A FUNCTIONAL LOGARITHMIC FORMULA FOR THE HYPERGEOMETRIC FUNCTION $_{3}F_{2}$

Abstract

We give a sufficient condition for the hypergeometric function $_{3}F_{2}$ to be a linear combination of the logarithm of algebraic functions.

1 Introduction

For $\unicode[STIX]{x1D6FC}_{i},\unicode[STIX]{x1D6FD}_{j}\in \mathbb{C}$ with $\unicode[STIX]{x1D6FD}_{j}\not \in \mathbb{Z}_{{\leqslant}0}$ , the generalized hypergeometric function is defined by a power series expansion

$$\begin{eqnarray}_{p}F_{p-1}{\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{p}\atop \unicode[STIX]{x1D6FD}_{1},\ldots ,\unicode[STIX]{x1D6FD}_{p-1}};x=\mathop{\sum }_{n=0}^{\infty }\frac{(\unicode[STIX]{x1D6FC}_{1})_{n}\cdots (\unicode[STIX]{x1D6FC}_{p})_{n}}{(\unicode[STIX]{x1D6FD}_{1})_{n}\cdots (\unicode[STIX]{x1D6FD}_{p-1})_{n}}\frac{x^{n}}{n!},\end{eqnarray}$$

where

$$\begin{eqnarray}(\unicode[STIX]{x1D6FC})_{0}:=1,\qquad (\unicode[STIX]{x1D6FC})_{n}:=\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D6FC}+1)\cdots (\unicode[STIX]{x1D6FC}+n-1)\quad \text{for }n\geqslant 1\end{eqnarray}$$

denotes the Pochhammer symbol. When $p=2$ , this is called the Gauss hypergeometric function. This has an analytic continuation to $\mathbb{C}$ , and then becomes a multivalued function which is holomorphic on $\mathbb{C}\setminus \{0,1\}$ . A number of formulas have been discovered since 19th century (e.g., [10, Chapters 15, 16]), and they have been applied in various areas in mathematics. At present, the theory of hypergeometric function is one of the most important tools in mathematics.

In [5], we discussed the special values of $_{3}F_{2}({1,1,q\atop a,b};x)$ at $x=1$ , and gave a sufficient condition for it to be a $\overline{\mathbb{Q}}$ -linear combination of logarithms of algebraic numbers, namely

$$\begin{eqnarray}\displaystyle & & \displaystyle \,_{3}F_{2}{1,1,q\atop a,b};1\in \overline{\mathbb{Q}}+\overline{\mathbb{Q}}\log \overline{\mathbb{Q}}^{\times }\nonumber\\ \displaystyle & & \displaystyle \quad :=\left\{\left.a+\mathop{\sum }_{i=1}^{n}b_{i}\log c_{i}\,\right|a,b_{i},c_{i}\in \overline{\mathbb{Q}},\,c_{i}\neq 0,\,n\in \mathbb{Z}_{{\geqslant}0}\right\}.\nonumber\end{eqnarray}$$

The goal of this paper is to give its functional version. To be precise, set

$$\begin{eqnarray}\displaystyle & & \displaystyle \overline{\mathbb{Q}(x)}+\overline{\mathbb{Q}(x)}\log \overline{\mathbb{Q}(x)}^{\times }\nonumber\\ \displaystyle & & \displaystyle \quad :=\left\{\left.f+\mathop{\sum }_{i=1}^{n}g_{i}\log h_{i}\,\right|f,g_{i},h_{i}\in \overline{\mathbb{Q}(x)},\,h_{i}\neq 0,\,n\in \mathbb{Z}_{{\geqslant}0}\right\}\nonumber\end{eqnarray}$$

where $\overline{\mathbb{Q}(x)}$ denotes the algebraic closure of the field of rational functions $\mathbb{Q}(x)$ . We say that the logarithmic formula holds for a function $F(x)$ if it belongs to the above set. The main theorem gives a sufficient condition on $(a,b,q)$ for $_{3}F_{2}({1,1,q\atop a,b};x)$ to satisfy a logarithmic formula. Recall that two proofs are presented in [5]. One of the proofs uses hypergeometric fibrations and the other uses Fermat surfaces. In this paper we follow the method of hypergeometric fibrations, while employing a new ingredient from [3]. It seems impossible to prove the functional log formula using the method of Fermat surfaces.

By developing the technique here, we can get explicit log formulas in some cases. For example, let

$$\begin{eqnarray}\displaystyle e_{1}(x) & := & \displaystyle \frac{1}{2}+x^{-1/3}\left(-\frac{1}{4}+\frac{x}{8}+\frac{1}{4}\sqrt{1-x}\right)^{1/3}\nonumber\\ \displaystyle & & \displaystyle +\,x^{-1/3}\left(-\frac{1}{4}+\frac{x}{8}-\frac{1}{4}\sqrt{1-x}\right)^{1/3}\nonumber\\ \displaystyle e_{2}(x) & := & \displaystyle \frac{1}{2}+e^{-2\unicode[STIX]{x1D70B}i/3}x^{-1/3}\left(-\frac{1}{4}+\frac{x}{8}+\frac{1}{4}\sqrt{1-x}\right)^{1/3}\nonumber\\ \displaystyle & & \displaystyle +\,e^{2\unicode[STIX]{x1D70B}i/3}x^{-1/3}\left(-\frac{1}{4}+\frac{x}{8}-\frac{1}{4}\sqrt{1-x}\right)^{1/3}\nonumber\\ \displaystyle e_{3}(x) & := & \displaystyle \frac{1}{2}+e^{2\unicode[STIX]{x1D70B}i/3}x^{-1/3}\left(-\frac{1}{4}+\frac{x}{8}+\frac{1}{4}\sqrt{1-x}\right)^{1/3}\nonumber\\ \displaystyle & & \displaystyle +\,e^{-2\unicode[STIX]{x1D70B}i/3}x^{-1/3}\left(-\frac{1}{4}+\frac{x}{8}-\frac{1}{4}\sqrt{1-x}\right)^{1/3}\nonumber\end{eqnarray}$$

$$\begin{eqnarray}p_{\pm }=p_{\pm }(x):=\left(\frac{1\pm \sqrt{1-x}}{\sqrt{x}}\right)^{2/3},\qquad q_{j}=q_{j}(x):=\frac{1-\sqrt{3x}\cdot e_{j}(x)}{1+\sqrt{3x}\cdot e_{j}(x)}.\end{eqnarray}$$

Then

However, there remain technical difficulties arising from algebraic cycles to obtain explicit log formulas in more general cases.

2 Main theorem

Let $\hat{\mathbb{Z}}=\mathop{\varprojlim }\nolimits_{n}\mathbb{Z}/n\mathbb{Z}$ be the completion, and $\hat{\mathbb{Z}}^{\times }=\mathop{\varprojlim }\nolimits_{n}(\mathbb{Z}/n\mathbb{Z})^{\times }$ the group of units. The ring $\hat{\mathbb{Z}}$ acts on the additive group $\mathbb{Q}/\mathbb{Z}$ in a natural way, and then it induces $\hat{\mathbb{Z}}^{\times }\cong \text{Aut}(\mathbb{Q}/\mathbb{Z})$ . We denote by $\{x\}:=x-\lfloor x\rfloor$ the fractional part of $x\in \mathbb{Q}$ . The map $\{-\}:\mathbb{Q}\rightarrow [0,1)$ factors through $\mathbb{Q}/\mathbb{Z}$ , which we denote by the same notation.

Theorem 2.1. (Logarithmic formula)

Let $q,a,b\in \mathbb{Q}$ satisfy the property that none of $q,a,b,q-a,q-b,q-a-b$ is an integer. Suppose

(2.1) $$\begin{eqnarray}\displaystyle & 1=\{sa\}+\{sb\}+2\{-sq\}-\{s(a-q)\}-\{s(b-q)\} & \displaystyle \nonumber\\ \displaystyle & (\Longleftrightarrow \quad \min (\{sa\},\{sb\})<\{sq\}<\max (\{sa\},\{sb\})) & \displaystyle\end{eqnarray}$$

for $\forall s\in \hat{\mathbb{Z}}^{\times }$ . Then

$$\begin{eqnarray}_{3}F_{2}{n_{1},n_{2},q\atop a,~b};x\in \overline{\mathbb{Q}(x)}+\overline{\mathbb{Q}(x)}\log \overline{\mathbb{Q}(x)}^{\times }\end{eqnarray}$$

for any integers $n_{i}>0$ .

As we shall see in Section 4, one can shift the indices $n_{i}$ , $q,a,b$ by arbitrary integers by applying differential operators. Thus it is enough to prove the log formula for $_{3}F_{2}({1,1,q\atop a,~b};x)$ .

Recall the main theorem of [5] which asserts that if

(2.2) $$\begin{eqnarray}2=\{sq\}+\{s(a-q)\}+\{s(b-q)\}+\{s(q-a-b)\}\end{eqnarray}$$

for $\forall s\in \hat{\mathbb{Z}}^{\times }$ , then

$$\begin{eqnarray}_{3}F_{2}{1,1,q\atop a,~b};1\in \overline{\mathbb{Q}}+\overline{\mathbb{Q}}\log \overline{\mathbb{Q}}^{\times }\end{eqnarray}$$

as long as it converges ( $\Leftrightarrow$ $a+b>q+2$ ). It is easy to see (2.1) $\Rightarrow$ (2.2) while the converse is no longer true (e.g., $(a,b,q)=(1/6,1/4,1/2)$ ). Therefore Theorem 2.1 does not imply all of the main theorem of [5].

