# Finite Groups for the Kummer Surface: The Genetic Code and a Quantum Gravity Analogy

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## Abstract

**:**

## 1. Introduction

## 2. The Hyperelliptic Curve and the Attached KUMMER Surface from Groups ${\mathit{G}}_{\mathit{6}}$ and ${\mathbf{G}}_{\mathbf{7}}$

#### 2.1. Excerpts about Elliptic and Hyperelliptic Curves

#### 2.2. The Group ${G}_{6}:=(288,69)\cong {\mathbb{Z}}_{6}\u22ca2O$

#### 2.3. The Group ${G}_{7}:=(336,118)\cong {\mathbb{Z}}_{7}\times 2O$

## 3. The Genetic Code Revisited

## 4. Kummer Surface and Quantum Gravity

## 5. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Table A1.**For the group ${G}_{5}:=(240,105)\cong {\mathbb{Z}}_{5}\u22ca2O$, the table provides the dimension of the representation, the rank of the Gram matrix obtained under the action of the 22 -dimensional Pauli group, the order of a group element in the class, the entries involved in the character and a good assignment to an amino acid according to its polar requirement value. Bold characters are for faithful representations. There is an ‘exception’ for the assignment of the sextet ‘Leu’ that is assumed to occupy two 4-dimensional slots. All characters are informationally complete except for the ones assigned to ‘Stop’, ‘Leu’, ‘Pyl’ and ‘Sec’. The notation in the entries is as follows: ${z}_{1}=-(\sqrt{5}+1)/2$, ${z}_{2}=\sqrt{5}-1$, ${z}_{3}=3(1+\sqrt{5})/2$, ${z}_{4}=\sqrt{2}$, ${z}_{5}=-2cos(\pi /15)$, compare [Table 7] of [1].

(240,105) | dimension | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |

${\mathbb{Z}}_{5}\u22ca({\mathbb{Z}}_{2}\xb7{S}_{4})$ | d-dit, d = 22 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ |

$\cong {\mathbb{Z}}_{5}\u22ca2O$ | amino acid | Met | Trp | Cys | Phe | Tyr | His | Gln | Asn | Lys | Glu | Asp |

order | 1 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 10 | |

char | Cte | Cte | Cte | ${z}_{1}$ | ${z}_{1}$ | ${z}_{4}$ | ${z}_{4}$ | ${z}_{1,5}$ | ${z}_{1,5}$ | ${z}_{1,5}$ | ${z}_{1,5}$ | |

polar req. | 5.3 | 5.2 | 4.8 | 5.0 | 5.4 | 8.4 | 8.6 | 10.0 | 10.1 | 12.5 | 13.0 | |

(240,105) | dimension | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 |

d-dit, d = 22 | ${d}^{2}$ | 475 | 483 | 480 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

amino acid | Ile | Stop | Leu,Pyl,Sec | Leu | Val | Pro | Thr | Ala | Gly | Ser | Arg | |

order | 10 | 15 | 15 | 15 | 15 | 20 | 20 | 30 | 30 | 30 | 30 | |

char | Cte | Cte | Cte | ${z}_{1,2}$ | ${z}_{1,2}$ | ${z}_{2,5}$ | ${z}_{2,5}$ | ${z}_{2,5}$ | ${z}_{2,5}$ | ${z}_{1,3}$ | ${z}_{1,3}$ | |

polar req. | 4.9 | 4.9 | 5.6 | 6.6 | 6.6 | 7.0 | 7.9 | 7.5 | 9.1 |

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**Figure 2.**(

**a**) A standard plot of the Kummer surface in its 3-dimensional projection, (

**b**) a section at constant ${x}_{4}$ of the Kummer surface defined in Section 2.3.

**Table 1.**For the group ${G}_{6}:=(288,69)\cong {\mathbb{Z}}_{6}\u22ca2O$, the table provides the dimension of the representation, the rank of the Gram matrix obtained under the action of the 30 -dimensional Pauli group and the entries involved in the characters. All characters are neither faithful nor informationally complete. The notation is $I=exp(2i\pi /4)$, ${z}_{1}=-\sqrt{2}$, ${z}_{2}=I\sqrt{2}$ and ${z}_{3}=-2$$cos(\pi /9)$.

