7,8,10,11,12,13\}\). Suppose \(\{a,b,c,d\}\) is a set of four distinct positive integers with the property \(D(4)\), meaning that for any two distinct elements \(x,y\in\{a,b,c,d\}\), the number \(xy+4\) is a perfect square, and assume moreover that \(b=ka\). Which conclusion about \(\{a,b,c,d\}\) is valid under these hypotheses?

The quadruple \(\{a,b,c,d\}\) is regular; equivalently, the fourth element satisfies \(d=d_{\pm}\).

Choices:
B. The quadruple \(\{a,b,c,d\}\) is regular for every positive integer \(k\ge 7\) with \(b=ka\); equivalently, one always has \(d=d_{\pm}\).
C. There exists a regular \(D(4)\)-quadruple having the form \(\{a,ka,c,d\}\); in particular, for such a quadruple the fourth element satisfies \(d=d_{\pm}\).
D. The quadruple \(\{a,b,c,d\}\) can be extended from the pair \(\{a,ka\}\) only when the third element \(c\) belongs to one of the admissible families depending on \(a\); consequently, \(d\) need not equal \(d_{\pm}\).
E. For each \(k\in\{7,8,10,11,12,13\}\), every \(D(4)\)-quadruple \(\{a,b,c,d\}\) with \(b=ka\) is uniquely determined by the pair \(\{a,b\}\), so in particular there is exactly one possible value of \(d\), namely \(d=d_{+}\).

universal conclusion for every such quadruple

final step forcing regularity from admissible c-families

regularity versus uniqueness of extension and exclusion of d_-


Assume that all roots $\lambda_k$  of $p$ have rational real and imaginary parts.  If  $\varphi$  is holomorphic at the roots  and all values $\varphi^{(\nu)}(\lambda_k)$ have also rational real and imaginary parts then the series expansions of $f_k$ have coefficients with rational real and imaginary parts only.

\bigskip

Using {\it multicentric calculus} \cite{Ne12a} it is possible to get stable and convergent power series based representations for piecewise constant holomorphic functions.  When these series expansions are truncated,  we obtain polynomials which formally agree with  polynomials obtained as Hermite interpolants.  In this paper we  present  different ways of creating the multicentric representations.  Then we comment different  ways of creating Hermite interpolants  for the same purpose.  All these different representations can be done with exact arithmetics, but we shall then compare their performance under assumptions where  we allow in evaluations of variables tiny errors.  In the representations obtained from the multicentric form the errors do not accumulate essentially at all, unlike in the other expressions.

While with Hermite interpolation we may form a basis from polynomials,  of which the interpolative  polynomial can be obtained as linear combination, in multicentric representation we use a fixed Lagrangian interpolation basis  but   consider the coefficients as functions. And it is the series expansions of these "coefficients" which   when truncated yield  formally the Hermite interpolant.  This  is the cause for a very different order of steps in  evaluating the polynomials and is visible in  the sensitivity in error propagation.

\bigskip
Different approaches to get the multicentric representations are surveyed in Section 2 but the point of departure is in {\it taking  a polynomial as a new variable}.    We assume that a polynomial $p$ of degree $d$ with simple roots $\lambda_k$ has been selected and  the lemniscate set
$$
V(\rho) =\{ z \in \mathbb C: \ |p(z)| \le \rho \}
$$
has several components.  In {\it multicentric calculus} we represent  scalar  functions
$$
\varphi:  V(\rho) \rightarrow \mathbb C, \ \  z \mapsto \varphi(z)
$$
by vector valued functions
$$
f: B(\rho) \rightarrow \mathbb C^d,   \ \   w \mapsto f(w)
$$
so that  if $z \in V(\rho)$ then the new variable $w= p(z)$ is in the disc $B(\rho)$ and within the lemniscate
\[
\gamma_\rho = \{ z \in \mathbb C: \ |p(z)| = \rho \}.
\]
The original function can then be recovered by
\begin{equation}\label{multicentric}
\varphi(z) = \sum_{k=1}^d \delta_k(z) f_k(p(z))
\end{equation}
where the polynomial $\delta_k$ is, as in the Lagrange interpolation, the polynomial of degree $d-1$ such that $\delta_k(\lambda_k)=1$, while $\delta_k(\lambda_j)=0$ for $j\not= k$.

 Jacobi \cite{Jac} considered (without the Cauchy integral) expansions of the form
 $$
 \varphi(z) \sim \sum_{j=0}^\infty c_j(z) p(z)^j
 $$
 where $c_j$ are polynomials of degree less than $d$,  see e.g.  \cite{Ki1906, Fe32, Wa69, Ma77}.   However, in this tradition  $w=p(z)$ was not  considered a truly new variable.
 Hence, the real potential of  utilising power series in discs for computational purposes was overlooked.  We call this multicentric calculus;  observe that $|p(z)|^{1/d}$ is the geometric average of distances from $z$ to the "local centers" $\lambda_k$.     On this development see \cite{Ne12a, Ne12b, Ap16, Ne16, An19},  but we aim here to be sufficiently  self-contained.

 An important application of multicentric calculus  is in holomorphic functional calculus, where one needs an accurate and and easily computable approximation of  holomorphic functions $\varphi$ near the spectrum $\sigma(A)$ of a given bounded operator $A$.  When the spectrum  lies inside the components bounded by the lemniscate, then the spectrum of $B= p(A)$ lies inside the corresponding disc and if we have power series
 $$ f_k(w)  = \sum_{j=0} ^\infty \alpha_{k,j} w^j
 $$
 available and converging fast, then only a moderate number of terms of the power series is needed to represent   $f_k(B)$, after which  an approximation of
 $$
 \varphi(A) = \sum_{k=1}^d \delta_k(A) f_k(B)
 $$
 can be computed.  Notice that if, say for a sparse large scale matrix $A$,  the  matrix $\varphi(A)$ is only needed at a vector $b$, then all  multiplications needed to create the approximate $\varphi(A)b$ can be obtained using matrix-vector multiplications only.

Here we concentrate on holomorphic functions which are piecewise constant. However, this can be done by further  narrowing the focus on {\it functions which are identically $1$ in one component and  vanish in the others}  as the general case can be obtained by superposing.
In this approach we have approximation errors mainly  from the truncation of power series of the components of $f$  and numerical errors from  evaluation of the  truncated expansions at the  operator or matrix $B=p(A)$.  However,  it is important to notice that we  have the liberty to choose the local centers $\lambda_k$ to have rational real and imaginary parts as then  in  the multicentric representation the coefficients in the power series of  the components of $f$ are all rational, and the computations to create them can be done with exact arithmetic.

Using {\it multicentric calculus} \cite{Ne12a} it is possible to get stable and convergent power series based representations for piecewise constant holomorphic functions. When these series expansions are truncated, we obtain polynomials which formally agree with polynomials obtained as Hermite interpolants. In this paper we present different ways of creating the multicentric representations. Then we comment different ways of creating Hermite interpolants for the same purpose. All these different representations can be done with exact arithmetics, but we shall then compare their performance under assumptions where we allow in evaluations of variables tiny errors. In the representations obtained from the multicentric form the errors do not accumulate essentially at all, unlike in the other expressions.

While with Hermite interpolation we may form a basis from polynomials, of which the interpolative polynomial can be obtained as linear combination, in multicentric representation we use a fixed Lagrangian interpolation basis but consider the coefficients as functions. And it is the series expansions of these "coefficients" which when truncated yield formally the Hermite interpolant. This is the cause for a very different order of steps in evaluating the polynomials and is visible in the sensitivity in error propagation.

\bigskip
Different approaches to get the multicentric representations are surveyed in Section 2 but the point of departure is in {\it taking a polynomial as a new variable}. We assume that a polynomial $p$ of degree $d$ with simple roots $\lambda_k$ has been selected and the lemniscate set
$$
V(\rho) =\{ z \in \mathbb C: \ |p(z)| \le \rho \}
$$
has several components. In {\it multicentric calculus} we represent scalar functions
$$
\varphi: V(\rho) \rightarrow \mathbb C, \ \ z \mapsto \varphi(z)
$$
by vector valued functions
$$
f: B(\rho) \rightarrow \mathbb C^d, \ \ w \mapsto f(w)
$$
so that if $z \in V(\rho)$ then the new variable $w= p(z)$ is in the disc $B(\rho)$ and within the lemniscate
\[
\gamma_\rho = \{ z \in \mathbb C: \ |p(z)| = \rho \}.
\]
The original function can then be recovered by
\begin{equation}\label{multicentric}
\varphi(z) = \sum_{k=1}^d \delta_k(z) f_k(p(z))
\end{equation}
where the polynomial $\delta_k$ is, as in the Lagrange interpolation, the polynomial of degree $d-1$ such that $\delta_k(\lambda_k)=1$, while $\delta_k(\lambda_j)=0$ for $j\not= k$.