Conjecture 2.2. (Cf. [5, Conjecture 5.2])

The converse of Theorem 2.1 is true.

In the seminal paper [7], Beukers and Heckman gave a necessary and sufficient condition for $_{p}F_{p-1}$ to be an algebraic function, or equivalently for its monodromy group to be finite. Let $a_{i},b_{j}\in \mathbb{Q}$ . Then their theorem states that, under the condition that $\{a_{i}\}\neq \{b_{j}\}$ and $\{a_{i}\}\neq 0$ ,

$$\begin{eqnarray}_{p}F_{p-1}{a_{1},\ldots ,a_{p}\atop b_{1},\ldots ,b_{p-1}};x\in \overline{\mathbb{Q}(x)}\end{eqnarray}$$

if and only if $(\{sa_{1}\},\ldots ,\{sa_{p}\})$ and $(0,\{sb_{1}\},\ldots ,\{sb_{p-1}\})$ interlace for all $s\in \hat{\mathbb{Z}}^{\times }$ [7, Theorem 4.8]. Here we say that two sets $(\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{p})$ and $(\unicode[STIX]{x1D6FD}_{1},\ldots ,\unicode[STIX]{x1D6FD}_{p})$ interlace if and only if

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{1}<\unicode[STIX]{x1D6FD}_{1}<\cdots <\unicode[STIX]{x1D6FC}_{p}<\unicode[STIX]{x1D6FD}_{p}\qquad \text{or}\qquad \unicode[STIX]{x1D6FD}_{1}<\unicode[STIX]{x1D6FC}_{1}<\cdots <\unicode[STIX]{x1D6FD}_{p}<\unicode[STIX]{x1D6FC}_{p}\end{eqnarray}$$

when ordering $\unicode[STIX]{x1D6FC}_{1}<\cdots <\unicode[STIX]{x1D6FC}_{p}$ and $\unicode[STIX]{x1D6FD}_{1}<\cdots <\unicode[STIX]{x1D6FD}_{p}$ . In this terminology, (2.1) is translated into that $(0,\{sq\})$ and $(\{sa\},\{sb\})$ interlace. Our main Theorem 2.1 is not directly related to their theorem, while they are obviously comparable.

3 Hypergeometric fibrations

We mean by a fibration over a ring $k$ a projective flat morphism of quasiprojective smooth $k$ -schemes.

3.1 Definition

Let $f:X\rightarrow \mathbb{P}^{1}$ be a fibration over a field $k$ . For simplicity we assume $k=\overline{k}$ and fix an embedding $k\subset \mathbb{C}$ . Let $R$ be a finite-dimensional semisimple commutative $\mathbb{Q}$ -algebra. We mean by a multiplication on $R^{1}f_{\ast }\mathbb{Q}$ by $R$ a homomorphism $\unicode[STIX]{x1D70C}:R\rightarrow \text{End}_{\text{VHS}}(R^{1}f_{\ast }\mathbb{Q}|_{U})$ of rings where $U\subset \mathbb{P}^{1}$ is the maximal Zariski open set such that $f$ is smooth over $U$ . Let $e:R\rightarrow E$ be a projection onto a number field $E$ . We say $f$ is a hypergeometric fibration with multiplication by $(R,e)$ (HG fibration) if the following conditions hold. We fix an inhomogeneous coordinate $t\in \mathbb{P}^{1}$ .

1. (a) $f$ is smooth over $\mathbb{P}^{1}\setminus \{0,1,\infty \}$ ,

2. (b) $\dim _{E}H^{1}(X_{t},\mathbb{Q})(e)=2$ where $X_{t}=f^{-1}(t)$ is a general fiber and we write $V(e):=E\otimes _{e,R}V$ the $e$ -part,

3. (c) Let $\text{Pic}_{f}^{0}\rightarrow \mathbb{P}^{1}\setminus \{0,1,\infty \}$ be the Picard fibration whose general fiber is the Picard variety $\text{Pic}^{0}(f^{-1}(t))$ , and let $\text{Pic}_{f}^{0}(e)$ be the component associated to the $e$ -part $R^{1}f_{\ast }\mathbb{Q}(e)$ (this is well defined up to isogeny). Then $\text{Pic}_{f}^{0}(e)\rightarrow \mathbb{P}^{1}\setminus \{0,1,\infty \}$ has totally degenerate semistable reduction at $t=1$ .

The last condition (c) is equivalent to saying that the local monodromy $T$ on $H^{1}(X_{t},\mathbb{Q})(e)$ at $t=1$ is unipotent and the rank of log monodromy $N:=\log (T)$ is maximal, namely $\text{rank}(N)={\textstyle \frac{1}{2}}\dim _{\mathbb{Q}}H^{1}(X_{t},\mathbb{Q})(e)$ ( $=[E:\mathbb{Q}]$ by condition (b)).

3.2 HG fibration of Gauss type

Let $f:X\rightarrow \mathbb{P}^{1}$ be a fibration over $\overline{\mathbb{Q}}$ whose general fiber $X_{t}=f^{-1}(t)$ is the nonsingular projective model of the affine curve

(3.1) $$\begin{eqnarray}y^{N}=x^{a}(1-x)^{b}(1-tx)^{N-b},\quad 0<a,b<N,\,\gcd (N,a,b)=1.\end{eqnarray}$$

$f$ is smooth outside $\{0,1,\infty \}$ so that the condition (a) is satisfied. The group $\unicode[STIX]{x1D707}_{N}$ of $N$ th roots of unity acts on $f^{-1}(t)$ by $(x,y,t)\mapsto (x,\unicode[STIX]{x1D701}y,t)$ for $\unicode[STIX]{x1D701}\in \unicode[STIX]{x1D707}_{N}$ , which gives rise to a multiplication on $R^{1}f_{\ast }\mathbb{Q}$ by the group ring $R_{0}:=\mathbb{Q}[\unicode[STIX]{x1D707}_{N}]$ .

Lemma 3.1. [4, Proposition 3.1] Let $e_{0}:R_{0}:=\mathbb{Q}[\unicode[STIX]{x1D707}_{N}]\rightarrow E_{0}$ be a projection onto a number field $E_{0}$ . Then $(R_{0},e_{0})$ satisfies the conditions (b) and (c) if and only if $ad\not \equiv 0$ and $bd\not \equiv 0$ modulo $N$ where $d:=\sharp \text{Ker}[\unicode[STIX]{x1D707}_{N}\rightarrow R_{0}^{\times }\overset{e_{0}}{\rightarrow }E_{0}^{\times }]$ .

Definition 3.2. We say that $f$ is a HG fibration of Gauss type with multiplication by $(\mathbb{Q}[\unicode[STIX]{x1D707}_{N}],e)$ if $ad\not \equiv 0$ and $bd\not \equiv 0$ modulo $N$ .

Let $\unicode[STIX]{x1D712}:R_{0}\rightarrow \overline{\mathbb{Q}}$ be a homomorphism of $\mathbb{Q}$ -algebras factoring through $e$ . Let $n$ be an integer such that $\unicode[STIX]{x1D712}(\unicode[STIX]{x1D701})=\unicode[STIX]{x1D701}^{-n}$ for all $\unicode[STIX]{x1D701}\in \unicode[STIX]{x1D707}_{N}$ . Note $\gcd (n,N)=1$ . By [1, p. 917, (13)], $H_{\text{dR}}^{1}(X_{t})(\unicode[STIX]{x1D712})\cap H^{1,0}$ is spanned by the 1-form

$$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D714}_{n}:={\displaystyle \frac{x^{a_{n}}(1-x)^{b_{n}}(1-tx)^{c_{n}}}{y^{n}}}\,dx, & \displaystyle \nonumber\\ \displaystyle & a_{n}:=\left\lfloor {\displaystyle \frac{an}{N}}\right\rfloor ,\qquad b_{n}:=\left\lfloor {\displaystyle \frac{bn}{N}}\right\rfloor ,\qquad c_{n}:=\left\lfloor {\displaystyle \frac{Nn-bn}{N}}\right\rfloor =n-b_{n}-1. & \displaystyle \nonumber\end{eqnarray}$$

Let $P_{1}$ (resp. $P_{2}$ ) be a point of $X_{t}$ above $x=0$ (resp. $x=1$ ). There are $\text{gcd}(N,a)$ -points above $x=0$ (resp. $\text{gcd}(N,b)$ -points above $x=1$ ). Let $u$ be a path from $P_{1}$ to $P_{2}$ above the real interval $x\in [0,1]$ . It defines a homology cycle $u\in H_{1}(X_{t},\{P_{1},P_{2}\};\mathbb{Z})$ with boundary. Put $d_{1}:=\text{gcd}(N,a)$ , $d_{2}:=\text{gcd}(N,b)$ . Let $\unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D707}_{N}$ be an automorphism. Since $\unicode[STIX]{x1D70E}^{d_{1}}P_{1}=P_{1}$ and $\unicode[STIX]{x1D70E}^{d_{2}}P_{2}=P_{2}$ , one has a homology cycle

(3.2) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70E}):=(1-\unicode[STIX]{x1D70E}^{d_{1}})(1-\unicode[STIX]{x1D70E}^{d_{2}})u\in H_{1}(X_{t},\mathbb{Z}).\end{eqnarray}$$

By an integral expression of Gauss hypergeometric functions (e.g., [6, p. 4, 1.5] or [11, p. 20, (1.6.6)]), one has

(3.3) $$\begin{eqnarray}\displaystyle \quad \qquad \int _{\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70E})}\unicode[STIX]{x1D714}_{n} & = & \displaystyle (1-\unicode[STIX]{x1D701}^{-nd_{1}})(1-\unicode[STIX]{x1D701}^{-nd_{2}})\int _{0}^{1}\unicode[STIX]{x1D714}_{n}\end{eqnarray}$$

