# Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

Get a physicist's constructive proof of Fuchs's theorem on hyperelliptic curves

Contributed by:
Bradley Klee

ResourceFunction["HyperellipticODE"][ returns the ODE constraining integral periods T( | |

ResourceFunction["HyperellipticODE"][ also returns a checked and checkable Association containing proof data, when |

The acronym ODE stands for "Ordinary Differential Equation".

Specification of variables {*p*,* q*, *α*} is optional if the potential is written in terms of the formal variable q.

For our purposes, a rational, elliptic or hyperelliptic level curve may be written in set-builder notation as:

𝒞(*α*)={(*p*,*q*)∈ℂ^{2}:*α*=2*H*(*p*,*q*)=*p*^{2}+2*V*(*q*):*V*(*q*)=∑*n*=1N*v*_{n}*q*^{n} :*v*_{n}∈ℝ }

Also, at least one *v*_{n} with *n*>1 must not equal to zero.

When viewed in four dimensional space, such a level curve is really a level surface with measurable genus.

Whether the curve is rational, elliptic, or hyperelliptic depends on the degree cutoff N as well as particular values of the *v*_{n}.

ResourceFunction["HyperellipticODE"] does not distinguish between harmonic, elliptic, or hyperelliptic curves.

ResourceFunction["HyperellipticODE"] cannot accept linear potentials as input because linear potentials do not admit periodic solutions.

Time *t* is defined in terms of the tangents D_{t}{*p*, *q*}={-∂_{q}*H*,∂_{p}*H*}.

The invariant time differential may then be written most directly as *dt*=*dq*/*p.*

Subsequent *α*-derivatives of the integrand *dt* are written as .

The primary task of ResourceFunction["HyperellipticODE"] is to compute a set of coefficients *c*_{n}(*α*) and a certificate Ξ(*q*,* p*) such that:

When this condition is satisfied, the exact differential can be integrated to zero around a contour, which implies:

This is the form of the minimal output, an ordinary differential equation constraining period functions T(*α*).

The proof data described above can be put into an Association:

Potential | V |

Hamiltonian | α = 2H |

Coordinates | {p, q,α} |

Tangents | {-∂_{q}H,∂_{p}H} |

Time Forms | (∂_{α})^{n}dt,n=0, 1, … |

ODE Coefficients | c_{n},n=0, 1, … |

Certificate Function | Ξ |

Truth Value |

The last item with Key "Truth Value" should auto-evaluate to 0 for valid data.

For more details refer to "An Update on the Computational Theory of Hamiltonian Period Functions", especially Chapter 3.

Find the period ODE coefficients of an elliptic curve:

In[1]:= |

Out[1]= |

Solve the ODE to find the real-valued hypergeometric period function:

In[2]:= |

Out[2]= |

Display the proof data in a table:

In[3]:= |

Out[3]= |

Write and check an unusual proof that the harmonic oscillator has a constant period:

In[4]:= |

Out[4]= |

Continue on to a random hyperelliptic curve with terms up to q^{6}:

In[5]:= |

Out[5]= |

Potentials can be specified with undetermined coefficients:

In[6]:= |

Out[6]= |

This sort of data leads quite readily to finding interesting cases:

In[7]:= |

Out[7]= |

Proof data is dynamically checkable in a short time:

In[8]:= |

In[9]:= |

Out[9]= |

Inputting a linear potential function returns a failure:

In[10]:= |

Out[10]= |

Discover an apparent-genus-degree relation:

In[11]:= |

Out[11]= |

- 1.0.0 – 08 April 2022

This work is licensed under a Creative Commons Attribution 4.0 International License