Jacobi \cite{Jac} considered (without the Cauchy integral) expansions of the form
 $$
 \varphi(z) \sim \sum_{j=0}^\infty c_j(z) p(z)^j
 $$
 where $c_j$ are polynomials of degree less than $d$, see e.g. \cite{Ki1906, Fe32, Wa69, Ma77}. However, in this tradition $w=p(z)$ was not considered a truly new variable.
 Hence, the real potential of utilising power series in discs for computational purposes was overlooked. We call this multicentric calculus; observe that $|p(z)|^{1/d}$ is the geometric average of distances from $z$ to the "local centers" $\lambda_k$. On this development see \cite{Ne12a, Ne12b, Ap16, Ne16, An19}, but we aim here to be sufficiently self-contained.

An important application of multicentric calculus is in holomorphic functional calculus, where one needs an accurate and and easily computable approximation of holomorphic functions $\varphi$ near the spectrum $\sigma(A)$ of a given bounded operator $A$. When the spectrum lies inside the components bounded by the lemniscate, then the spectrum of $B= p(A)$ lies inside the corresponding disc and if we have power series
 $$ f_k(w) = \sum_{j=0} ^\infty \alpha_{k,j} w^j
 $$
 available and converging fast, then only a moderate number of terms of the power series is needed to represent $f_k(B)$, after which an approximation of
 $$
 \varphi(A) = \sum_{k=1}^d \delta_k(A) f_k(B)
 $$
 can be computed. Notice that if, say for a sparse large scale matrix $A$, the matrix $\varphi(A)$ is only needed at a vector $b$, then all multiplications needed to create the approximate $\varphi(A)b$ can be obtained using matrix-vector multiplications only.

Here we concentrate on holomorphic functions which are piecewise constant. However, this can be done by further narrowing the focus on {\it functions which are identically $1$ in one component and vanish in the others} as the general case can be obtained by superposing.
In this approach we have approximation errors mainly from the truncation of power series of the components of $f$ and numerical errors from evaluation of the truncated expansions at the operator or matrix $B=p(A)$. However, it is important to notice that we have the liberty to choose the local centers $\lambda_k$ to have rational real and imaginary parts as then in the multicentric representation the coefficients in the power series of the components of $f$ are all rational, and the computations to create them can be done with exact arithmetic.

\bigskip
Different approaches to get the multicentric representations are surveyed in Section 2 but the point of departure is in {\it taking a polynomial as a new variable}. We assume that a polynomial $p$ of degree $d$ with simple roots $\lambda_k$ has been selected and the lemniscate set
$$
V(\rho) =\{ z \in \mathbb C: \ |p(z)| \le \rho \}
$$
has several components. In {\it multicentric calculus} we represent scalar functions
$$
\varphi: V(\rho) \rightarrow \mathbb C, \ \ z \mapsto \varphi(z)
$$
by vector valued functions
$$
f: B(\rho) \rightarrow \mathbb C^d, \ \ w \mapsto f(w)
$$
so that if $z \in V(\rho)$ then the new variable $w= p(z)$ is in the disc $B(\rho)$ and within the lemniscate
\[
\gamma_\rho = \{ z \in \mathbb C: \ |p(z)| = \rho \}.
\]
The original function can then be recovered by
\begin{equation}\label{multicentric}
\varphi(z) = \sum_{k=1}^d \delta_k(z) f_k(p(z))
\end{equation}
where the polynomial $\delta_k$ is, as in the Lagrange interpolation, the polynomial of degree $d-1$ such that $\delta_k(\lambda_k)=1$, while $\delta_k(\lambda_j)=0$ for $j\not= k$.

Jacobi \cite{Jac} considered (without the Cauchy integral) expansions of the form
 $$
 \varphi(z) \sim \sum_{j=0}^\infty c_j(z) p(z)^j
 $$
 where $c_j$ are polynomials of degree less than $d$, see e.g. \cite{Ki1906, Fe32, Wa69, Ma77}. However, in this tradition $w=p(z)$ was not considered a truly new variable.
 Hence, the real potential of utilising power series in discs for computational purposes was overlooked. We call this multicentric calculus; observe that $|p(z)|^{1/d}$ is the geometric average of distances from $z$ to the "local centers" $\lambda_k$. On this development see \cite{Ne12a, Ne12b, Ap16, Ne16, An19}, but we aim here to be sufficiently self-contained.

An important application of multicentric calculus is in holomorphic functional calculus, where one needs an accurate and and easily computable approximation of holomorphic functions $\varphi$ near the spectrum $\sigma(A)$ of a given bounded operator $A$. When the spectrum lies inside the components bounded by the lemniscate, then the spectrum of $B= p(A)$ lies inside the corresponding disc and if we have power series
 $$ f_k(w) = \sum_{j=0} ^\infty \alpha_{k,j} w^j
 $$
 available and converging fast, then only a moderate number of terms of the power series is needed to represent $f_k(B)$, after which an approximation of
 $$
 \varphi(A) = \sum_{k=1}^d \delta_k(A) f_k(B)
 $$
 can be computed. Notice that if, say for a sparse large scale matrix $A$, the matrix $\varphi(A)$ is only needed at a vector $b$, then all multiplications needed to create the approximate $\varphi(A)b$ can be obtained using matrix-vector multiplications only.

Here we concentrate on holomorphic functions which are piecewise constant. However, this can be done by further narrowing the focus on {\it functions which are identically $1$ in one component and vanish in the others} as the general case can be obtained by superposing.
In this approach we have approximation errors mainly from the truncation of power series of the components of $f$ and numerical errors from evaluation of the truncated expansions at the operator or matrix $B=p(A)$. However, it is important to notice that we have the liberty to choose the local centers $\lambda_k$ to have rational real and imaginary parts as then in the multicentric representation the coefficients in the power series of the components of $f$ are all rational, and the computations to create them can be done with exact arithmetic.

We use the following notation for truncating power series. If
\begin{equation} \label{katkaisu}
a(x) \sim \sum_{j=0} ^\infty c_j x^j , \ \ \text { then } \lfloor a \rfloor_n (x) = \sum_{j=0}^n c_j x^j.
\end{equation}
Thus, when we truncate the series for $f_k(w)$ we can write at $w=p(z)$
\begin{equation}\label{katkaisunmerkitys}
\lfloor f_k \rfloor_n (p(z)) = \sum_{j=0}^n \alpha_{k,j}\ p(z)^j.
\end{equation}

We decompose the Cauchy kernel $\frac{1}{\lambda - z}$ writing
\begin{equation}\label{decompCauchy}
\frac{1}{\lambda - z} = \frac{1}{p(\lambda)- p(z)} \ \sum_{k=1}^d p'(\lambda_k) \delta_k(\lambda) \delta_k(z).
\end{equation}
Denoting
\begin{equation}\label{decompkernel}
K_k(\lambda, w) = \frac{1}{\lambda - \lambda_k} \frac{p(\lambda)}{p(\lambda)-w}
\end{equation} we obtain for $f_k$ the expression
\begin{equation}\label{fexpressed}
f_k(w) = \frac{1}{2\pi i } \int_ {\gamma_\rho} K_k(\lambda, w) \varphi(\lambda)d\lambda,
\end{equation}
where $\varphi$ is holomorphic inside $\gamma_\rho$ and continuous up to boundary. Here we are interested in piecewise continuous $\varphi$ which means that if the lemniscate would pass through a critical point, then $\varphi$ would have to take the same value in each component touching this critical point. Without loss on generality we assume that $\rho$ is such that the lemniscate contains no critical points and in particular the lemniscate consists of smooth curves, each surrounding at least one $\lambda_k$. Then, if $|w| < |p(\lambda)| = \rho$, the series
$$
f_k(w) = \sum_{n=0} ^\infty \alpha_{k, n} w^n
$$
converges where
\begin{equation}\label{expressderivatives}
\alpha_{k, n} \ = \ \frac{1}{2\pi i} \int_{\gamma_\rho} \frac{1}{ \lambda - \lambda_k } \ \frac{\varphi(\lambda)} {p(\lambda)^n} \ d \lambda.
\end{equation}

If we differentiate the holomorphic $\varphi$ in (\ref{multicentric})
and assume the data $\varphi^{(\nu)} (\lambda_k)$ to be given, then an explicit recursion for $\alpha_{j, \nu} = f_j^{(\nu)}(0)/ \nu !$ is obtained from
\begin{equation}\label{explicitrecursion}
p'(\lambda_j)^{\nu} f_j^{(\nu)}(0) = \varphi^{(\nu)}(\lambda_j)- t_{j,\nu},
\end{equation}
where
$$
t_{j,\nu}= \sum_{k=1}^d \sum_{\mu=0}^{\nu-1 } \binom{\nu}{\mu} \delta_k^{\nu-\mu}(\lambda_j) \sum_{l=0}^{\mu} b_{\mu,l}(\lambda_j) f_k^{(l)}(0) + \sum_{l=0}^{\nu-1} b_{\nu,l}(\lambda_j) f_j^{(l)}(0)
$$
and the polynomials $b_{\nu,l}$ are obtained from a triangular table, see \cite{Ne12a, An22}.