(3.4) $$\begin{eqnarray}\displaystyle & = & \displaystyle (1-\unicode[STIX]{x1D701}^{-nd_{1}})(1-\unicode[STIX]{x1D701}^{-nd_{2}})B(\unicode[STIX]{x1D6FC}_{n},\unicode[STIX]{x1D6FD}_{n})_{2}F_{1}(\unicode[STIX]{x1D6FC}_{n},\unicode[STIX]{x1D6FD}_{n},\unicode[STIX]{x1D6FC}_{n}+\unicode[STIX]{x1D6FD}_{n};t),\end{eqnarray}$$

where $B(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}):=\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D6FC})\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D6FD})/\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD})$ is the beta function, $\unicode[STIX]{x1D701}$ is defined by $\unicode[STIX]{x1D70E}(y)=\unicode[STIX]{x1D701}y$ and we put

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{n}:=\left\{\frac{-an}{N}\right\},\qquad \unicode[STIX]{x1D6FD}_{n}:=\left\{\frac{-bn}{N}\right\}.\end{eqnarray}$$

This shows that the monodromy on the 2-dimensional $H_{1}(X_{t},\mathbb{C})(\unicode[STIX]{x1D712})$ is isomorphic to the monodromy of the hypergeometric equation

$$\begin{eqnarray}(D_{t}(D_{t}+\unicode[STIX]{x1D6FC}_{n}+\unicode[STIX]{x1D6FD}_{n}-1)-t(D_{t}+\unicode[STIX]{x1D6FC}_{n})(D_{t}+\unicode[STIX]{x1D6FD}_{n}))(y)=0,\quad D_{t}:=t\frac{d}{dt}\end{eqnarray}$$

with the Riemann scheme

(3.5) $$\begin{eqnarray}\left\{\begin{array}{@{}ccc@{}}t=0 & t=1 & t=\infty \\ 0 & 0 & \unicode[STIX]{x1D6FC}_{n}\\ 1-\unicode[STIX]{x1D6FC}_{n}-\unicode[STIX]{x1D6FD}_{n} & 0 & \unicode[STIX]{x1D6FD}_{n}\end{array}\right\}\end{eqnarray}$$

In particular, the monodromy is irreducible as $\unicode[STIX]{x1D6FC}_{n},\unicode[STIX]{x1D6FD}_{n}\not \in \mathbb{Z}$ .

3.3 Hodge numbers

Let $f:X\rightarrow \mathbb{P}^{1}$ be a HG fibration with multiplication by $(R_{0},e_{0})$ . Following [3, Section 4.1], we consider motivic sheaves $\mathscr{M}$ and $\mathscr{H}$ which are defined in the following way. Let $S:=\mathbb{A}_{\overline{\mathbb{Q}}}^{1}\setminus \{0,1\}$ be defined over $\overline{\mathbb{Q}}$ with coordinate $\unicode[STIX]{x1D706}$ . Let $\mathbb{P}_{S}^{1}:=\mathbb{P}^{1}\times S$ and denote the coordinates by $(t,\unicode[STIX]{x1D706})$ . Put $\mathbb{P}_{S}^{1}\supset \mathscr{U}:=(\mathbb{A}_{\overline{\mathbb{Q}}}^{1}\setminus \{0,1\}\times S)\setminus \unicode[STIX]{x1D6E5}$ where $\unicode[STIX]{x1D6E5}$ is the diagonal subscheme. Let $l\geqslant 1$ be an integer. Let $\unicode[STIX]{x1D70B}:\mathbb{P}_{S}^{1}\rightarrow \mathbb{P}_{S}^{1}$ be a morphism over $S$ given by $(t,\unicode[STIX]{x1D706})\mapsto (\unicode[STIX]{x1D706}-t^{l},\unicode[STIX]{x1D706})$ . Then we define

$$\begin{eqnarray}\mathscr{M}:=\unicode[STIX]{x1D70B}_{\ast }\mathbb{Q}\otimes \text{pr}_{1}^{\ast }R^{1}f_{\ast }\mathbb{Q}|_{\mathscr{U}},\qquad \text{pr}_{1}:\mathbb{P}_{S}^{1}=\mathbb{P}^{1}\times S\rightarrow \mathbb{P}^{1}\end{eqnarray}$$

a variation of Hodge–de Rham structures (VHdR) on $\mathscr{U}$ and

$$\begin{eqnarray}\mathscr{H}:=R^{1}\text{pr}_{2\ast }\mathscr{M},\qquad \text{pr}_{2}:\mathscr{U}\rightarrow S\end{eqnarray}$$

a variation of mixed Hodge–de Rham structures (VMHdR) on $S$ , where the terminology is as in [3, Section 2.1] or [4, Section 2.1]. For the reader’s convenience, we give a description of the stalk $H_{a}=\mathscr{H}|_{\{a\}}$ and $\mathscr{M}_{a}=\mathscr{M}|_{\text{pr}_{2}^{-1}(a)}=\mathscr{M}|_{\mathbb{A}^{1}\setminus \{0,1,a\}}$ at $a\in S$ is given in the following way. Let $\unicode[STIX]{x1D70B}_{a}:\mathbb{P}^{1}\rightarrow \mathbb{P}^{1}$ be the map given by $t\mapsto a-t^{l}$ . Let

be a Cartesian diagram, and $i$ a desingularization along the singular fibers. Put $U_{a}:=\unicode[STIX]{x1D70B}_{a}^{-1}(\mathbb{A}^{1}\setminus \{0,1,a\})$ . Then

(3.6) $$\begin{eqnarray}\displaystyle \mathscr{M}_{a} & = & \displaystyle \unicode[STIX]{x1D70B}_{a\ast }\mathbb{Q}\otimes R^{1}f_{\ast }\mathbb{Q}|_{\mathbb{A}^{1}\setminus \{0,1,a\}}=\unicode[STIX]{x1D70B}_{a\ast }\unicode[STIX]{x1D70B}_{a}^{\ast }R^{1}f_{\ast }\mathbb{Q}|_{\mathbb{A}^{1}\setminus \{0,1,a\}}\nonumber\\ \displaystyle & \cong & \displaystyle \unicode[STIX]{x1D70B}_{a\ast }R^{1}f_{a\ast }\mathbb{Q}|_{\mathbb{A}^{1}\setminus \{0,1,a\}},\nonumber\\ \displaystyle H_{a} & = & \displaystyle H^{1}(\mathbb{A}^{1}\setminus \{0,1,a\},\mathscr{M}_{a})\nonumber\\ \displaystyle & \cong & \displaystyle H^{1}(U_{a},R^{1}f_{a\ast }\mathbb{Q})\subset H^{2}(f_{a}^{-1}(U_{a}),\mathbb{Q}).\end{eqnarray}$$

The weights of $\mathscr{H}$ are at most $2,3,4$ , and hence there is an exact sequence

(3.7) $$\begin{eqnarray}0\longrightarrow W_{2}\mathscr{H}\longrightarrow \mathscr{H}\longrightarrow \mathscr{H}/W_{2}\longrightarrow 0\end{eqnarray}$$

of VMHdR structures on $S$ . By (3.6), there is a canonical surjective map

(3.8) $$\begin{eqnarray}H^{2}(X_{a},\mathbb{Q})_{0}\longrightarrow W_{2}H_{a}\end{eqnarray}$$

where we put $H^{2}(X_{a},\mathbb{Q})_{0}:=\text{Ker}[H^{2}(X_{a},\mathbb{Q})\rightarrow H^{2}(f_{a}^{-1}(t),\mathbb{Q})]$ , $t\in U_{a}$ .

Let $\unicode[STIX]{x1D707}_{l}$ be the group of $l$ th roots of unity which acts on $\unicode[STIX]{x1D70B}_{\ast }\mathbb{Q}$ in a natural way. Then $\mathscr{M}$ has multiplication by the group ring $R:=R_{0}[\unicode[STIX]{x1D707}_{l}]$ . Let $e:R\rightarrow E$ be a projection onto a number field $E$ such that $\text{Ker}(e)\supset \text{Ker}(e_{0})$ . There is a unique embedding $E_{0}{\hookrightarrow}E$ making the diagram

commutative.

For $\unicode[STIX]{x1D712}:R\rightarrow \overline{\mathbb{Q}}$ factoring through $e$ , we denote by $V(\unicode[STIX]{x1D712})$ the $\unicode[STIX]{x1D712}$ -part which is defined to be the subspace on which $r\in R$ acts as multiplication by $\unicode[STIX]{x1D712}(r)$ .

