Intersection numbers and a period integral associated to a family of K3 surfaces

Data

1. shiga3g2-imat45.rr : Intersection matrix in the format of the computer algebra system Risa/Asir .
2. tperiod.m , fancy.m : Twisted period relation in the format of Mathematica. The relation [with the truncation at degree 2] is stored in the variable ans of tperiod.m.

References

1. An algorithm of computing cohomology intersection number of hypergeometric integrals , (paper).
2. Rational function solution and intersection numbers , (programs, more data).

 Last update: March 27, 2020

The small parameter $\varepsilon$ in the paper is $b$ below. $$\pmatrix{ \frac{ - 32} { ( {b}- 4) ( {b}+ 4)}& \frac{ ( 4) ( 4 {b}- 1)} { ( {b}- 4) ( {b}+ 4) ( {x}_{5})}& 0& \frac{ ( - 4) ( {b}+ 1) ( 4 {b}- 1)} { ( {b}- 4) ( {b}+ 4) ( {x}_{5})^{ 2} } \cr \frac{ ( - 4) ( 4 {b}- 1)} { ( {b}- 4) ( {b}+ 4) ( {x}_{5})}& \frac{ ( 4) ( {b}) ( 4 {b}- 1)} { ( {b}- 4) ( {b}+ 4) ( {x}_{5})^{ 2} }& 0& \frac{ ( - 1) ( 4 {b}- 1) ( ( 64 {b}^{ 2} + 48 {b}- 4) {x}_{5}^{ 2} + ( ( - 32 {b}^{ 2} - 36 {b}- 1) {x}_{4}^{ 2} - 128 {b}^{ 2} - 112 {b}+ 4) {x}_{5}+ ( 4 {b}^{ 2} + 4 {b}) {x}_{4}^{ 4} + ( - 32 {b}^{ 2} - 32 {b}) {x}_{4}^{ 2} + 64 {b}^{ 2} + 64 {b})} { ( {b}- 4) ( {b}+ 4) ( {x}_{5})^{ 3} ( 4 {x}_{5}- {x}_{4}^{ 2} - 4 {x}_{4}- 4) ( 4 {x}_{5}- {x}_{4}^{ 2} + 4 {x}_{4}- 4)} \cr 0& 0& 0& \frac{ ( - 4) ( 4 {b}- 1) ( 4 {b}+ 1) ( {x}_{4})} { ( {b}- 4) ( {b}+ 4) ( {x}_{5}) ( 4 {x}_{5}- {x}_{4}^{ 2} - 4 {x}_{4}- 4) ( 4 {x}_{5}- {x}_{4}^{ 2} + 4 {x}_{4}- 4)} \cr \frac{ ( - 4) ( {b}- 1) ( 4 {b}- 1)} { ( {b}- 4) ( {b}+ 4) ( {x}_{5})^{ 2} }& \frac{ ( 4 {b}- 1) ( ( 64 {b}^{ 2} - 80 {b}- 4) {x}_{5}^{ 2} + ( ( - 32 {b}^{ 2} + 28 {b}- 1) {x}_{4}^{ 2} - 128 {b}^{ 2} + 144 {b}+ 4) {x}_{5}+ ( 4 {b}^{ 2} - 4 {b}) {x}_{4}^{ 4} + ( - 32 {b}^{ 2} + 32 {b}) {x}_{4}^{ 2} + 64 {b}^{ 2} - 64 {b})} { ( {b}- 4) ( {b}+ 4) ( {x}_{5})^{ 3} ( 4 {x}_{5}- {x}_{4}^{ 2} - 4 {x}_{4}- 4) ( 4 {x}_{5}- {x}_{4}^{ 2} + 4 {x}_{4}- 4)}& \frac{ ( 4) ( 4 {b}- 1) ( 4 {b}+ 1) ( {x}_{4})} { ( {b}- 4) ( {b}+ 4) ( {x}_{5}) ( 4 {x}_{5}- {x}_{4}^{ 2} - 4 {x}_{4}- 4) ( 4 {x}_{5}- {x}_{4}^{ 2} + 4 {x}_{4}- 4)}& \frac{ ( - 2) ( 4 {b}- 1) ( ( 512 {b}^{ 3} - 256 {b}^{ 2} - 512 {b}+ 16) {x}_{5}^{ 4} + ( ( - 512 {b}^{ 3} + 64 {b}^{ 2} + 560 {b}+ 8) {x}_{4}^{ 2} - 2048 {b}^{ 3} + 768 {b}^{ 2} + 2112 {b}- 32) {x}_{5}^{ 3} + ( ( 192 {b}^{ 3} + 16 {b}^{ 2} - 200 {b}- 3) {x}_{4}^{ 4} + ( 512 {b}^{ 3} + 128 {b}^{ 2} - 448 {b}+ 8) {x}_{4}^{ 2} + 3072 {b}^{ 3} - 768 {b}^{ 2} - 3200 {b}+ 16) {x}_{5}^{ 2} + ( ( - 32 {b}^{ 3} - 4 {b}^{ 2} + 31 {b}) {x}_{4}^{ 6} + ( 128 {b}^{ 3} + 48 {b}^{ 2} - 116 {b}) {x}_{4}^{ 4} + ( 512 {b}^{ 3} - 192 {b}^{ 2} - 560 {b}) {x}_{4}^{ 2} - 2048 {b}^{ 3} + 256 {b}^{ 2} + 2112 {b}) {x}_{5}+ ( 2 {b}^{ 3} - 2 {b}) {x}_{4}^{ 8} + ( - 32 {b}^{ 3} + 32 {b}) {x}_{4}^{ 6} + ( 192 {b}^{ 3} - 192 {b}) {x}_{4}^{ 4} + ( - 512 {b}^{ 3} + 512 {b}) {x}_{4}^{ 2} + 512 {b}^{ 3} - 512 {b})} { ( {b}- 4) ( {b}+ 4) ( {x}_{5})^{ 4} ( ( 4 {x}_{5}- {x}_{4}^{ 2} - 4 {x}_{4}- 4))^{ 2} ( ( 4 {x}_{5}- {x}_{4}^{ 2} + 4 {x}_{4}- 4))^{ 2} } \cr }$$