We can use the integral representation from Section~\ref{cauchysection} to derive bounds for the accumulated errors.
Let $\varphi$ be holomorphic for $|p(z)| < r$ and denote for $\rho<r$
$$
M(\rho) = \max_{|p(z)| \le \rho} |\varphi (z)|.
$$
We assume here that the exact values of coefficients $\alpha_{k,j}$ can be used, as this is the case e.g. when the roots $\lambda_k$ have rational real and imaginary parts and $\varphi$ takes only integer values.

\begin{proof}
This follows immediately from the explicit recursion, see \cite{Ne12a, An19} or  (\ref{explicitrecursion}) below.
\end{proof}

We  use the following notation for truncating power series.  If
\begin{equation} \label{katkaisu}
a(x) \sim \sum_{j=0} ^\infty c_j x^j , \ \  \text { then }   \lfloor a \rfloor_n (x) = \sum_{j=0}^n c_j x^j.
\end{equation}
Thus, when we truncate  the series for $f_k(w)$ we can write at $w=p(z)$
\begin{equation}\label{katkaisunmerkitys}
\lfloor f_k \rfloor_n (p(z)) = \sum_{j=0}^n \alpha_{k,j}\  p(z)^j.
\end{equation}

\bigskip
Consider now the Hermite interpolation problem: at $d$ distinct  complex points $\lambda_k$  to find polynomial $P$ such that
\begin{equation}\label{hermiteyleinen}
P^{(\nu)}(\lambda_k) = a_{k,\nu} \ \text{ for } \nu= 0, \dots, n; \ \  k=1,\dots,d.
\end{equation}
\begin{proposition}
Let $\varphi$ be holomorphic at interpolation points $\lambda_k$. Then  the truncated polynomial of the multicentric representation  of $\varphi$
$$
P(z)= \sum_{k=1}^d \delta_k(z) \lfloor f_k\rfloor_n (p(z))
$$ is the unique polynomial of degree $N \le d(n+1)-1$ satisfying the interpolation problem (\ref{hermiteyleinen}) when $a_{k,\nu}= \varphi^{(\nu)}(\lambda_k)$.
\end{proposition}

\begin{proof}
Clearly, $P$ is of at most degree $d(n+1)-1$. Consider the difference $\varphi-P$. We have
$$
\varphi(z) - P(z) = p(z)^{n+1} \; \sum_{k=1}^d \delta_k(z) \negthickspace \sum_{j=n+1}^\infty \! \alpha_{k,j} \, p(z)^{j-n-1}
$$
which shows that the interpolation conditions are all satisfied.
\end{proof}

\bigskip

Denote by $\delta_{k,j,n}$  the solution to the special case of (\ref{hermiteyleinen} )  with $a_{k,j}=1$, while $a_{m, \nu} =0$ when $(m,\nu) \not=(k,j)$.  Then $(\delta_{k,j,n})$  is a basis  such that the general problem (\ref{hermiteyleinen}) is solved by
$$
P(z) = \sum_{k=1}^d \sum_{j=0}^n a_{k,j} \  \delta_{k,j,n}(z).
$$

We shall keep writing $\varphi$ and $f$ when we discuss general  formulae but use $\chi$ and $g$  in the  particular case where $\chi(z) = 1$ in the component containing the root $\lambda_1$  and vanishing in the components containing  other roots, so that
$$
\chi(z) = \sum_{k=1}^d \delta_k(z) g_k(p(z)).
$$
When we truncate   here the power series of $g_k$'s we  get the basis function $\delta_{1,0,n}$  which  has the multicentric representation
\begin{equation}\label{kantapolynom}
\delta_{1,0,n}(z) = \sum_{k=1}^d \delta_k(z) \lfloor g_k \rfloor_n (p(z)).
\end{equation}
This  representation of  the polynomial $\delta_{1,0,n}$  is later denoted by $M_n$, letter $M$ for multicentric.

In Section 3 we indicate different direct ways to evaluate this basis polynomial.  With $H_n$ we denote the direct representation  as powers of $z$,  $S_n$ shall denote a special representation of  the basis polynomial and $T_n$ is reserved for a particular formulation of the basis polynomial suitable for parallel computing.

In Section 4  we  consider the numerical stability of these different ways and in particular see that the order of computational steps obtained from truncation the multicentric representation is superior, as we let $n$  grow  while keeping the interpolation points, the roots of $p$ fixed.  More specifically,   we shall demonstrate their  different sensitivity using the following  two polynomials, one with real zeros symmetrically around he origin
  \begin{equation}\label{symmetric}
  p(z)= z (1-z^2)
  \end{equation}
  and  another, nonsymmetric one:
  \begin{equation}\label{nonsymmetric}
  p(z) = z( z^2+4z+5).
  \end{equation}
  In both polynomials we choose $\lambda_1=0$ to be the root  lying in the component where the piecewise holomorphic function would be identically 1.  Thus, the first one  models a situation where we would be using Riesz functional calculus to concentrate in the middle of the spectrum while the second could be thought as modelling a situation where one would like to identify the invariant subspace of an operator corresponding to the unstable part  of the spectrum. Notice that the behaviors are independent of translating and scaling of the  basic variable $z$.

This is stated to “follow[] immediately from the explicit recursion,” with references to \cite{Ne12a, An19} or “(\ref{explicitrecursion}) below.”

Question:
Let $p$ be a polynomial with simple roots $\lambda_1,\dots,\lambda_d$, and let $\delta_k$ be the Lagrange interpolation polynomials of degree $d-1$ characterized by $\delta_k(\lambda_k)=1$ and $\delta_k(\lambda_j)=0$ for $j\ne k$. Suppose a function $\varphi$ admits the multicentric representation
\[
\varphi(z)=\sum_{k=1}^d \delta_k(z)\,f_k(p(z)),
\]
where each component function $f_k$ has a power-series expansion at $0$ of the form
\[
f_k(w)=\sum_{n=0}^\infty \alpha_{k,n}w^n.
\]
Assume that every root $\lambda_k$ has rational real and imaginary parts, and that $\varphi$ is holomorphic in a neighborhood of each root $\lambda_k$ with the property that for every $k$ and every derivative order $\nu\ge 0$, the value $\varphi^{(\nu)}(\lambda_k)$ has rational real and imaginary parts. Which conclusion about the series expansions of the functions $f_k$ is valid under these hypotheses?

For every $k$ and every $n\ge 0$, the coefficient $\alpha_{k,n}$ has rational real and imaginary parts; equivalently, the power-series expansion of each $f_k$ has coefficients whose real and imaginary parts are all rational.

Choices:
B. For every $k$ and every $n\ge 0$, the coefficient $\alpha_{k,n}$ is an algebraic complex number; in general one cannot conclude that its real and imaginary parts are rational without additionally assuming that $\varphi$ is globally holomorphic on a lemniscate neighborhood.
C. For every $k$, the power-series expansion of $f_k$ at $0$ has coefficients whose real and imaginary parts are algebraic numbers.
D. For every fixed derivative order $\nu\ge 0$, there exists a choice of multicentric component functions $f_1,\dots,f_d$ such that the coefficients $\alpha_{k,n}$ have rational real and imaginary parts for all $k$ and all $0\le n\le \nu$.
E. For every $k$ and every $n\ge 0$, the coefficient $\alpha_{k,n}$ has rational real and imaginary parts provided only that each root $\lambda_k$ has rational real and imaginary parts and that the values $\varphi(\lambda_k)$ themselves have rational real and imaginary parts.


\begin{theorem}\label{thm_specialcase}
    Let $C$ be a genus $2$ curve with Jacobian that has multiplication by the maximal quaternion order with discriminant $6$,
    and has field of moduli a number field $L$ with at least one real embedding.
    Assume $C$ has potentially smooth stable reduction at primes above $2$ and $3$. Then its Jacobian has superspecial reduction at infinitely many primes.
\end{theorem}

\subsection{Background}

Elkies proved in \cite{MR903384} that an elliptic curve defined over a number field of odd degree has infinitely many primes of supersingular reduction, and extended the result in \cite{MR1030140} to elliptic curves over any number field with at least one real embedding. This has been generalized to various families of abelian varieties such as abelian surfaces with multiplication by the quaternion algebra of discriminant $6$, with field of moduli $\mathbb{Q}$ and which is a Jacobian in characteristic $2$ and $3$ in \cite{MR2414789}. For a detailed discussion of related works, see \cite{chen2025infinitelysupersingularprimesmumfords}.