Theorem 3.3. Let $T_{p}$ denote the local monodromy on $R^{1}f_{\ast }\overline{\mathbb{Q}}(\unicode[STIX]{x1D712})$ at $t=p$ . Let $\unicode[STIX]{x1D6FC}_{j}^{\unicode[STIX]{x1D712}}$ (resp. $\unicode[STIX]{x1D6FD}_{j}^{\unicode[STIX]{x1D712}}$ ) for $j=1,2$ be rational numbers such that $e^{2\unicode[STIX]{x1D70B}i\unicode[STIX]{x1D6FC}_{j}^{\unicode[STIX]{x1D712}}}$ (resp. $e^{2\unicode[STIX]{x1D70B}i\unicode[STIX]{x1D6FD}_{j}^{\unicode[STIX]{x1D712}}}$ ) are eigenvalues of $T_{0}$ (resp. $T_{\infty }$ ). Let $k$ be an integer such that $\unicode[STIX]{x1D712}(\unicode[STIX]{x1D701}_{l})=\unicode[STIX]{x1D701}_{l}^{k}$ for $\unicode[STIX]{x1D701}_{l}\in \unicode[STIX]{x1D707}_{l}$ . Suppose that $k/l,-k/l+\unicode[STIX]{x1D6FD}_{j}^{\unicode[STIX]{x1D712}}\not \in \mathbb{Z}$ and $\unicode[STIX]{x1D6FC}_{1}^{\unicode[STIX]{x1D712}}\in \mathbb{Z}$ . Write $h_{\unicode[STIX]{x1D712}}^{p,2-p}:=\dim _{\overline{\mathbb{Q}}}\text{Gr}_{F}^{p}W_{2}\mathscr{H}(\unicode[STIX]{x1D712})$ . Put

$$\begin{eqnarray}d_{\unicode[STIX]{x1D712}}:=2\{-k/l\}+\mathop{\sum }_{i=1}^{2}\{\unicode[STIX]{x1D6FD}_{i}^{\unicode[STIX]{x1D712}}\}-\{\unicode[STIX]{x1D6FD}_{i}^{\unicode[STIX]{x1D712}}-k/l\}.\end{eqnarray}$$

Then

$$\begin{eqnarray}(h_{\unicode[STIX]{x1D712}}^{2,0},h_{\unicode[STIX]{x1D712}}^{1,1},h_{\unicode[STIX]{x1D712}}^{0,2})=\left\{\begin{array}{@{}ll@{}}(1,1,0)\quad & \text{if }d_{\unicode[STIX]{x1D712}}=2,\\ (0,2,0)\quad & \text{if }d_{\unicode[STIX]{x1D712}}=1,\\ (0,1,1)\quad & \text{if }d_{\unicode[STIX]{x1D712}}=0.\end{array}\right.\end{eqnarray}$$

Note that $d_{\unicode[STIX]{x1D712}}$ takes values only in $0,1$ or $2$ . Indeed

$$\begin{eqnarray}d_{\unicode[STIX]{x1D712}}=\overbrace{\{\unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}}\}+\{-k/l\}-\{\unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}}-k/l\}}^{\unicode[STIX]{x1D6FF}_{1}}+\overbrace{\{\unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}}\}+\{-k/l\}-\{\unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}}-k/l\}}^{\unicode[STIX]{x1D6FF}_{2}}\end{eqnarray}$$

and each $\unicode[STIX]{x1D6FF}_{i}$ is either $0$ or $1$ .

Proof. We first note that $\dim _{E}W_{2}\mathscr{H}(e)=2$ [3, Section 4.3]. We employ two results from [2] and [9], respectively. First of all, it follows from [2, Theorem 4.2] that one has the Hodge numbers of the determinant $D:=\text{det}_{E}W_{2}\mathscr{H}(e)=\bigwedge _{E}^{2}W_{2}\mathscr{H}(e)$ . The result is

$$\begin{eqnarray}(D_{\unicode[STIX]{x1D712}}^{4,0},D_{\unicode[STIX]{x1D712}}^{3,1},D_{\unicode[STIX]{x1D712}}^{2,2},D_{\unicode[STIX]{x1D712}}^{1,3},D_{\unicode[STIX]{x1D712}}^{0,4})=\left\{\begin{array}{@{}ll@{}}(0,1,0,0,0)\quad & \text{if }d_{\unicode[STIX]{x1D712}}=2,\\ (0,0,1,0,0)\quad & \text{if }d_{\unicode[STIX]{x1D712}}=1,\\ (0,0,0,1,0)\quad & \text{if }d_{\unicode[STIX]{x1D712}}=0\end{array}\right.\end{eqnarray}$$

where we put $D_{\unicode[STIX]{x1D712}}^{p,4-p}:=\dim \text{Gr}_{F}^{p}D(\unicode[STIX]{x1D712})$ . Since $D_{\unicode[STIX]{x1D712}}^{p,4-p}=1$ $\Leftrightarrow$ $2h_{\unicode[STIX]{x1D712}}^{2,0}+h_{\unicode[STIX]{x1D712}}^{1,1}=p$ , this implies

$$\begin{eqnarray}(h_{\unicode[STIX]{x1D712}}^{2,0},h_{\unicode[STIX]{x1D712}}^{1,1},h_{\unicode[STIX]{x1D712}}^{0,2})=\left\{\begin{array}{@{}ll@{}}(1,1,0)\quad & \text{if }d_{\unicode[STIX]{x1D712}}=2,\\ (0,2,0)\text{ or }(1,0,1)\quad & \text{if }d_{\unicode[STIX]{x1D712}}=1,\\ (0,1,1)\quad & \text{if }d_{\unicode[STIX]{x1D712}}=0\end{array}\right.\end{eqnarray}$$

which completes the proof in the case $d_{\unicode[STIX]{x1D712}}\neq 1$ . Suppose $d_{\unicode[STIX]{x1D712}}=1$ . We want to show that $(h_{\unicode[STIX]{x1D712}}^{2,0},h_{\unicode[STIX]{x1D712}}^{1,1},h_{\unicode[STIX]{x1D712}}^{0,2})=(1,0,1)$ cannot happen. By [3, Theorem 5.8], the underlying connection of $W_{2}\mathscr{H}(\unicode[STIX]{x1D712})$ is defined by the hypergeometric differential operator as in loc. cit. One can apply the main theorem in [9] and then the possible triplets of the Hodge numbers are at most $(2,0,0),(0,2,0),(0,0,2)$ . In particular, the case $(h_{\unicode[STIX]{x1D712}}^{2,0},h_{\unicode[STIX]{x1D712}}^{1,1},h_{\unicode[STIX]{x1D712}}^{0,2})=(1,0,1)$ is excluded. This completes the proof in case $d_{\unicode[STIX]{x1D712}}=1$ .◻

Remark 3.4. For the latter half of the proof of Theorem 3.3, there is an alternative discussion without using the main theorem of [9]. Let $\unicode[STIX]{x1D70B}_{0}:\mathbb{P}^{1}\rightarrow \mathbb{P}^{1}$ be the map given by $t\mapsto -t^{l}$ . Let $\mathscr{M}_{0}:=\unicode[STIX]{x1D70B}_{0\ast }\mathbb{Q}\otimes R^{1}f_{\ast }\mathbb{Q}$ be a VHdR on $\mathbb{P}^{1}\setminus \{0,1,\infty \}$ . Put $H_{0}:=H^{1}(\mathbb{P}^{1}\setminus \{0,1,\infty \},\mathscr{M}_{0})$ . Let $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D706}=0}$ denote the nearby cycle functor. Then one can construct an injection

of mixed Hodge–de Rham structures. The cohomology group $W_{2}H_{0}(e)$ is studied in detail in [4]. In particular, if $d_{\unicode[STIX]{x1D712}}=1$ , then the Hodge type of $W_{2}H_{0}(\unicode[STIX]{x1D712})$ is $(1,1)$ . Hence $h_{\unicode[STIX]{x1D712}}^{1,1}>0$ by the above injection, which excludes the case $(h_{\unicode[STIX]{x1D712}}^{2,0},h_{\unicode[STIX]{x1D712}}^{1,1},h_{\unicode[STIX]{x1D712}}^{0,2})=(1,0,1)$ .

Corollary 3.5. $W_{2}\mathscr{H}(e)$ is a Tate VHdR of type $(1,1)$ if and only if $d_{\unicode[STIX]{x1D712}}=1$ for all $\unicode[STIX]{x1D712}:R\rightarrow \overline{\mathbb{Q}}$ , equivalently

$$\begin{eqnarray}\displaystyle & 2\{-sk_{0}/l\}+\mathop{\sum }_{i=1}^{2}\{s\unicode[STIX]{x1D6FD}_{i}^{\unicode[STIX]{x1D712}_{0}}\}-\{s(\unicode[STIX]{x1D6FD}_{i}^{\unicode[STIX]{x1D712}_{0}}-k_{0}/l)\}=1 & \displaystyle \nonumber\\ \displaystyle & \Longleftrightarrow \quad \{s\unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}_{0}}\}<\{sk_{0}/l\}<\{s\unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}_{0}}\}\qquad \text{or}\qquad \{s\unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}_{0}}\}<\{sk_{0}/l\}<\{s\unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}_{0}}\} & \displaystyle \nonumber\end{eqnarray}$$

for $\forall s\in \hat{\mathbb{Z}}^{\times }$ where $\unicode[STIX]{x1D712}_{0}$ is a fixed one and $\unicode[STIX]{x1D6FD}_{j}^{\unicode[STIX]{x1D712}_{0}},k_{0}$ are the rational numbers arising from $\unicode[STIX]{x1D712}_{0}$ .

3.4 Beilinson regulator

Let $\unicode[STIX]{x1D713}_{t=1}$ be the nearby cycle functor along the function $t-1$ on $\mathscr{U}$ , and put

$$\begin{eqnarray}C:=\text{Gr}_{2}^{W}\unicode[STIX]{x1D713}_{t=1}\mathscr{M}\cong \unicode[STIX]{x1D70B}_{\ast }\mathbb{Q}|_{\{1\}\times S}\otimes (\text{Gr}_{2}^{W}\unicode[STIX]{x1D713}_{t=1}R^{1}f_{\ast }\mathbb{Q})\end{eqnarray}$$

a VHdR on $S$ . The condition (c) in Section 3.1 implies that the $e$ -part $C(e)$ is of Hodge type $(1,1)$ . Recall from [3, Proposition 4.2] that there is a natural embedding

$$\begin{eqnarray}C(e)\otimes \mathbb{Q}(-1)\longrightarrow \mathscr{H}(e)/W_{2}.\end{eqnarray}$$

This gives a 1-extension

(3.9)

of VMHdR with multiplication by $E$ which is induced from (3.7). Note $C(e)$ is one-dimensional over $E$ and endowed with Hodge type $(1,1)$ by (c) in Section 3.1.