We put $\alpha=\frac{\pi^3}{\sin^2\pi\varepsilon\cos\pi(2\varepsilon)}$ and $\beta=-\frac{\pi^3}{\sin^2\pi\varepsilon}$. As to the definition of $\Phi_\varepsilon$ below, see the paper above. $$\Phi_\varepsilon \begin{pmatrix} \alpha&0&0&0\\ 0&\beta&0&0\\ 0&0&\beta&0\\ 0&0&0&\alpha \end{pmatrix} {}^t\Phi^\vee_\varepsilon =\frac{64}{(1-16\varepsilon^2)}.$$ We introduce a $4\times 4$ matrix $Q$ defined by $$Q= \begin{pmatrix} 1&-\varepsilon^{-1}&-\varepsilon^{-1}&\varepsilon^{-2}\\ 0&\varepsilon^{-1}&0&-\varepsilon^{-2}\\ 0&0&\varepsilon^{-1}&-\varepsilon^{-2}\\ 0&0&0&\varepsilon^{-2} \end{pmatrix}.$$ We put $\tilde{\Phi}_\varepsilon=\Phi_\varepsilon Q$ and $\tilde{\Phi}^\vee_\varepsilon=\Phi^\vee_\varepsilon Q$. A simple computation shows that $\tilde{\Phi}=\tilde{\Phi}_\varepsilon\restriction_{\varepsilon\rightarrow 0}$ and $\tilde{\Phi}^\vee_\varepsilon=\tilde{\Phi}^\vee\restriction_{\varepsilon\rightarrow 0}$ are convergent and we have a limit formula $$\tilde{\Phi} \begin{pmatrix} 4\pi^3&0&0&\pi\\ 0&0&\pi&0\\ 0&\pi&0&0\\ \pi&0&0&0 \end{pmatrix} {}^t\tilde{\Phi}^\vee =64.$$ We present the explicit form of the series in this formula in the following two pages. These are obtained by Mathematica programs tperiod.m and fancy.m from our formula. Let $\gamma$ be the Euler constant and $\psi^{(k)}(z) = d^{k+1} \log \Gamma(z)/dz^{k+1}$ be the polygamma function. The elements of the vector ${\tilde \Phi}$ are as follows. $${\tilde \Phi}_1= \frac{1}{\sqrt{\pi } \sqrt{z_4}} \left(1+\frac{3}{2 z_4^2}+O\left(\left(\frac{1}{z_4}\right)^4\right) \right)$$ \begin{eqnarray} {\tilde \Phi}_2 &=& {\tilde \Phi}_3 = \frac{1}{\sqrt{\pi } \sqrt{z_4}} (\phi'_{20} + (\log z_4) \phi'_{21}) \end{eqnarray} where \begin{eqnarray} \phi'_{20}&=&\left(\frac{364288}{45045}-2 \gamma -2 \psi ^{(0)}\left(-\frac{15}{2}\right)\right)+\frac{\frac{169093}{30030}-3 \gamma -3 \psi ^{(0)}\left(-\frac{15}{2}\right)}{z_4^2}+O\left(\left(\frac{1}{z_4}\right)^4\right)\\ \phi'_{21}&=&2+\frac{3}{z_4^2}+O\left(\left(\frac{1}{z_4}\right)^4\right) \end{eqnarray} \begin{eqnarray} {\tilde \Phi}_4 &=& \frac{1}{\sqrt{\pi } \sqrt{z_4}}(\phi'_{40}+(\log z_4) \phi'_{41} +(\log z_4)^2 \phi'_{42})\end{eqnarray} where \begin{eqnarray} \phi_{40}&=& 2 \left(66352873472-32818705920 \gamma +4058104050 \gamma ^2-2029052025 \pi ^2-32818705920 \psi ^{(0)}\left(-\frac{15}{2}\right)\right.\nonumber\\ & &\left.+8116208100 \gamma \psi ^{(0)}\left(-\frac{15}{2}\right)+4058104050 \psi ^{(0)}\left(-\frac{15}{2}\right)^2\right)/{(2029052025)} \nonumber\\ &&+\frac{1}{676350675 z_4^2}\left(-11395170344-9645846210 \gamma +4058104050 \gamma ^2-2029052025 \pi ^2\right.