\subsection{The main result}
We extend the result of \cite{MR2414789} to number fields with at least one real embedding, and weaken the local conditions at the bad reduction primes $2$ and $3$. We expect that similar methods can be applied to the settings in \cites{MR2704678, MR2705896, li2025infinitelyprimesbasicreduction,chen2025infinitelysupersingularprimesmumfords}.

A special case of our main theorem is as follows.

Elkies proved in \cite{MR903384} that an elliptic curve defined over a number field of odd degree has infinitely many primes of supersingular reduction, and extended the result in \cite{MR1030140} to elliptic curves over any number field with at least one real embedding. This has been generalized to various families of abelian varieties such as abelian surfaces with multiplication by the quaternion algebra of discriminant $6$, with field of moduli $\mathbb{Q}$ and which is a Jacobian in characteristic $2$ and $3$ in \cite{MR2414789}. For a detailed discussion of related works, see \cite{chen2025infinitelysupersingularprimesmumfords}.

\subsection{The main result}
We extend the result of \cite{MR2414789} to number fields with at least one real embedding, and weaken the local conditions at the bad reduction primes $2$ and $3$. We expect that similar methods can be applied to the settings in \cites{MR2704678, MR2705896, li2025infinitelyprimesbasicreduction,chen2025infinitelysupersingularprimesmumfords}.

A special case of our main theorem is as follows.

\begin{theorem}\label{thm_main}
Let $C$ be a genus $2$ curve with Jacobian that has multiplication by the maximal quaternion order with discriminant $6$,
 and has field of moduli a number field $L$ with at least one real embedding.
 Write $j_0:=j(C)\in L$ and $\Nm_{L/\bQ}(j_0) = \frac{n}{d}$ with $n,d\in \bZ, (n,d,6) = 1, d>0$.
 Assume at least one of the following conditions:
 \begin{enumerate}
 \item $v_{\mathfrak{p}}(j_0) \leq 0$ for $\mathfrak{p}$ above $2,3$, and $[L:\bQ]+v_3(d\Nm_{L/\bQ}(27j_0+16))$ is odd or $j_0$ has at least one real conjugate in $(-\frac{16}{27},0)$;
 \item $v_{\mathfrak{p}}(j_0) \geq 0$ for $\mathfrak{p}$ above $2$, $v_{\mathfrak{p}}(j_0) \leq 0$ for $\mathfrak{p}$ above $3$, and $j_0$ has at least one real conjugate in $(0, \infty)$;
 \item $[L:\bQ]$ is even, $v_{\mathfrak{p}}(j_0) \geq 0$ for $\mathfrak{p}$ above $2,3$, $j_0$ has at least one real conjugate in $(-\infty, -\frac{16}{27})\cup (0, \infty)$.
 \end{enumerate}
 Then the Jacobian of $C$ has superspecial reduction at infinitely many primes.
\end{theorem}

An abelian surface with action of a maximal order in a rational quaternion algebra $B$ has either ordinary or superspecial reduction modulo primes not dividing the discriminant of $B$ (\cite{Clark2003}*{p.70} and \cite{MR1097624}*{p.23}). It therefore suffices to construct primes of supersingular reduction, for which we follow Elkies' general strategy. Given a genus $2$ curve $C$, we find supersingular reduction of its Jacobian from its intersection with some Heegner cycle $\cP_D$ of discriminant $D$. The intersection at a prime $p$ is captured by $v_p(P_D(j_0))>0$, where $P_D(x)$ is the integral minimal polynomial of the $j$-invariants of the points in $\cP_D$, and when this occurs, supersingular reduction at $p$ is detected by $\left(\frac{D}{p}\right) \neq 1$.

In each case, we are going to find some discriminant $D$ with a rational prime $p\notin \Nm_{L/\bQ}(S)$ such that $v_p(P_{D}(j_0)) > 0$ and $\left(\frac{D}{p}\right) \neq 1$, then as in \cite{MR2414789}, there is a prime $\fp \notin S$ above $p$, such that the Jacobian of $C$ has supersingular, and hence superspecial reduction.
 \begin{enumerate}
 \item Choose a prime $l$ satisfying the conditions:
 \begin{itemize}
 \item $l\equiv 13 \pmod{24}$,
 \item $\left(\frac{-l}{q}\right) = 1$ for every prime $q\in \Nm_{L/\bQ}(S) \backslash \{2,3\}$,
 \item $P_{-4l}(x)$ has a real root $j(a_i)$ in a subinterval of $(-\frac{16}{27},0)$ to be specified later.
 \end{itemize}
 Let $D = -4l$. By \cite{MR2414789}*{Lemma 6} and \Cref{lem:red23}, \begin{align*}
 P_{-4l}(x) &\equiv x^{h'} \pmod{4}, \\
 P_{-4l}(x) &\equiv \pm x^{h'} \pmod{3}, \\
 P_{-4l}(x) &\equiv (27x+16)S(x)^2 \pmod{l} \text{ for some }S(x) \in\bZ[x].
 \end{align*}
 Since $v_{\mathfrak{p}}(j_0) \leq 0$ for all $\mathfrak{p}$ above $2,3$, we can choose $d$ such that
 $d\prod_{\sigma\in T} \sigma(j_0)$ is integral for any $T\subset \Hom(L, \overline{\bQ})$ and any prime dividing $d$ lies in $\Nm_{L/\bQ}(S)$, then
 $N, dQ\in \bN$ and $(N, 6) = 1,\, (dQ,2) = 1$.
 Suppose $l\nmid N$, then \begin{align*}
 \left(\frac{-4l}{N}\right) &= \left(\frac{-1}{N}\right)\left(\frac{l}{N}\right)= \left(\frac{-1}{N}\right)\left(\frac{N}{l}\right) \\
 &= \left(\frac{-1}{N}\right)\left(\frac{dQ}{l}\right) = \left(\frac{-1}{N}\right)\left(\frac{l}{dQ}\right)\\
 &= \left(\frac{-1}{N}\right)\left(\frac{-1}{dQ}\right) \left(\frac{-l}{dQ}\right) \\ &= \left(\frac{-1}{s(d\Nm_{L/\bQ} j_0)^{h'}}\right)\left(\frac{-1}{s'3^{[L:\bQ]}d\Nm_{L/\bQ}(j_0)}\right)\left(\frac{-l}{3^{v_3(dQ)}}\right) \\
 &= ss'(-1)^{[L:\bQ]+v_3(dQ)}.\end{align*}
 If $[L:\bQ]+v_3(dQ)$ is odd, let \[j(a_1)\in\left(-\frac{16}{27}, -\frac{16}{27} + \min_{1\leq i \leq r} \left|j_i + \frac{16}{27}\right|\right)\]
 so that $(27j_i + 16) P_{-4l}(j_i) > 0$ for each $1\leq i\leq r$.
 If $[L:\bQ]+v_3(dQ)$ is even, by assumption let $j_t$ be the minimal real conjugate of $j_0$ in $( -\frac{16}{27} , 0)$, and \[j(a_1)\in\left(j_t, j_{t+1}\right)\] so that $(27j_i + 16) P_{-4l}(j_i) > 0$ for each $1\leq i\leq r, i\neq t$ and $(27j_t + 16) P_{-4l}(j_t) < 0$. In either case we have \[\left(\frac{-4l}{N}\right) = -1\] and there is a prime divisor $p$ of $N$ such that \[\left(\frac{-4l}{p}\right) = -1.\] The conditions on $l$ and $(N,6) = 1$ imply that $p\notin \Nm_{L/\bQ}(S)$ and $v_p(\Nm_{L/\bQ}(P_D(j_0))) = v_p(N) > 0$.
 If $l | N$, then since $l\notin \Nm_{L/\bQ}(S)$ by construction, we can choose $p = l$.
 \item Choose a prime $l$ satifying the conditions:
 \begin{itemize}
 \item $l\equiv 19\pmod{24}$,
 \item $\left(\frac{q}{l}\right)=\left(\frac{-l}{q}\right) = 1$ for every prime $q\in \Nm_{L/\bQ}(S) \backslash \{2,3\}$,
 \item $P_{-l}(x)$ has a real root $j(a_1)$ in a subinterval of $(0, \infty)$ to be specified later.
 \end{itemize} Let $D = -l$. By \Cref{lem:red23}, \Cref{lem:pair}, and \cite{MR2414789}*{Lemma 4},
 \begin{align*}
 P_{-l}(x) &\equiv 1 \pmod{2}, \\
 P_{-l}(x) &\equiv \pm x^{h'} \pmod{3}, \\
 P_{-l}(x) &\equiv 3(27x+16)S(x)^2 \pmod{l} \text{ for some }S(x) \in\bZ[x].
 \end{align*}
 Since $v_{\mathfrak{p}}(j_0) \geq 0$ for $\mathfrak{p}$ above $2$ and $v_{\mathfrak{p}}(j_0) \leq 0$ for $\mathfrak{p}$ above $3$, we can choose $d$ such that
 $d\prod_{\sigma\in T} \sigma(j_0)$ is integral for any $T\subset \Hom(L, \overline{\bQ})$ and any prime dividing $d$ lies in $\Nm_{L/\bQ}(S)\backslash\{2\}$, then
 $N, dQ\in \bN$ and $(N, 6) = 1$ .
 Suppose $l\nmid N$, then
 \begin{align*}
 \left(\frac{-l}{N}\right) &= \left(\frac{N}{l}\right) \\
 &= \left(\frac{s(d\Nm_{L/\bQ}(3(27j_0+16)))}{l}\right)\\
 &= \left(\frac{ss'3^{[L:\bQ]}dQ}{l}\right) \\
 &= ss'(-1)^{[L:\bQ]+v_3(dQ)+v_2(Q)}
 \end{align*}
 Let $j_t$ be the minimal real conjugate of $j_0$ in $(0, \infty)$ and $n$ ($0\leq n < r$) be the number of real conjugates of $j_0$ in $(-\frac{16}{27}, 0)$. If $[L:\bQ]+v_3(dQ)+v_2(Q)$ is odd, let \[\twocase{j(a_1)\in}{(j_t, j_{t+1})}{$n$ is odd}{(0, j_t)}{$n$ is even}\] so that $\Nm_{L/\bQ}(P_{-l}(j_0))\Nm_{L/\bQ}(27j_0+16)>0$. If $[L:\bQ]+v_3(dQ)+v_2(Q)$ is even, let \[\twocase{j(a_1)\in}{(j_t, j_{t+1})}{$n$ is even}{(0, j_t)}{$n$ is odd}\] so that $\Nm_{L/\bQ}(P_{-l}(j_0))\Nm_{L/\bQ}(27j_0+16)<0$. In either case we have \[\left(\frac{-l}{N}\right) = -1\] and there is a prime divisor $p$ of $N$ such that \[\left(\frac{-l}{p}\right) = -1.\] The conditions on $l$ and $(N,6) = 1$ imply that $p\notin \Nm_{L/\bQ}(S)$ and $v_p(\Nm_{L/\bQ}(P_D(j_0))) = v_p(N) > 0$.
 If $l | N$, then since $l\notin \Nm_{L/\bQ}(S)$ by construction, we can choose $p = l$.