In [3, Section 5] we discussed the extension data of (3.9). More precisely, let $\mathscr{O}^{\text{zar}}$ be the Zariski sheaf of polynomial functions (with coefficients in $\overline{\mathbb{Q}}$ ) on $S=\mathbb{A}_{\overline{\mathbb{Q}}}^{1}\setminus \{0,1\}$ with coordinate $\unicode[STIX]{x1D706}$ . Let $\mathscr{O}^{an}$ be the sheaf of analytic functions on $S^{an}=\mathbb{C}^{an}\setminus \{0,1\}$ . Let $a:S^{an}\rightarrow S^{\text{zar}}$ be the canonical morphism from the analytic site to the Zariski site. Set

$$\begin{eqnarray}\mathscr{J}:=\text{Coker}[a^{-1}F^{2}W_{2}\mathscr{H}_{\text{dR}}\oplus \unicode[STIX]{x1D704}(W_{2}\mathscr{H}_{B})\rightarrow \mathscr{O}^{an}\otimes _{a^{-1}\mathscr{O}^{\text{zar}}}a^{-1}W_{2}\mathscr{H}_{\text{dR}}]\end{eqnarray}$$

a sheaf on the analytic site $\mathbb{C}^{an}\setminus \{0,1\}$ where $\unicode[STIX]{x1D704}:\mathscr{H}_{B}\rightarrow a^{-1}\mathscr{H}_{\text{dR}}$ is the comparison map. Let $h:\widetilde{S}\rightarrow S$ be a generically finite and dominant map such that $\sqrt[l]{\unicode[STIX]{x1D706}-1}\in \overline{\mathbb{Q}}(\widetilde{S})$ . Then $h^{\ast }C(e)$ is a direct sum of copies of the constant VHdR $\mathbb{Q}(-1)$ . The connecting homomorphism arising from (3.9) gives a map

$$\begin{eqnarray}h^{\ast }C(e)\otimes \mathbb{Q}(1)\longrightarrow \text{Ext}_{\text{VMHdR}}^{1}(\mathbb{Q},W_{2}\mathscr{H}(e)\otimes \mathbb{Q}(2))\end{eqnarray}$$

to the Yoneda extension group of VMHdR’s on $S$ where we simply write $h^{\ast }C(e)\otimes \mathbb{Q}(1)=\unicode[STIX]{x1D6E4}(\widetilde{S},h^{\ast }C(e)\otimes \mathbb{Q}(1))$ . Combining this with the Carlson isomorphism (cf. [3, Proposition 2.1]), we have

(3.10) $$\begin{eqnarray}\unicode[STIX]{x1D70C}:h^{\ast }C(e)\otimes \mathbb{Q}(1)\longrightarrow \unicode[STIX]{x1D6E4}(\widetilde{S}^{an},h^{\ast }\mathscr{J}(e)).\end{eqnarray}$$

A down-to-earth description of $\unicode[STIX]{x1D70C}$ is the following. Let $x\in h^{\ast }C(e)\otimes \mathbb{Q}(1)$ . Let $e_{\text{dR},x}\in \mathscr{H}^{\prime }(e)_{\text{dR}}\otimes \mathbb{Q}(2)$ and $e_{B,x}\in \mathscr{H}^{\prime }(e)_{B}\otimes \mathbb{Q}(2)$ be liftings of $x$ . Then $\unicode[STIX]{x1D70C}(x)=\pm (\unicode[STIX]{x1D704}(e_{B,x})-e_{\text{dR},x})$ (see also [3, Section 5.2]).

The map $\unicode[STIX]{x1D70C}$ agrees with the Beilinson regulator map on the motivic cohomology supported on singular fibers up to sign in the following sense. Let $\widetilde{\unicode[STIX]{x1D70B}}:\mathbb{P}_{\widetilde{S}}^{1}:=\mathbb{P}^{1}\times _{\overline{\mathbb{Q}}}\widetilde{S}\rightarrow \mathbb{P}^{1}$ be given by $(s,\unicode[STIX]{x1D706}^{\prime })\mapsto h(\unicode[STIX]{x1D706}^{\prime })-s^{l}$ . Consider the diagram

with $i$ desingularization and $p$ the 2nd projection. Let

$$\begin{eqnarray}\text{reg}:H_{\mathscr{M}}^{3}(X_{\widetilde{S}},\mathbb{Q}(2))\longrightarrow H_{\mathscr{D}}^{3}(X_{\widetilde{S}},\mathbb{Q}(2))=\text{Ext}_{\text{MHM}(X_{\widetilde{S}})}^{3}(\mathbb{Q},\mathbb{Q}(2))\end{eqnarray}$$

be the Beilinson regulator map where $\text{MHM}(\widetilde{S})$ denotes the category of mixed Hodge modules on $\widetilde{S}$ . There is a canonical surjective map

$$\begin{eqnarray}\text{Ext}_{\text{MHM}(X_{\widetilde{S}})}^{3}(\mathbb{Q},\mathbb{Q}(2))\longrightarrow \text{Ext}_{\text{VMHdR}(\widetilde{S})}^{1}(\mathbb{Q},R^{2}g_{\ast }\mathbb{Q}(2)).\end{eqnarray}$$

Let $U_{\widetilde{S}}\subset \mathbb{P}_{\widetilde{S}}^{1}$ be a Zariski open set on which $f_{\widetilde{S}}$ is smooth and projective. Put

$$\begin{eqnarray}H_{\mathscr{M}}^{3}(X_{\widetilde{S}},\mathbb{Q}(2))_{0}:=\text{Ker}[H_{\mathscr{M}}^{3}(X_{\widetilde{S}},\mathbb{Q}(2))\longrightarrow H_{\mathscr{M}}^{3}(f_{\widetilde{S}}^{-1}(U_{\widetilde{S}}),\mathbb{Q}(2))]\end{eqnarray}$$

and $(R^{2}g_{\ast }\mathbb{Q}(2))_{0}:=\text{Ker}[R^{2}g_{\ast }\mathbb{Q}(2)\rightarrow p_{\ast }(R^{2}(f_{\widetilde{S}})_{\ast }\mathbb{Q}(2)|_{U_{\widetilde{S}}})]$ . Then the regulator map induces a map

$$\begin{eqnarray}H_{\mathscr{M}}^{3}(X_{\widetilde{S}},\mathbb{Q}(2))_{0}\longrightarrow \text{Ext}_{\text{VMHdR}(\widetilde{S})}^{1}(\mathbb{Q},(R^{2}g_{\ast }\mathbb{Q}(2))_{0}).\end{eqnarray}$$

Recall from (3.8) that there is a canonical surjective map $(R^{2}g_{\ast }\mathbb{Q}(2))_{0}\rightarrow h^{\ast }W_{2}\mathscr{H}(2)$ . We thus have a composition

$$\begin{eqnarray}\text{reg}_{0}:H_{\mathscr{M}}^{3}(X_{\widetilde{S}},\mathbb{Q}(2))_{0}\longrightarrow \text{Ext}_{\text{VMHdR}(\widetilde{S})}^{1}(\mathbb{Q},h^{\ast }W_{2}\mathscr{H}(2))\longrightarrow \unicode[STIX]{x1D6E4}(\widetilde{S}^{an},h^{\ast }\mathscr{J})\end{eqnarray}$$

of the maps. The compatibility with (3.10) is given by the commutate diagram

(3.11)

where $D_{\widetilde{S}}:=X_{\widetilde{S}}\setminus U_{\widetilde{S}}$ .

3.5 Regulator formula for HG fibrations of Gauss type

One of the main results in [3] (which we call regulator formula) is an explicit description of the map $\unicode[STIX]{x1D70C}$ in (3.10). Here we apply [3, Theorem 5.9] (=a precise version of regulator formula) to the case that $f$ is a HG fibration of Gauss type (see Definition 3.2).

Let $f:X\rightarrow \mathbb{P}^{1}$ be a HG fibration of Gauss type with multiplication by $(R_{0}:=\mathbb{Q}[\unicode[STIX]{x1D707}_{N}],e_{0})$ as in Definition 3.2. Let $\unicode[STIX]{x1D712}:E_{0}\rightarrow \overline{\mathbb{Q}}$ be a homomorphism such that $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D701})=\unicode[STIX]{x1D701}^{-n}$ . Recall from Section 3.2 that $F^{1}H_{\text{dR}}^{1}(X_{t})(\unicode[STIX]{x1D712})$ is one-dimensional and spanned by a 1-form

$$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D714}_{n}:={\displaystyle \frac{x^{a_{n}}(1-x)^{b_{n}}(1-tx)^{c_{n}}}{y^{n}}}\,dx, & \displaystyle \nonumber\\ \displaystyle & a_{n}:=\left\lfloor {\displaystyle \frac{an}{N}}\right\rfloor ,\qquad b_{n}:=\left\lfloor {\displaystyle \frac{bn}{N}}\right\rfloor ,\qquad c_{n}:=\left\lfloor {\displaystyle \frac{Nn-bn}{N}}\right\rfloor =n-b_{n}-1, & \displaystyle \nonumber\end{eqnarray}$$

where $n\in \{1,2,\ldots ,N-1\}$ such that $\unicode[STIX]{x1D712}(\unicode[STIX]{x1D701})=\unicode[STIX]{x1D701}^{-n}$ for $\forall \unicode[STIX]{x1D701}\in \unicode[STIX]{x1D707}_{N}$ .