(288,69) | dimension | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |

${\mathbb{Z}}_{6}\u22ca({\mathbb{Z}}_{2}\xb7{S}_{4})$ | d-dit, d = 30 | 31 | 796 | 867 | 867 | 882 | 882 | 880 | 897 | 897 | 880 |

char | Cte | Cte | I | I | Cte | Cte | ${z}_{1}$ | ${z}_{2}$ | ${z}_{2}$ | ${z}_{1}$ | |

(288,69) | dimension | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |

d-dit, d = 30 | 885 | 885 | 885 | 885 | 885 | 885 | 876 | 878 | 899 | 899 | |

char | ${z}_{3}$ | ${z}_{3}$ | ${z}_{3}$ | ${z}_{3}$ | ${z}_{3}$ | ${z}_{3}$ | Cte | Cte | I | I | |

(288,69) | dimension | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 |

d-dit, d = 30 | 877 | 878 | 885 | 885 | 885 | 885 | 885 | 885 | 880 | 880 | |

char | Cte | Cte | ${z}_{3}$ | ${z}_{3}$ | ${z}_{3}$ | ${z}_{3}$ | ${z}_{3}$ | ${z}_{3}$ | Cte | Cte |

**Table 2.**The structure of the addition table for the 16 singular Jacobian points of the hyperelliptic curves ${\mathcal{C}}_{6}$ and ${\mathcal{C}}_{7}$.

A | B | C | D |

B | A | D | C |

C | D | A | B |

D | C | B | A |

**Table 3.**The algebra for the character table of group ${G}_{7}:=(336,118)$. In column 1 are the characters in question. Column 2 provides the powers of the entries ${z}_{i}$, $i=1$, 2, 3 or 5. The ${z}_{i}$ are ${z}_{1}=2cos(2\pi /7)$, ${z}_{2}=2{z}_{1}$, ${z}_{3}=-6cos(\pi /7)$, ${z}_{4}=\sqrt{2}$ and ${z}_{5}=2cos(2\pi /21)$. Column 3 explains the polynomial $f\left(x\right)$ whose roots are the powers of a selected ${z}_{i}$. When $f\left(x\right)$ is an elliptic curve defined over the rationals the Cremona reference is in column 4. If $f\left(x\right)$ is a sextic polynomial it leads to a Kummer surface.

Character | ${\mathit{z}}_{\mathit{i}}$ Powers | $\mathit{f}\left(\mathit{x}\right)$ Polynomial | Cremona Ref. |
---|---|---|---|

4–6 | ${z}_{1}:[1,2,3]$ | ${x}^{3}+{x}^{2}-2x-1$ | $784{i}_{1}$ |

18–20 | ${z}_{1}:[1,2,3]$ | . | . |

. | ${z}_{2}:[1,2,3]$ | ${x}^{3}+2{x}^{2}-8x-8$ | $3136{x}_{1}$ |

. | ${z}_{1,2}$ | ${x}^{6}+3{x}^{5}-8{x}^{4}-\phantom{\rule{0ex}{0ex}}21{x}^{3}+6{x}^{2}+24x+8$ | Kummer |

27–29 | ${z}_{1}:[1,2,3]$ | . | . |

. | ${z}_{3}:[1,2,3]$ | ${x}^{3}+3{x}^{2}-18x-27$ | $1764{j}_{1}$ |

. | ${z}_{1,3}:[1,2,3]$ | ${x}^{6}+4{x}^{5}-17{x}^{4}-\phantom{\rule{0ex}{0ex}}52{x}^{3}+6{x}^{2}+72x+27$ | Kummer |

9–14 & 21–26 | ${z}_{1,2}$ | . | . |

${z}_{5}:[1,2,4,5,8,10]$ | ${x}^{6}-{x}^{5}-6{x}^{4}+\phantom{\rule{0ex}{0ex}}6{x}^{3}+8{x}^{2}-8x+1$ | Kummer |