\begin{rem} \label{rem:conclude} One can try to further weaken the local conditions by considering Heegner cycles with different forms of discriminant $D$.
For primes $p$ dividing $D$, the unpaired roots of $P_D(x)$ modulo $p$ can be predicted by checking whether a maximal order in a definite quaternion algebra ramified at $2,3,p$ contains two anticommuting CM orders of some given discriminants, as computed in \cite{MR2704678}*{3.4.2}. One can then impose conditions on the number of prime divisors of $D$ and the congruence class modulo $24$ of primes dividing $D$, so that $P_D(x)$ has no unpaired roots or $P_D(x)$ has a single unpaired root from the same elliptic point modulo each $p\nmid D$. In addition, one imposes a congruence condition on $D$ so that $\cP_{D, \cE}$ avoids intersection with one of $\cP_{-3, \cE}$ and $\cP_{-4, \cE}$ by \Cref{lem:avoidintersection}.
The local conditions on $j_0$ are determined so that it avoids intersection with the these $\cP_D$ at $p = 2,3$, and some real conjugate of $j_0$ and some real root of $P_D$ lie in the subinterval corresponding to one geodesic. For $D$ not of the form $-Np$ with $N = 1, 3, 4, 8, 12, 24$, it is possible that $P_D(x)$ has multiple real roots in each subinterval, but we still expect an equidistribution result.
\end{rem}
\bibliographystyle{amsalpha}
\bibliography{references}

More generally, the coarse moduli variety of principally polarized abelian surfaces with potential multiplication by the maximal quaternion order of discriminant $6$ is isomorphic to $\mathbb{P}^1$, and an arithmetic $j$-function (see \eqref{eq:j} in \cref{sec:coord}) is defined in \cite{MR2423455}. In terms of this coordinate, the assumption at primes $\mathfrak{p}$ above $2$ and $3$ in \Cref{thm_specialcase} corresponds to the case $v_{\mathfrak{p}}(j(C)) = 0$ by \cite{MR2414789}*{Proposition 2}, and our main theorem can be stated as follows.

\begin{theorem}\label{thm_main}
Let $C$ be a genus $2$ curve with Jacobian that has multiplication by the maximal quaternion order with discriminant $6$,
    and has field of moduli a number field $L$ with at least one real embedding.
    Write $j_0:=j(C)\in L$ and $\Nm_{L/\mathbb{Q}}(j_0) = \frac{n}{d}$ with $n,d\in \mathbb{Z}, (n,d,6) = 1, d>0$.
    Assume at least one of the following conditions:
    \begin{enumerate}
        \item $v_{\mathfrak{p}}(j_0) \leq 0$ for $\mathfrak{p}$ above $2,3$, and $[L:\mathbb{Q}]+v_3(d\Nm_{L/\mathbb{Q}}(27j_0+16))$ is odd or $j_0$ has at least one real conjugate in $(-\frac{16}{27},0)$;
        \item $v_{\mathfrak{p}}(j_0) \geq 0$ for $\mathfrak{p}$ above $2$, $v_{\mathfrak{p}}(j_0) \leq 0$ for $\mathfrak{p}$ above $3$, and $j_0$ has at least one real conjugate in $(0, \infty)$;
        \item $[L:\mathbb{Q}]$ is even, $v_{\mathfrak{p}}(j_0) \geq 0$ for $\mathfrak{p}$ above $2,3$, $j_0$ has at least one real conjugate in $(-\infty, -\frac{16}{27})\cup (0, \infty)$.
    \end{enumerate}
    Then the Jacobian of $C$ has superspecial reduction at infinitely many primes.
\end{theorem}

An abelian surface with action of a maximal order in a rational quaternion algebra $B$ has either ordinary or superspecial reduction modulo primes not dividing the discriminant of $B$ (\cite{Clark2003}*{p.70} and \cite{MR1097624}*{p.23}). It therefore suffices to construct primes of supersingular reduction, for which we follow Elkies' general strategy. Given a genus $2$ curve $C$, we find supersingular reduction of its Jacobian from its intersection with some Heegner cycle $\cP_D$ of discriminant $D$. The intersection at a prime $p$ is captured by $v_p(P_D(j_0))>0$, where $P_D(x)$ is the integral minimal polynomial of the $j$-invariants of the points in $\cP_D$, and when this occurs, supersingular reduction at $p$ is detected by $\left(\frac{D}{p}\right) \neq 1$.

As in \cite{MR2414789}, we need to consider certain elliptic point in addition to the Heegner cycle in order to pair the roots of $P_D(x)$ modulo primes dividing $D$. We use \cite{MR2704678} to generalize \cite{MR2414789} through a more detailed, case-by-case study, where in each case the discriminant $D$ is chosen in a form adapted to the conditions.
In characteristic $2$ and $3$, \cite{MR2414789} shows that the reduction of any Heegner cycle lies in the superspecial locus $\{j = 0, \infty\}$, and the intersection formula of Heegner divisors in \cite{MR2441697} provides the new input that determines the specific superspecial point for each chosen Heegner cycle. The local conditions on $j_0$ at primes above  $2$ and $3$ ensure that it avoids intersection with the chosen Heegner cycles at these primes.
For more on the choice of local conditions and Heegner cycles, and for potential further weakenings of these conditions, see \Cref{rem:conclude}.