\nonumber\\ &&\left.-7616794185 \psi ^{(0)}\left(-\frac{15}{2}\right)+4058104050 \gamma \psi ^{(0)}\left(-\frac{15}{2}\right)+2029052025 \psi ^{(0)}\left(-\frac{15}{2}\right)^2\right.\nonumber\\ &&\left.-2029052025 \psi ^{(0)}\left(-\frac{3}{2}\right)+4058104050 \gamma \psi ^{(0)}\left(-\frac{3}{2}\right)+2029052025 \psi ^{(0)}\left(-\frac{3}{2}\right)^2 \right) \nonumber\\ &&+O\left(\left(\frac{1}{z_4}\right)^4\right) \\ \phi'_{41}&=& (\left(\frac{1457152}{45045}-8 \gamma -8 \psi ^{(0)}\left(-\frac{15}{2}\right)\right)+\frac{\frac{214138}{15015}-12 \gamma -6 \psi ^{(0)}\left(-\frac{15}{2}\right)-6 \psi ^{(0)}\left(-\frac{3}{2}\right)}{z_4^2}+O\left(\left(\frac{1}{z_4}\right)^4\right)\\ \phi'_{42}&=& 4+\frac{6}{z_4^2}+O\left(\left(\frac{1}{z_4}\right)^4\right) \end{eqnarray} [By ppDual0 of fancy.m] The elements of the vector ${\tilde \Phi}^\vee$ are as follows. $${\tilde \Phi}_1^\vee=\frac{2 \sqrt{z_4}}{\sqrt{\pi }}\left(1-\frac{1}{2 z_4^2}+O\left(\left(\frac{1}{z_4}\right)^4\right)\right)$$ \begin{eqnarray} {\tilde \Phi}_2^\vee&=&{\tilde \Phi}_3^\vee=\frac{2 \sqrt{z_4}}{\sqrt{\pi }}(\phi_{20} + (\log z_4)\phi_{21}) \end{eqnarray} where \begin{eqnarray} \phi_{20}&=&\left(-\frac{172096}{45045}+2 \gamma +2 \psi ^{(0)}\left(-\frac{13}{2}\right)\right)+\frac{\frac{1}{2}-\gamma -\psi ^{(0)}\left(-\frac{1}{2}\right)}{z_4^2}+O\left(\left(\frac{1}{z_4}\right)^4\right) \\ \phi_{21}&=&-2+\left(\frac{1}{z_4}\right)^2+O\left(\left(\frac{1}{z_4}\right)^4\right).\\ \end{eqnarray} \begin{eqnarray} {\tilde \Phi}_4^\vee&=&\frac{2 \sqrt{z_4}}{\sqrt{\pi }}(\phi_{40} + (\log z_4)\phi_{41}+(\log z_4)^2 \phi_{42}). \end{eqnarray} where \begin{eqnarray} \phi_{40}&=&2 \left(31040932808-15504128640 \gamma +4058104050 \gamma ^2-2029052025 \pi ^2-15504128640 \psi ^{(0)}\left(-\frac{13}{2}\right)\right.\nonumber\\ &&\left.+8116208100 \gamma \psi ^{(0)}\left(-\frac{13}{2}\right)+4058104050 \psi ^{(0)}\left(-\frac{13}{2}\right)^2\right)/{(2029052025)}\nonumber\\ &&+\frac{1}{z_4^2} \left( \frac{4952125736}{2029052025}+\frac{262186 \gamma }{45045}-2 \gamma ^2+\pi ^2+\frac{217141 \psi ^{(0)}\left(-\frac{13}{2}\right)}{45045}-2 \gamma \psi ^{(0)}\left(-\frac{13}{2}\right)-\psi ^{(0)}\left(-\frac{13}{2}\right)^2\right.\nonumber\\ &&\left.+\psi ^{(0)}\left(-\frac{1}{2}\right)-2 \gamma \psi ^{(0)}\left(-\frac{1}{2}\right)-\psi ^{(0)}\left(-\frac{1}{2}\right)^2 \right)\nonumber\\ &&+O\left(\left(\frac{1}{z_4}\right)^4\right). \\ \phi_{41}&=&\left(\frac{688384}{45045}-8 \gamma -8 \psi ^{(0)}\left(-\frac{13}{2}\right)\right)+\frac{-\frac{262186}{45045}+4 \gamma +2 \psi ^{(0)}\left(-\frac{13}{2}\right)+2 \psi ^{(0)}\left(-\frac{1}{2}\right)}{z_4^2}+O\left(\left(\frac{1}{z_4}\right)^4\right). \\ \phi_{42}&=&4-\frac{2}{z_4^2}+O\left(\left(\frac{1}{z_4}\right)^4\right). \end{eqnarray}