Lemma 3.6. Let $D_{0},D_{1}$ be the reduced singular fibers over $t=0,1$ . We assume that $D_{0}+D_{1}$ is a normal crossing divisor (abbreviated NCD). Then $t\unicode[STIX]{x1D714}_{n}\in \unicode[STIX]{x1D6E4}(\mathbb{P}^{1}\setminus \{\infty \},f_{\ast }\unicode[STIX]{x1D6FA}_{X/\mathbb{P}^{1}}^{1}(\log D_{0}+D_{1}))$ .

Proof. Put $S=\mathbb{P}^{1}\setminus \{0,1,\infty \}$ and $U=f^{-1}(S)$ . Let $\mathscr{V}:=H_{\text{dR}}^{1}(U/S)$ be the bundle and $\unicode[STIX]{x1D6FB}:\mathscr{V}\rightarrow \unicode[STIX]{x1D6FA}_{S}^{1}\otimes \mathscr{V}$ the Gauss–Manin connection. Let $D_{\infty }$ be the reduced singular fibers over $t=\infty$ and assume that it is a NCD. Put $T:=\{0,1,\infty \}$ . Recall that the sheaf $\unicode[STIX]{x1D6FA}_{X/\mathbb{P}^{1}}^{1}(\log D)$ ( $D:=D_{0}+D_{1}+D_{\infty }$ ) is defined by the exact sequence

$$\begin{eqnarray}0\longrightarrow f^{\ast }\unicode[STIX]{x1D6FA}_{\mathbb{P}^{1}}^{1}(\log T)\longrightarrow \unicode[STIX]{x1D6FA}_{X}^{1}(\log D)\longrightarrow \unicode[STIX]{x1D6FA}_{X/\mathbb{P}^{1}}^{1}(\log D)\longrightarrow 0.\end{eqnarray}$$

Let $\mathscr{V}_{e}$ be Deligne’s canonical extension over $\mathbb{P}^{1}$ . This is characterized as the subbundle $\mathscr{V}_{e}\subset j_{\ast }\mathscr{V}$ ( $j:S{\hookrightarrow}\mathbb{P}^{1}$ ) which satisfies

• ∙ $\unicode[STIX]{x1D6FB}$ has at most log poles, $\unicode[STIX]{x1D6FB}:\mathscr{V}_{e}\rightarrow \unicode[STIX]{x1D6FA}_{\mathbb{P}^{1}}^{1}(\log (0+1+\infty ))\otimes \mathscr{V}_{e}$ ,

• ∙ The eigenvalues of residue $\text{Res}(\unicode[STIX]{x1D6FB})$ at $t=0,1,\infty$ belong to $[0,1)$ .

Then there is an isomorphism

$$\begin{eqnarray}\mathscr{V}_{e}\cong R^{1}f_{\ast }\unicode[STIX]{x1D6FA}_{X/\mathbb{P}^{1}}^{\bullet }(\log D)\end{eqnarray}$$

[12, 2.20] and $F^{1}\mathscr{V}_{e}:=\mathscr{V}_{e}\cap j_{\ast }F^{1}\mathscr{V}\cong f_{\ast }\unicode[STIX]{x1D6FA}_{X/\mathbb{P}^{1}}^{1}(\log D)$ (loc. cit. 4.20 (ii)). Hence the desired assertion is equivalent to

(3.12) $$\begin{eqnarray}t\unicode[STIX]{x1D714}_{n}\in \unicode[STIX]{x1D6E4}(\mathbb{P}^{1}\setminus \{\infty \},\mathscr{V}_{e}).\end{eqnarray}$$

To show this, we give a local frame of $\mathscr{V}_{e}$ at $t=0,1$ explicitly. Let

$$\begin{eqnarray}\unicode[STIX]{x1D702}_{n}:=\frac{x^{a_{n}}(1-x)^{b_{n}+1}(1-tx)^{c_{n}}}{y^{n}}\,dx,\end{eqnarray}$$

and put

$$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}}:=\left\{\frac{-an}{N}\right\},\qquad \unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}}:=\left\{\frac{-bn}{N}\right\}.\end{eqnarray}$$

Recall from (3.2) a homology cycle $\unicode[STIX]{x1D6FF}:=(1-\unicode[STIX]{x1D70E}^{d_{1}})(1-\unicode[STIX]{x1D70E}^{d_{2}})u\in H_{1}(X_{t},\mathbb{Z})$ . Then

(3.13) $$\begin{eqnarray}\int _{\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D714}_{n}=(1-\unicode[STIX]{x1D701}^{-nd_{1}})(1-\unicode[STIX]{x1D701}^{-nd_{2}})B(\unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}},\unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}})_{2}F_{1}(\unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}},\unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}},\unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}}+\unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}};t),\end{eqnarray}$$

(3.14) $$\begin{eqnarray}\int _{\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D702}_{n}=(1-\unicode[STIX]{x1D701}^{-nd_{1}})(1-\unicode[STIX]{x1D701}^{-nd_{2}})B(\unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}},\unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}}+1)_{2}F_{1}(\unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}},\unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}},1+\unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}}+\unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}};t).\end{eqnarray}$$

This shows that $\unicode[STIX]{x1D714}_{n}$ and $\unicode[STIX]{x1D702}_{n}$ are basis of the $\unicode[STIX]{x1D712}$ -part $\mathscr{V}(\unicode[STIX]{x1D712})$ of the bundle (over a Zariski open set of $\mathbb{P}^{1}\setminus \{0,1,\infty \}$ ). Denote by $\mathscr{V}(\unicode[STIX]{x1D712})^{\ast }$ the dual connection, and by $\{\unicode[STIX]{x1D714}_{n}^{\ast },\unicode[STIX]{x1D702}_{n}^{\ast }\}$ the dual basis. Then

$$\begin{eqnarray}\left(\int _{\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D714}_{n}\right)\unicode[STIX]{x1D714}_{n}^{\ast }+\left(\int _{\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D702}_{n}\right)\unicode[STIX]{x1D702}_{n}^{\ast }\end{eqnarray}$$

is annihilated by the dual connection, and hence

(3.15) $$\begin{eqnarray}d\left(\int _{\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D714}_{n}\right)\unicode[STIX]{x1D714}_{n}^{\ast }+d\left(\int _{\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D702}_{n}\right)\unicode[STIX]{x1D702}_{n}^{\ast }+\left(\int _{\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D714}_{n}\right)\unicode[STIX]{x1D6FB}(\unicode[STIX]{x1D714}_{n}^{\ast })+\left(\int _{\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D702}_{n}\right)\unicode[STIX]{x1D6FB}(\unicode[STIX]{x1D702}_{n}^{\ast })=0.\end{eqnarray}$$

Now (3.13)–(3.15) together with the formulas

$$\begin{eqnarray}(1-t)\frac{d}{dt}_{2}F_{1}(a,b,a+b;t)=\frac{ab}{a+b}_{2}F_{1}(a,b,a+b+1;t),\end{eqnarray}$$

$$\begin{eqnarray}\displaystyle & & \displaystyle t\frac{d}{dt}_{2}F_{1}(a,b,a+b+1;t)\nonumber\\ \displaystyle & & \displaystyle \quad =(a+b)(\,_{2}F_{1}(a,b,a+b;t)-_{2}F_{1}(a,b,a+b+1;t))\nonumber\end{eqnarray}$$

imply

$$\begin{eqnarray}\displaystyle & (\unicode[STIX]{x1D6FB}(\unicode[STIX]{x1D714}_{n}^{\ast }),\unicode[STIX]{x1D6FB}(\unicode[STIX]{x1D702}_{n}^{\ast }))=dt\otimes (\unicode[STIX]{x1D714}_{n}^{\ast },\unicode[STIX]{x1D702}_{n}^{\ast })\left(\begin{array}{@{}cc@{}}0 & -\unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}}/(1-t)\\ -\unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}}/t & (\unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}}+\unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}})/t\end{array}\right) & \displaystyle \nonumber\\ \displaystyle & \Longleftrightarrow \quad (\unicode[STIX]{x1D6FB}(\unicode[STIX]{x1D714}_{n}),\unicode[STIX]{x1D6FB}(\unicode[STIX]{x1D702}_{n}))=dt\otimes (\unicode[STIX]{x1D714}_{n},\unicode[STIX]{x1D702}_{n})\left(\begin{array}{@{}cc@{}}0 & \unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}}/t\\ \unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}}/(1-t) & -(\unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}}+\unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}})/t\end{array}\right). & \displaystyle \nonumber\end{eqnarray}$$

Then it is an elementary linear algebra to compute local frames of $\mathscr{V}_{e}$ :

$$\begin{eqnarray}\displaystyle & \mathscr{V}_{e}(\unicode[STIX]{x1D712})|_{t=0}=\left\{\begin{array}{@{}ll@{}}\langle \unicode[STIX]{x1D714}_{n},t(\unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}}\unicode[STIX]{x1D714}_{n}+(\unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}}+\unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}})\unicode[STIX]{x1D702}_{n})\rangle \quad & \unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}}+\unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}}\leqslant 1,\\ \langle t\unicode[STIX]{x1D714}_{n},(\unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}}+\unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}}-1)\unicode[STIX]{x1D714}_{n}+t\unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}}\unicode[STIX]{x1D702}_{n}\rangle \quad & \unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}}+\unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}}>1,\end{array}\right. & \displaystyle \nonumber\\ \displaystyle & \mathscr{V}_{e}(\unicode[STIX]{x1D712})|_{t=1}=\langle \unicode[STIX]{x1D714}_{n},\unicode[STIX]{x1D702}_{n}\rangle . & \displaystyle \nonumber\end{eqnarray}$$