\subsection{Notation and conventions}

Assume the following unless specified otherwise.

Let $B = B_{\Delta}$ be an indefinite quaternion algebra over $\mathbb{Q}$ of discriminant $\Delta$,  $\Lambda = \Lambda_{\Delta}$ be a maximal order of $B$, and $\Lambda^1 = \Lambda_{\Delta}^1$ be the group of units in $\Lambda$ of norm $1$. Fix an element $\mu \in \Lambda_{\Delta}$ such that $\mu^2 = -\Delta$,\footnote{The element $\mu$ exists because the field $\mathbb{Q}(\sqrt{-\Delta})$ embeds in $B$ by the local-global principle, and any two maximal orders in $B$ are conjugate to each other by strong approximation.} then the involution $\alpha \mapsto \alpha' = \mu^{-1} \bar{\alpha} \mu$ is a positive anti-involution on $B$.

An abelian surface with action of a maximal order in a rational quaternion algebra $B$ has either ordinary or superspecial reduction modulo primes not dividing the discriminant of $B$ (\cite{Clark2003}*{p.70} and \cite{MR1097624}*{p.23}), so “it therefore suffices to construct primes of supersingular reduction,” following “Elkies' general strategy.” Given $C$, one finds supersingular reduction of its Jacobian “from its intersection with some Heegner cycle $\cP_D$ of discriminant $D$.” The intersection at a prime $p$ is “captured by $v_p(P_D(j_0))>0$,” where $P_D(x)$ is “the integral minimal polynomial of the $j$-invariants of the points in $\cP_D$,” and when this occurs, supersingular reduction at $p$ is “detected by $\left(\frac{D}{p}\right) \neq 1$.”

As in \cite{MR2414789}, one must “consider certain elliptic point in addition to the Heegner cycle in order to pair the roots of $P_D(x)$ modulo primes dividing $D$.” Using \cite{MR2704678}, the argument proceeds via “a more detailed, case-by-case study,” where “in each case the discriminant $D$ is chosen in a form adapted to the conditions.” In characteristic $2$ and $3$, \cite{MR2414789} shows “the reduction of any Heegner cycle lies in the superspecial locus $\{j = 0, \infty\}$,” and “the intersection formula of Heegner divisors in \cite{MR2441697} provides the new input that determines the specific superspecial point for each chosen Heegner cycle.” Finally, “the local conditions on $j_0$ at primes above $2$ and $3$ ensure that it avoids intersection with the chosen Heegner cycles at these primes,” yielding infinitely many primes of superspecial reduction.

Question: Let $C$ be a genus $2$ curve whose Jacobian $J(C)$ admits an action of the maximal order in the rational quaternion algebra of discriminant $6$, and suppose the field of moduli of $C$ is a number field $L$ having at least one real embedding. Assume moreover that for every prime of $L$ above $2$ or $3$, the curve $C$ has potentially smooth stable reduction there (that is, after a finite extension, its stable reduction becomes smooth). Under these hypotheses, which statement about the reduction of $J(C)$ holds? Here, saying that $J(C)$ has superspecial reduction at a prime means that its reduction modulo that prime is a superspecial abelian surface.

The Jacobian $J(C)$ has superspecial reduction at infinitely many primes.

Choices:
B. The Jacobian $J(C)$ has superspecial reduction at all but finitely many primes.
C. The Jacobian $J(C)$ has superspecial reduction at at least one prime.
D. There exists a finite extension $L'/L$ such that the base change $J(C)_{L'}$ has superspecial reduction at infinitely many primes of $L'$.
E. The Jacobian $J(C)$ has infinitely many primes of supersingular reduction.


\label{examples}
In this section, we provide a concrete example to show how specific  high  rank  elliptic curve of the form
$y^2=x(x^2+B)$  with a known set of rational points correspond to a single

\begin{thm} \label{main_theorem}
		Let $k$ be a number field containing a primitive $s$-th root of unity. Assume the strong version of Lang's conjecture (Conjecture \ref{conj1}) holds for varieties defined over $k$. Then the rank $r(J_C(k))$ is uniformly bounded as $J_C$ varies over the Jacobian varieties of all smooth curves $C$ defined over $k$ by equation \eqref{curve_main} with genus $g_{r,s}(C) \geq 1$.
	\end{thm}

A central theme in arithmetic geometry concerns the structure of the group of rational points on algebraic varieties defined over number fields. For an abelian variety $A$ over a number field $k$, the celebrated Mordell-Weil theorem asserts that the group $A(k)$ is finitely generated \cite{Silverman1986}. Thus, $A(k) \cong A(k)_{\text{tors}} \oplus \mathbb{Z}^r$, where $A(k)_{\text{tors}}$ is the finite torsion subgroup and $r = \rk A(k)$ is the Mordell-Weil rank. Understanding the behavior of this rank, particularly for Jacobians $J_C$ of algebraic curves $C$, remains a profound challenge. A fundamental open question asks whether $r(J_C(k))$ is uniformly bounded as $C$ varies over all curves of a fixed genus $g \geq 1$ defined over $k$.

Even for elliptic curves ($g=1$) over $k=\mathbb{Q}$, this question is unresolved. While computational evidence reveals curves of high rank, theoretical heuristics suggest boundedness, possibly with rank exceeding 21 being rare \cite{Park2019}. Deep conjectures in Diophantine geometry provide conditional evidence for rank boundedness. Notably, Lang's conjectures on rational points on varieties of general type \cite{Lang1986, Lang1991} imply, via the work of Caporaso, Harris, and Mazur \cite{Caporaso1997, caporaso2022update}, strong uniformity statements about the number of rational points on curves. Furthermore, Pasten demonstrated that conjectures on Diophantine approximation imply Honda's conjecture on rank boundedness for elliptic curves \cite{Pasten2019}. The link between point counts and rank was solidified by Dimitrov, Gao, and Habegger \cite{Dimitrov2021}, making uniform point boundedness equivalent to uniform rank boundedness.

An alternative approach, pioneered by Yamagishi \cite{Yamagishi2003} for the family $y^2=ax^4+bx^2+c$, employs geometric methods. By constructing parameter spaces for curves with multiple rational points and demonstrating that these spaces (or relevant subvarieties) become varieties of general type, one can invoke Lang's conjectures to deduce finiteness and uniformity properties, ultimately leading to rank bounds. This geometric strategy was adapted by the author in \cite{Salami2025toappear} to study the uniformity of Mordell-Weil rank of Jacobian variety of the family of curves
 $ y^s = ax^r+b$.

This paper applies the same geometric methodology to investigate the closely related two-parameter family of curves defined by the affine equation
 \begin{equation} \label{curve_main}
 C_{a,b}: y^s=x(ax^r+b),
 \end{equation}
 where $r \geq 1, s \geq 2$ are fixed integers, and $k$ is a number field containing $\zeta_s$, a primitive $s$-th root of unity. We focus on smooth curves $C_{a,b}$ with genus $g_{r,s}(C_{a,b}) \geq 1$. Our main result, conditional on Lang's conjecture, establishes uniform rank boundedness for this family.

\begin{conj}[Lang, \cite{Lang1986, Lang1991}]
		\label{conj1}
		Let $k$ be a number field and $X$ a smooth variety of general type over $k$.
		\begin{enumerate}
			\item[(a)] \textbf{(Weak)} The set of $k$-rational points $X(k)$ is not Zariski dense in $X$.
			\item[(b)] \textbf{(Strong)} There exists a proper Zariski-closed subset $Z \subset X$ such that for any finite extension $K$ of $k$, the set of $K$-rational points on $X \setminus Z$ is finite.
		\end{enumerate}
	\end{conj}

\begin{equation} \label{curve_main}
		C_{a,b}: y^s=x(ax^r+b),
	\end{equation}

\begin{abstract}
 Let $k$ be a number field. We investigate the Mordell-Weil ranks of Jacobian varieties $J_C$ associated with algebraic curves $C$ of genus $g \geq 1$ defined by affine equations of the form $y^s=x(ax^r+b)$, where $a, b \in k$ ($ab \neq 0$), and $r \geq 1, s \geq 2$ are fixed integers. Assuming the strong version of Lang's conjecture concerning rational points on varieties of general type, we establish that the ranks $r(J_C(k))$ are uniformly bounded as $C$ varies within this family.
 Our methodology builds upon the geometric approach employed by H. Yamagishi and subsequently adapted by the author for the family $y^s=ax^r+b$. We construct a parameter space $\Wc_n$ for curves possessing $n+1$ specified rational points and analyze its birational model $\Xc_n$, a complete intersection variety. The geometric properties of the fibers of $\Xc_n \to \Sym^{n+1}(\Pp^1)$, specifically their genus and gonality, are studied. Combining these geometric insights with Faltings' theorem, uniformity conjectures stemming from Lang's work, and recent results connecting rank with the number of rational points, we deduce the main boundedness result. In the case of genus one curves $C$, it
 states that the rank of elliptic curves $y^2=x (x^2+B)$
 is uniformly bounded subject to the strong version of Lang's conjecture.
\end{abstract}