**Table 4.**For the group ${G}_{7}:=(336,118)\cong {\mathbb{Z}}_{7}\u22ca2O$, the table provides the dimension of the representation, the rank of the Gram matrix obtained under the action of the 29 -dimensional Pauli group, the order of a group element in the class, the angles involved in the character and a good assignment to an amino acid according to its polar requirement value. Bold characters are for faithful representations. All characters are informationally complete except for the trivial character and the one assigned to ‘Met’. The entries involved in the characters are ${z}_{1}=2cos(2\pi /7)$, ${z}_{2}=2{z}_{1}$, ${z}_{3}=-6cos(\pi /7)$, ${z}_{4}=\sqrt{2}$ and ${z}_{5}=2cos(2\pi /21)$ featuring the angles $2\pi /8$ (in ${z}_{4}$), $2\pi /7$ and $2\pi /21$.

(336,118) | dimension | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |

${\mathbb{Z}}_{7}\u22ca({\mathbb{Z}}_{2}\xb7{S}_{4})$ | d-dit, d = 29 | 29 | 785 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ |

$\cong {\mathbb{Z}}_{7}\u22ca2O$ | amino acid | . | Met | Trp | Cys | Phe | Tyr | . | . | His | Gln |

order | 1 | 2 | 3 | 4 | 4 | 6 | 7 | 7 | 7 | 8 | |

char | Cte | Cte | Cte | ${z}_{1}$ | ${z}_{1}$ | ${z}_{1}$ | ${z}_{4}$ | ${z}_{4}$ | ${z}_{1,5}$ | ${z}_{1,5}$ | |

polar req. | . | 5.3 | 5.2 | 4.8 | 5.0 | 5.4 | . | . | 8.4 | 8.6 | |

(336,118) | dimension | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 |

d-dit, d = 29 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

amino acid | Asn | Lys | Glu | Asp | Ile | Stop | . | . | . | . | |

order | 14 | 14 | 14 | 21 | 21 | 21 | 21 | 21 | 21 | 21 | |

char | ${z}_{1,5}$ | ${z}_{1,5}$ | ${z}_{1,5}$ | ${z}_{1,5}$ | Cte | Cte | Cte | ${z}_{1,2}$ | ${z}_{1,2}$ | ${z}_{1,2}$ | |

polar req. | 10.0 | 10.1 | 12.5 | 13.0 | 10 | 15 | . | . | . | . | |

(336,118) | dimension | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | |

d-dit, d = 29 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ||

amino acid | Val | Pro | Thr | Ala | Gly | . | Leu | Ser | Arg | ||

order | 28 | 28 | 28 | 42 | 42 | 42 | 42 | 42 | 42 | ||

char | ${z}_{2,5}$ | ${z}_{2,5}$ | ${z}_{2,5}$ | ${z}_{2,5}$ | ${z}_{2,5}$ | ${z}_{2,5}$ | ${z}_{1,3}$ | ${z}_{1,3}$ | ${z}_{1,3}$ | ||

polar req. | 5.6 | 6.6 | 6.6 | 7.0 | 7.9 | . | 4.9 | 7.5 | 9.1 |

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**MDPI and ACS Style**

Planat, M.; Chester, D.; Aschheim, R.; Amaral, M.M.; Fang, F.; Irwin, K.
Finite Groups for the Kummer Surface: The Genetic Code and a Quantum Gravity Analogy. *Quantum Rep.* **2021**, *3*, 68-79.
https://doi.org/10.3390/quantum3010005

**AMA Style**

Planat M, Chester D, Aschheim R, Amaral MM, Fang F, Irwin K.
Finite Groups for the Kummer Surface: The Genetic Code and a Quantum Gravity Analogy. *Quantum Reports*. 2021; 3(1):68-79.
https://doi.org/10.3390/quantum3010005

**Chicago/Turabian Style**

Planat, Michel, David Chester, Raymond Aschheim, Marcelo M. Amaral, Fang Fang, and Klee Irwin.
2021. "Finite Groups for the Kummer Surface: The Genetic Code and a Quantum Gravity Analogy" *Quantum Reports* 3, no. 1: 68-79.
https://doi.org/10.3390/quantum3010005