Now (3.12) is immediate.◻

Let $e_{0}:\unicode[STIX]{x1D707}_{N}\rightarrow E_{0}^{\times }$ be an injective homomorphism. Then the condition in Lemma 3.1 is satisfied. Let $e:R:=\mathbb{Q}[\unicode[STIX]{x1D707}_{l},\unicode[STIX]{x1D707}_{N}]\rightarrow E$ be a projection such that $\text{Ker}(e)\supset \text{Ker}(e_{0})$ . Let $\unicode[STIX]{x1D712}:R\rightarrow \overline{\mathbb{Q}}$ be a homomorphism factoring through $e$ . Fix integers $k,n$ such that

$$\begin{eqnarray}\unicode[STIX]{x1D712}(\unicode[STIX]{x1D701}_{1},\unicode[STIX]{x1D701}_{2})=\unicode[STIX]{x1D701}_{1}^{k}\unicode[STIX]{x1D701}_{2}^{n},\quad \forall (\unicode[STIX]{x1D701}_{1},\unicode[STIX]{x1D701}_{2})\in \unicode[STIX]{x1D707}_{l}\times \unicode[STIX]{x1D707}_{N}.\end{eqnarray}$$

Note $\text{gcd}(n,N)=1$ as $e_{0}:\unicode[STIX]{x1D707}_{N}\rightarrow E_{0}^{\times }$ is injective. Let

(3.16) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}}:=\left\{\frac{-na}{N}\right\},\qquad \unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}}:=\left\{\frac{-nb}{N}\right\},\qquad \unicode[STIX]{x1D6FC}_{1}^{\unicode[STIX]{x1D712}}:=0,\qquad \unicode[STIX]{x1D6FC}_{2}^{\unicode[STIX]{x1D712}}:=1-\unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}}-\unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}}\end{eqnarray}$$

which do not depend on the choice of $n$ . Then $e^{2\unicode[STIX]{x1D70B}i\unicode[STIX]{x1D6FC}_{j}^{\unicode[STIX]{x1D712}}}$ (resp. $e^{2\unicode[STIX]{x1D70B}i\unicode[STIX]{x1D6FD}_{j}^{\unicode[STIX]{x1D712}}}$ ) are eigenvalues of the local monodromy $T_{0}$ at $t=0$ (resp. $T_{\infty }$ at $t=\infty$ ) on $R^{1}f_{\ast }\mathbb{C}(\unicode[STIX]{x1D712})\cong \mathbb{C}^{2}$ (see (3.5)). The relative 1-form $\unicode[STIX]{x1D714}:=t\unicode[STIX]{x1D714}_{n}$ satisfies the conditions (P1), (P2) in [3, Section 4.5]:

1. (P1) $\int _{\unicode[STIX]{x1D6FE}_{t}}\unicode[STIX]{x1D714}(\unicode[STIX]{x1D6FE}_{t}\in H_{1}(X_{t}))$ is spanned by $t_{2}F_{1}(\unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}},\unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}},1;1-t)$ and $t_{2}F_{1}(\unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}},\unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}},\unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}}+\unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}};t)$ . (This follows from (3.4).)

2. (P2) $\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6E4}(\mathbb{P}^{1}\setminus \{\infty \},f_{\ast }\unicode[STIX]{x1D6FA}_{X/\mathbb{P}^{1}}^{1}(\log D))$ . (This is Lemma 3.6.)

We thus can apply the regulator formula [3, Theorem 5.9]. In our particular case, it is stated as follows (the notation is slightly changed for the use in below).

Theorem 3.7. Let $e_{0},e,\unicode[STIX]{x1D712}$ be as above, and let $\unicode[STIX]{x1D6FC}_{i}^{\unicode[STIX]{x1D712}}$ , $\unicode[STIX]{x1D6FD}_{j}^{\unicode[STIX]{x1D712}}$ be as in (3.16). Assume that $k/l,k/l-\unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}},k/l-\unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}},k/l-\unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}}-\unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}}\not \in \mathbb{Z}$ . Put

$$\begin{eqnarray}\displaystyle \mathscr{F}_{1}(\unicode[STIX]{x1D706}) & := & \displaystyle (1-\unicode[STIX]{x1D706})^{k/l-1}~_{3}F_{2}{1,1,1-k/l\atop 2-\unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}},2-\unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}}};(1-\unicode[STIX]{x1D706})^{-1},\nonumber\\ \displaystyle \mathscr{F}_{2}(\unicode[STIX]{x1D706}) & := & \displaystyle (1-\unicode[STIX]{x1D706})^{k/l-1}~_{3}F_{2}{1,1,2-k/l\atop 2-\unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}},2-\unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}}};(1-\unicode[STIX]{x1D706})^{-1}.\nonumber\end{eqnarray}$$

Let $\unicode[STIX]{x1D70C}(\!\,^{t}\unicode[STIX]{x1D712})$ be the $^{t}\unicode[STIX]{x1D712}$ -part of the map $\unicode[STIX]{x1D70C}$ in (3.10). Let

$$\begin{eqnarray}\unicode[STIX]{x1D70C}(\!\,^{t}\unicode[STIX]{x1D712})=(\unicode[STIX]{x1D719}_{1}(\unicode[STIX]{x1D706}),\unicode[STIX]{x1D719}_{2}(\unicode[STIX]{x1D706}))\in (\mathscr{O}^{an})^{\oplus 2}\cong \mathscr{O}^{an}\otimes W_{2}\mathscr{H}_{\text{dR}}(\!\,^{t}\unicode[STIX]{x1D712})\end{eqnarray}$$

be a local lifting where the above isomorphism is with respect to $\overline{\mathbb{Q}}$ -frame of $W_{2}\mathscr{H}_{\text{dR}}(\!\,^{t}\unicode[STIX]{x1D712})$ . Define rational functions $E_{i}^{(r)}=E_{i}^{(r)}(\unicode[STIX]{x1D706})\in \mathbb{Q}(\unicode[STIX]{x1D706})$ for $r\in \mathbb{Z}_{{\geqslant}-1}$ in the following way. Write $a:=2-\unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}}$ , $b:=2-\unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}}$ . Put

$$\begin{eqnarray}\displaystyle & & \displaystyle A(s):=\frac{s(a+b+2s-3-s(1-\unicode[STIX]{x1D706})^{-1})}{(a+s-1)(b+s-1)},\nonumber\\ \displaystyle & & \displaystyle \qquad B(s):=\frac{s(1-s)(1-(1-\unicode[STIX]{x1D706})^{-1})}{(a+s-1)(b+s-1)}.\nonumber\end{eqnarray}$$

Define $C_{i}(s)$ and $D_{i}(s)$ by

$$\begin{eqnarray}\left(\begin{array}{@{}c@{}}C_{i+1}(s)\\ D_{i+1}(s)\end{array}\right)=\left(\begin{array}{@{}cc@{}}A(s) & 1\\ B(s) & 0\end{array}\right)\left(\begin{array}{@{}c@{}}C_{i}(s+1)\\ D_{i}(s+1)\end{array}\right),\qquad \left(\begin{array}{@{}c@{}}C_{-1}(s)\\ D_{-1}(s)\end{array}\right):=\left(\begin{array}{@{}c@{}}0\\ 1\end{array}\right),\end{eqnarray}$$

and define $E_{i}^{(r)}$ by

(3.17) $$\begin{eqnarray}\displaystyle & & \displaystyle E_{1}^{(r)}=\unicode[STIX]{x1D706}C_{r}(k/l)+(1-\unicode[STIX]{x1D706})C_{r+1}(k/l),\nonumber\\ \displaystyle & & \displaystyle \qquad E_{2}^{(r)}=\unicode[STIX]{x1D706}D_{r}(k/l)+(1-\unicode[STIX]{x1D706})D_{r+1}(k/l).\end{eqnarray}$$

Then for infinitely many integers $r>0$ , we have

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}_{1}(\unicode[STIX]{x1D706}) & \equiv & \displaystyle C_{1}(1-\unicode[STIX]{x1D706})^{r}[E_{1}^{(r)}(\unicode[STIX]{x1D706})\mathscr{F}_{1}(\unicode[STIX]{x1D706})+E_{2}^{(r)}(\unicode[STIX]{x1D706})\mathscr{F}_{2}(\unicode[STIX]{x1D706})],\nonumber\\ \displaystyle \unicode[STIX]{x1D719}_{2}(\unicode[STIX]{x1D706}) & \equiv & \displaystyle C_{2}(1-\unicode[STIX]{x1D706})^{r-1}[E_{1}^{(r-1)}(\unicode[STIX]{x1D706})\mathscr{F}_{1}(\unicode[STIX]{x1D706})+E_{2}^{(r-1)}(\unicode[STIX]{x1D706})\mathscr{F}_{2}(\unicode[STIX]{x1D706})]\nonumber\end{eqnarray}$$

modulo $\overline{\mathbb{Q}(\unicode[STIX]{x1D706})}$ with some $C_{1},C_{2}\in \overline{\mathbb{Q}}^{\times }$ .

We note that $(N,l,k,n,a,b)$ in Theorem 3.7 can run over the set of all 6-tuples of integers satisfying

• ∙ $0<a,b<N$ , $\gcd (N,a,b)=1$ and $\gcd (n,N)=1$ ,

• ∙ $k/l,k/l-\unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}},k/l-\unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}},k/l-\unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}}-\unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}}\not \in \mathbb{Z}$ (see (3.16) for definition of $\unicode[STIX]{x1D6FD}_{j}^{\unicode[STIX]{x1D712}}$ ).