A central theme in arithmetic geometry concerns the structure of the group of rational points on algebraic varieties defined over number fields. For an abelian variety $A$ over a number field $k$, the celebrated Mordell-Weil theorem asserts that the group $A(k)$ is finitely generated \cite{Silverman1986}. Thus, $A(k) \cong A(k)_{\text{tors}} \oplus \mathbb{Z}^r$, where $A(k)_{\text{tors}}$ is the finite torsion subgroup and $r = \rk A(k)$ is the Mordell-Weil rank. Understanding the behavior of this rank, particularly for Jacobians $J_C$ of algebraic curves $C$, remains a profound challenge. A fundamental open question asks whether $r(J_C(k))$ is uniformly bounded as $C$ varies over all curves of a fixed genus $g \geq 1$ defined over $k$.

This paper applies the same geometric methodology to investigate the closely related two-parameter family of curves defined by the affine equation
 \begin{equation} \label{curve_main}
 C_{a,b}: y^s=x(ax^r+b),
 \end{equation}
 where $r \geq 1, s \geq 2$ are fixed integers, and $k$ is a number field containing $\zeta_s$, a primitive $s$-th root of unity. We focus on smooth curves $C_{a,b}$ with genus $g_{r,s}(C_{a,b}) \geq 1$. Our main result, conditional on Lang's conjecture, establishes uniform rank boundedness for this family.

The main result and its consequences provide new geometric evidence for rank boundedness conjectures for elliptic curves similar to that given in author's previous work \cite{Salami2025toappear}
different from that of given in \cite{Pasten2019}.
To be explicit, let us consider
the case $g_{r,s}(C) = 1$ and $k=\Q$, where the curve $C $ is an elliptic curves of the form $y^2= x(x^2+ b)$ with $\Z$.
By Theorem\ref{main_theorem}, one can conclude that the rank of elliptic curves
 is uniformly bounded subject to the strong version of Lang's conjecture, see Section \ref{sec:applications}.
 We note that the record of rank in this family is 14 due to M. Watkins obtained in 2002, according to \cite{DujellaRankRecord}.
 As a particular case of the above family, is the congruent number elliptic curves. There are only 27 congruent number $N$ for which the rank of corresponding elliptic curves $y^2=x (x^2-N^2)$ is equal to 7. It is also suspected that there does not exist any congruent number elliptic curve of rank 8, see \cite{Watkins2014}.
Theorem\ref{main_theorem} implies the uniformity of rank in family of congruent number elliptic curves, subject to the strong version of Lang's conjecture, see Section \ref{sec:applications}.

\begin{thm} \label{uniformity_theorem}
 Assume the weak version of Lang's conjecture (Conjecture \ref{conj1}) holds over $k$. Let $r \geq 1, s \geq 2$ such that $g_{r,s}(C) \geq 1$, and let $n_0$ be as in Theorem \ref{finiteness_theorem}. Then for $n \geq n_0$, there exists a uniform bound $M_0 = M_0(k, r, s, n)$ such that $\#\Cc_{\ba_n}(k) < M_0$ for all valid choices of $\ba_n$. Furthermore, if $g_{r,s}(C) \geq 2$, there exists an integer $m_0 > n_0$ such that $\Cc_{\ba_m}(k) = \emptyset$ for all $m \geq m_0$ and all valid $\ba_m$. (The strong Lang conjecture implies $M_0$ and $m_0$ can be chosen independently of $k$.)
 \end{thm}

\begin{proof}[Proof of Theorem \ref{main_theorem}]
 Let $k$ be a number field, $[k:\Q]=d$. Let $\mathcal{F}$ be the family of smooth curves $C: y^s=x(ax^r+b)$ over $k$ with genus $g=g_{r,s}(C) \geq 1$. Assume, for contradiction, that the rank $r(J_C(k))$ is unbounded for $C \in \mathcal{F}$.
 Then there exists a sequence of non-isomorphic curves $C_j \in \mathcal{F}$ such that $r_j = r(J_{C_j}(k)) \to \infty$ as $j \to \infty$.
 By the result of Dimitrov, Gao, and Habegger (Theorem \ref{dgh}), there is a constant $c=c(g,d)$ such that $\#C_j(k) \leq c^{1+r_j}$. Since $r_j \to \infty$, it follows that $\#C_j(k) \to \infty$.

Now, assume the Strong Lang Conjecture (Conjecture \ref{conj1}). This implies the strong version of Theorem \ref{uniformity_theorem}: there exists an integer $m_0$ (depending only on $r, s$, possibly $g$, but not $k$) such that for any $m \geq m_0$, $\Cc_{\ba_m}(K) = \emptyset$ for any number field $K$ and any valid $\ba_m \in \Uc_{r,m}(K)$. This means no smooth curve in the family \eqref{curve_main} over any number field $K$ can possess $m+1$ points $(\alpha_i, \beta_i)$ where the $\alpha_i$ are distinct, non-zero, and satisfy $\alpha_i^r \neq \alpha_j^r$, provided $m \ge m_0$.

\begin{thm}
 \label{CN-rank1}
 Assume the strong version of Lang's conjecture (Conjecture \ref{conj1}) holds. Then the Mordell-Weil ranks
$r(E_B(\Q))$ of elliptic curves $E_B: y^2=x (x^2+B)$ are uniformly bounded as $B$ varies over all integers.
\end{thm}

The main  result and its consequences provide new geometric evidence for rank boundedness  conjectures  for elliptic curves  similar to  that given in author's previous work \cite{Salami2025toappear}
different from that of given in \cite{Pasten2019}.
To be explicit, let us consider
the case $g_{r,s}(C) = 1$ and $k=\mathbb{Q}$, where the curve $C $ is an elliptic curves of the form  $y^2= x(x^2+ b)$ with  $\mathbb{Z}$.
By  Theorem\ref{main_theorem}, one can conclude that the  rank of elliptic curves
 is  uniformly bounded subject to the strong version of Lang's conjecture,   see Section \ref{sec:applications}.
    We note that  the record of rank in this family is 14 due to  M. Watkins obtained in 2002, according to \cite{DujellaRankRecord}.
 As a particular case of the above family, is the congruent number elliptic curves. There are only 27 congruent number $N$ for which the rank of corresponding elliptic curves $y^2=x (x^2-N^2)$ is equal to 7. It is also suspected that there does not exist any congruent number elliptic  curve of rank 8, see   \cite{Watkins2014}.
Theorem\ref{main_theorem}  implies the uniformity of rank  in family of congruent number elliptic  curves, subject to the strong version of Lang's conjecture,   see Section \ref{sec:applications}.

	The proof of Theorem\ref{main_theorem} relies crucially on studying the set $\Cc_{\ba_n}(k)$ consisting of smooth curves $C$ in the family \eqref{curve_main} that pass through a given set of $n+1$ points with $x$-coordinates $\alpha_i \in k$ ($i=0,\dots,n$) where $\ba_n$ represents the collection $\{\alpha_0, \dots, \alpha_n\}$ of distinct, non-zero elements of $k$ satisfying $\alpha_i^r \neq \alpha_j^r$ for $i \neq j$. We establish the following finiteness and uniformity results for these sets.

	\begin{thm} \label{finiteness_theorem}
		Let $k$ be a number field containing $\zeta_s$. Let $n_0=4$ if $s=2$ and $n_0=3$ otherwise. For any choice of $\ba_n = \{\alpha_0, \ldots, \alpha_n\}$ consisting of $n+1$ distinct, non-zero elements in $k$ with $\alpha_i^r \neq \alpha_j^r$ for $i \neq j$, and for integers $r \geq 1, s \geq 2$ such that $g_{r,s}(C) \geq 1$, the set $\Cc_{\ba_n}(k)$ is infinite if $n < n_0$ and finite if $n \geq n_0$. Consequently, for a fixed $\ba_n$ with $n \geq n_0$, the rank $r(J_C(k))$ is bounded for $C \in \Cc_{\ba_n}(k)$.
	\end{thm}

We use the uniformity results given this non-conditional result to discuss  a solution for the problem of arithmetic progression  on the points on the elliptic curves $y^2=x (x^2+B)$ as well as the congruent number elliptic curves, in Section~\ref{sec:applications}.

	Using the uniformity theorems on the number of $k$-rational points on curve of genus $\geq 2$ gibem in \cite{Caporaso1997}, as
	consequences of weak version of Lang's conjecture, we showed the cardinal number of $\Cc_{\ba_n}(k)$ does not depends on the choice of ${\ba_n}$ and  is   uniformly bounded and eventually empty for large  n.

	\begin{thm} \label{uniformity_theorem}
		Assume the weak version of Lang's conjecture (Conjecture \ref{conj1}) holds over $k$. Let $r \geq 1, s \geq 2$ such that $g_{r,s}(C) \geq 1$, and let $n_0$ be as in Theorem \ref{finiteness_theorem}. Then for $n \geq n_0$, there exists a uniform bound $M_0 = M_0(k, r, s, n)$ such that $\#\Cc_{\ba_n}(k) < M_0$ for all valid choices of $\ba_n$. Furthermore, if $g_{r,s}(C) \geq 2$, there exists an integer $m_0 > n_0$ such that $\Cc_{\ba_m}(k) = \emptyset$ for all $m \geq m_0$ and all valid $\ba_m$. (The strong Lang conjecture implies $M_0$ and $m_0$ can be chosen independently of $k$.)
	\end{thm}

	The structure of the paper mirrors the logical progression of the proof. Section \ref{sec_geom_constr} details the construction of the parameter space $\Wc_n$ using twists and its birational model $\Xc_n$, a complete intersection variety, establishing their relationship. Section \ref{sec_fiber_analysis} focuses on the geometry of the fibers $\Xc_{\ba_n}$ of $\Xc_n \to \Sym^{n+1}(\mathbb{P}^1)$, calculating their genus and gonality, analyzing rational points in low-genus cases, and invoking the theory of towers of curves for the general case, integrating necessary background results. Section \ref{sec_finiteness_uniformity_proofs} provides proofs for Theorems \ref{finiteness_theorem} and \ref{uniformity_theorem}, incorporating Faltings' theorem and Lang's conjectures. In Section \ref{sec_main_theorem_proof}, we prove  the main result, Theorem \ref{main_theorem}, using the uniformity theorem and the work of Dimitrov, Gao, and Habegger.
	 Throughout, we cite analogous arguments from \cite{Salami2025toappear} where applicable.
	In Section~\ref{sec:applications}, we  have provided consequences of our results for   elliptic curves of the form $y^2=x(x^2+B)$ and  in particular the congruent number elliptic curves.
	 Finally, in   Section \ref{examples}, we have provided a computational example of the correspondence between
	the set of curves  $\Cc_{\ba_n}$ and rational points on the fiber $\Xc_{\ba_n}$.

The post-theorem introduction gives the following proof outline for Theorem\ref{main_theorem}.

- The proof \emph{“relies crucially on studying the set $\Cc_{\ba_n}(k)$ consisting of smooth curves $C$ in the family \eqref{curve_main} that pass through a given set of $n+1$ points with $x$-coordinates $\alpha_i\in k$”}, where $\ba_n=\{\alpha_0,\dots,\alpha_n\}$ are distinct nonzero elements with $\alpha_i^r\neq \alpha_j^r$ for $i\neq j$.

- One first \emph{“establish[es]… finiteness and uniformity results for these sets”} via Theorems \ref{finiteness_theorem} and \ref{uniformity_theorem} (finiteness for $n\ge n_0$; uniform bounds $\#\Cc_{\ba_n}(k)<M_0$ for all valid $\ba_n$, and eventual emptiness for large $n$ when $g_{r,s}(C)\ge 2$). The proofs of these results are said to be in Section~\ref{sec_finiteness_uniformity_proofs}, \emph{“incorporating Faltings' theorem and Lang's conjectures.”}

- Geometric input is developed beforehand: Section~\ref{sec_geom_constr} \emph{“details the construction of the parameter space $\Wc_n$ using twists and its birational model $\Xc_n$, a complete intersection variety”}; Section~\ref{sec_fiber_analysis} \emph{“focuses on the geometry of the fibers $\Xc_{\ba_n}$…, calculating their genus and gonality, analyzing rational points in low-genus cases, and invoking the theory of towers of curves for the general case.”}

- Finally, \emph{“In Section \ref{sec_main_theorem_proof}, we prove the main result, Theorem \ref{main_theorem}, using the uniformity theorem and the work of Dimitrov, Gao, and Habegger.”}

\label{main_theorem}
		Let $k$ be a number field containing a primitive $s$-th root of unity. Assume the strong version of Lang's conjecture holds for varieties defined over $k$, namely:
		\begin{conj}[Lang, \cite{Lang1986, Lang1991}]
		\label{conj1}
		Let $k$ be a number field and $X$ a smooth variety of general type over $k$.
		\begin{enumerate}
			\item[(a)] \textbf{(Weak)} The set of $k$-rational points $X(k)$ is not Zariski dense in $X$.
			\item[(b)] \textbf{(Strong)} There exists a proper Zariski-closed subset $Z \subset X$ such that for any finite extension $K$ of $k$, the set of $K$-rational points on $X \setminus Z$ is finite.
		\end{enumerate}
	\end{conj}
		Then the rank $r(J_C(k))$ is uniformly bounded as $J_C$ varies over the Jacobian varieties of all smooth curves $C$ defined over $k$ by the equation
		\begin{equation} \label{curve_main}
		C_{a,b}: y^s=x(ax^r+b),
	\end{equation}
		with genus $g_{r,s}(C) \geq 1%.

Question:
Let $k$ be a number field containing a primitive $s$-th root of unity, and fix integers $r\ge 1$ and $s\ge 2$. Assume the strong version of Lang's conjecture over $k$: for every smooth variety $X$ of general type over $k$, there exists a proper Zariski-closed subset $Z\subset X$ such that for every finite extension $K/k$, the set $(X\setminus Z)(K)$ is finite. For each smooth curve $C=C_{a,b}$ over $k$ given by
\[
C_{a,b}: y^s=x(ax^r+b),
\]
with genus $g_{r,s}(C)\ge 1$, let $J_C$ be its Jacobian and let $r(J_C(k))$ denote the Mordell-Weil rank of the finitely generated abelian group $J_C(k)$. Which conclusion about these ranks is valid under these hypotheses?

The Mordell-Weil ranks are uniformly bounded in this family: there exists a constant $M$ such that for every smooth curve $C$ over $k$ of the form $y^s=x(ax^r+b)$ with $g_{r,s}(C)\ge 1$, one has
\[
r(J_C(k))\le M.
\]
Equivalently, as $J_C$ varies over the Jacobians of all such curves, the ranks $r(J_C(k))$ remain bounded by a single constant depending only on the fixed data of the family.

Choices:
B.
The Mordell-Weil ranks are uniformly bounded in this family after allowing finite extensions: there exists a constant $M$ such that for every smooth curve $C$ over $k$ of the form $y^s=x(ax^r+b)$ with $g_{r,s}(C)\ge 1$, and for every finite extension $K/k$, one has
\[
r(J_C(K))\le M.
\]
Equivalently, as $J_C$ varies over the Jacobians of all such curves and $K$ varies over finite extensions of $k$, the ranks remain bounded by a single constant depending only on the fixed data of the family.
C. For each smooth curve $C$ over $k$ of the form $y^s=x(ax^r+b)$ with $g_{r,s}(C)\ge 1$, the Mordell-Weil group $J_C(k)$ is finitely generated, so in particular the rank $r(J_C(k))$ is finite for every individual curve.
D.
The Mordell-Weil ranks are uniformly bounded in this family only for the subfamily of curves with genus $g_{r,s}(C)\ge 2$: there exists a constant $M$ such that every smooth curve $C$ over $k$ of the form $y^s=x(ax^r+b)$ with $g_{r,s}(C)\ge 2$ satisfies
\[
r(J_C(k))\le M,
\]
while no conclusion is asserted for the genus-one members of the family.
E.
There exists a constant $M$ such that for every smooth curve $C$ over $k$ of the form $y^s=x(ax^r+b)$ with $g_{r,s}(C)\ge 1$, one has a uniform bound on rational points
\[
\#C(k)\le M.
\]
Equivalently, the family has uniformly bounded numbers of $k$-rational points, and hence the Jacobian ranks are bounded as well.