4 Proof of main theorem

We are now in a position to prove Theorem 2.1 (log formula).

There are the following formulas

$$\begin{eqnarray}\displaystyle (b_{1}-1)_{3}F_{2}{a_{1},a_{2},a_{3}\atop b_{1}-1,b_{2}};x & = & \displaystyle \left(b_{1}-1+x\frac{d}{dx}\right)_{3}F_{2}{a_{1},a_{2},a_{3}\atop b_{1},b_{2}};x\!,\nonumber\\ \displaystyle {a_{1}\cdot \,}_{3}F_{2}{a_{1}+1,a_{2},a_{3}\atop b_{1},b_{2}};x & = & \displaystyle \left(a_{1}+x\frac{d}{dx}\right)_{3}F_{2}{a_{1},a_{2},a_{3}\atop b_{1},b_{2}};x\!,\nonumber\end{eqnarray}$$

$$\begin{eqnarray}\displaystyle (a_{2}-b_{1})(a_{1}-b_{1})(a_{3}-b_{1})_{3}F_{2}{a_{1},a_{2},a_{3}\atop b_{1}+1,b_{2}};x & = & \displaystyle \unicode[STIX]{x1D703}_{1}\left(\,_{3}F_{2}{a_{1},a_{2},a_{3}\atop b_{1},b_{2}};x\!\right)\!,\nonumber\\ \displaystyle (a_{1}-b_{1})(a_{1}-b_{2})_{3}F_{2}{a_{1}-1,a_{2},a_{3}\atop b_{1},b_{2}};x & = & \displaystyle \unicode[STIX]{x1D703}_{2}\left(\,_{3}F_{2}{a_{1},a_{2},a_{3}\atop b_{1},b_{2}};x\!\right)\!,\nonumber\end{eqnarray}$$

where

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}_{1} & := & \displaystyle -a_{1}a_{2}a_{3}+(a_{2}-b_{1})(a_{1}-b_{1})(a_{3}-b_{1})\nonumber\\ \displaystyle & & \displaystyle +\,b_{1}(b_{2}+(b_{1}-a_{1}-a_{2}-a_{3}-1)x)\frac{d}{dx}+b_{1}(x-x^{2})\frac{d^{2}}{dx^{2}}\nonumber\\ \displaystyle \unicode[STIX]{x1D703}_{2} & := & \displaystyle (a_{1}-b_{1})(a_{1}-b_{2})-a_{2}a_{3}x\nonumber\\ \displaystyle & & \displaystyle +\,((b_{1}+b_{2}-a_{1})-(a_{2}+a_{3}+1)x)x\frac{d}{dx}+(1-x)x^{2}\frac{d^{2}}{dx^{2}}.\nonumber\end{eqnarray}$$

Therefore if one can prove the log formula for $_{3}F_{2}({1,1,q\atop a,~b};x)$ then one immediately has the log formula for $_{3}F_{2}({n_{1},n_{2},q+n_{3}\atop a+n_{4},~b+n_{5}};x)$ for arbitrary integers $n_{1},n_{2}>0$ and $n_{3},n_{4},n_{5}\in \mathbb{Z}$ .

We keep the setting and the notation in Section 3.5. Suppose that

(4.1) $$\begin{eqnarray}1=2\{-sk/l\}+\mathop{\sum }_{i=1}^{2}\{s\unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}}\}-\{s(\unicode[STIX]{x1D6FD}_{i}^{\unicode[STIX]{x1D712}}-k/l)\},\quad \forall s\in \hat{\mathbb{Z}}^{\times }.\end{eqnarray}$$

Then it follows from Corollary 3.5 that $W_{2}\mathscr{H}(e)$ is a Tate HdR structure of type $(1,1)$ . Let us look at the map $\unicode[STIX]{x1D70C}(\!\,^{t}\unicode[STIX]{x1D712})$ in Theorem 3.7. This turns out to be the Beilinson regulator by the diagram (3.11). Since $W_{2}\mathscr{H}(e)$ is Tate, it is generated by the divisor classes of the geometric generic fiber $X_{\overline{\unicode[STIX]{x1D702}}}$ of $f_{\widetilde{S}}$ . This implies that the image of $\text{reg}_{0}$ in (3.11) is generated by the images of $H_{\mathscr{M}}^{1}(\widetilde{D}_{i},\mathbb{Q}(1))$ where $D_{i}$ runs over the generators of the Neron–Severi group $\text{NS}(X_{\overline{\unicode[STIX]{x1D702}}})\otimes \mathbb{Q}$ and $\widetilde{D}_{i}\rightarrow D_{i}$ is the desingularization. As is well known, $H_{\mathscr{M}}^{1}(\widetilde{D}_{i},\mathbb{Q}(1))\cong \overline{\unicode[STIX]{x1D702}}^{\times }\otimes \mathbb{Q}$ as $\widetilde{D}_{i}$ is smooth projective, and the Beilinson regulator on it is given by the logarithmic function. Therefore we have

(4.2) $$\begin{eqnarray}\unicode[STIX]{x1D719}_{1}(\unicode[STIX]{x1D706}),\,\unicode[STIX]{x1D719}_{2}(\unicode[STIX]{x1D706})\in \overline{\mathbb{Q}(\unicode[STIX]{x1D706})}+\overline{\mathbb{Q}(\unicode[STIX]{x1D706})}\log \overline{\mathbb{Q}(\unicode[STIX]{x1D706})}^{\times }.\end{eqnarray}$$

We now apply Theorem 3.7. If one can show that

$$\begin{eqnarray}\left|\begin{array}{@{}cc@{}}E_{1}^{(r)} & E_{2}^{(r)}\\ E_{1}^{(r-1)} & E_{2}^{(r-1)}\end{array}\right|\neq 0\end{eqnarray}$$

for almost all $r>0$ , then we have $\mathscr{F}_{i}(\unicode[STIX]{x1D706})\in \overline{\mathbb{Q}(\unicode[STIX]{x1D706})}+\overline{\mathbb{Q}(\unicode[STIX]{x1D706})}\log \overline{\mathbb{Q}(\unicode[STIX]{x1D706})}^{\times }$ , which would finish the proof of Theorem 2.1. To do this, recall (3.17). Letting

$$\begin{eqnarray}\displaystyle & & \displaystyle E_{1}^{(r)}(s):=\unicode[STIX]{x1D706}C_{r}(s)+(1-\unicode[STIX]{x1D706})C_{r+1}(s),\nonumber\\ \displaystyle & & \displaystyle \qquad E_{2}^{(r)}(s):=\unicode[STIX]{x1D706}D_{r}(s)+(1-\unicode[STIX]{x1D706})D_{r+1}(s),\nonumber\end{eqnarray}$$

we want to show

(4.3) $$\begin{eqnarray}\left|\begin{array}{@{}cc@{}}E_{1}^{(r)}(k/l) & E_{2}^{(r)}(k/l)\\ E_{1}^{(r-1)}(k/l) & E_{2}^{(r-1)}(k/l)\end{array}\right|\neq 0\end{eqnarray}$$

for almost all $r>0$ . Since

$$\begin{eqnarray}\left(\begin{array}{@{}cc@{}}E_{1}^{(r+1)}(s) & E_{1}^{(r)}(s)\\ E_{2}^{(r+1)}(s) & E_{2}^{(r)}(s)\end{array}\right)=\left(\begin{array}{@{}cc@{}}A(s) & 1\\ B(s) & 0\end{array}\right)\left(\begin{array}{@{}cc@{}}E_{1}^{(r)}(s+1) & E_{1}^{(r-1)}(s+1)\\ E_{2}^{(r)}(s+1) & E_{2}^{(r-1)}(s+1)\end{array}\right)\end{eqnarray}$$

(4.3) is reduced to showing that

$$\begin{eqnarray}\left|\begin{array}{@{}cc@{}}E_{1}^{(0)}(k/l+r) & E_{2}^{(0)}(k/l+r)\\ E_{1}^{(-1)}(k/l+r) & E_{2}^{(-1)}(k/l+r)\end{array}\right|\neq 0\end{eqnarray}$$

for all integers $r$ . However, this follows from

$$\begin{eqnarray}\displaystyle \left|\begin{array}{@{}cc@{}}E_{1}^{(0)}(s) & E_{2}^{(0)}(s)\\ E_{1}^{(-1)}(s) & E_{2}^{(-1)}(s)\end{array}\right| & = & \displaystyle \left|\begin{array}{@{}cc@{}}\unicode[STIX]{x1D706}+(1-\unicode[STIX]{x1D706})A(s) & (1-\unicode[STIX]{x1D706})B(s)\\ 1-\unicode[STIX]{x1D706} & \unicode[STIX]{x1D706}\end{array}\right|\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D706}\frac{(a-1)(b-1)\unicode[STIX]{x1D706}+s(a+b-2)}{(s+a-1)(s+b-1)},\nonumber\\ \displaystyle & & \displaystyle (a:=2-\unicode[STIX]{x1D6FD}_{1}^{\unicode[STIX]{x1D712}},b:=2-\unicode[STIX]{x1D6FD}_{2}^{\unicode[STIX]{x1D712}})\nonumber\end{eqnarray}$$

and the fact $\unicode[STIX]{x1D6FD}_{i}^{\unicode[STIX]{x1D712}}\not \in \mathbb{Z}$ (see (3.16)) and $k/l-\unicode[STIX]{x1D6FD}_{i}^{\unicode[STIX]{x1D712}}\not \in \mathbb{Z}$ as is assumed. This completes the proof of Theorem 2.1.