Geodesics, periods, and equations of real hyperelliptic curves.

*(English)*Zbl 1019.30042The classical uniformization theorem states that any Riemann surface has a universal covering conformally equivalent to either the Riemann sphere \(\mathbb P^1\), the complex plane \(\mathbb C\) or the hyperbolic upper half plane \(\mathbb H\). One of the consequences is that every smooth complex algebraic curve \(C\) of genus \(g>1\) is conformally equivalent to the quotient of \(\mathbb H\) by a Fuchsian group \(G\subset PSL_2(\mathbb R)\), and conversely.

The problem of how to realize this correspondence in an explicit fashion is, however, a difficult one, and it is still partially unsolved. In this paper, the authors consider one of the two possible directions of the correspondence, namely going from the Fuchsian group \(G\) to the algebraic curve in the case of real hyperelliptic curves.

A real curve \(C\) is an algebraic curve equipped with an anti-holomorphic involution \(\sigma\) whose fixed points comprise the set of real points \(C(\mathbb R)\) of \(C\). The set of real points has at most \(g+1\) components if \(g\) is the genus of \(C\), and the authors consider the case when this bound is attained. Thus \(C\) is hyperelliptic of genus \(g\) with \(C(\mathbb R)\) having \(g+1\) components. This amounts to say that the polynomial \(P\) in the equation \(y^2=P(x)\) is real with real roots \(x_1,\dots, x_{2g+1}\).

The authors’ method is as follows. First, they show that for a curve \(C\) of the above type its period matrix can be reconstructed analytically from the data of hyperbolic capacities of a set of harmonic functions. The equation of the curve is then reconstructed from the period matrix via theta characteristics. The crucial point is to obtain for \(C\) a set of harmonic functions – with their boundary conditions – depending only on \(G\), without reference to the algebraic structure.

Let \(M\) be a domain in the hyperbolic plane, and let \(h\) be a harmonic function on \(M\). The capacity associated to it is \[ \int_M {||\nabla h||}^2 dM = \int_{\partial M} h \nu [h] d\mu \] where \(\nu [h]\) is the normal derivative and \(d\mu\) is the measure induced on the boundary.

A hyperelliptic curve \(C\) with the above real structure can be obtained by gluing together two spheres with \(g+1\) disks removed. Each sphere can in turn be obtained by gluing together two \(2g+2\)-gons. The authors show that starting from the data of the hyperelliptic curve in the form \(y^2=P(x)\), a set of hyperbolic capacities \(\{h_i\}\) can be obtained satisfying very simple boundary conditions on the \(2g+2\)-gons.

At this point the algorithm works in the following way. Given the lengths of a \(2g+2\) and the boundary conditions, the capacities \(\{h_i\}\) can be reconstructed numerically via harmonic polynomials. In turn this determines the period matrix, and then the equation of the curve via theta characteristics.

The second half of the paper is devoted to a series of cases where the correspondence is actually exact. For certain specific families of real hyperelliptic genus \(2\) curves, the equation can be obtained explicitly in exact form. In particular, the authors consider the following subspace of the real moduli space of genus \(2\) real hyperelliptic curves. If \(g=2\), the \(2g+2\)-gon is a hexagon characterized by three lengths \(l_1,l_2,l_3\). If two of them are equal, say \(l_2=l_3\), an involution is obtained acting on the corresponding curve. Since there is also the hyperelliptic involution, the group \({\mathbb Z}/2 \times {\mathbb Z}/2\) acts on these curves by automorphisms preserving the real structure. Conversely, if \({\mathbb Z}/2 \times {\mathbb Z}/2\) is contained in the group of automorphisms of \(C\) preserving the real structure, the resulting hexagon will have two lengths equal. For families in this subspace, the authors construct an action of the dihedral group \(D_5\) and show in several explicit examples how for the various fixed points of the \(D_5\) action it is possible to obtain explicit equations starting from the lengths \(l_1\), \(l_2\).

A final section is devoted to a compilation of tables and examples. Other interesting points addressed in the body of the paper, concern the application of the procedure to one-punctured tori, and to curves with half twists.

The problem of how to realize this correspondence in an explicit fashion is, however, a difficult one, and it is still partially unsolved. In this paper, the authors consider one of the two possible directions of the correspondence, namely going from the Fuchsian group \(G\) to the algebraic curve in the case of real hyperelliptic curves.

A real curve \(C\) is an algebraic curve equipped with an anti-holomorphic involution \(\sigma\) whose fixed points comprise the set of real points \(C(\mathbb R)\) of \(C\). The set of real points has at most \(g+1\) components if \(g\) is the genus of \(C\), and the authors consider the case when this bound is attained. Thus \(C\) is hyperelliptic of genus \(g\) with \(C(\mathbb R)\) having \(g+1\) components. This amounts to say that the polynomial \(P\) in the equation \(y^2=P(x)\) is real with real roots \(x_1,\dots, x_{2g+1}\).

The authors’ method is as follows. First, they show that for a curve \(C\) of the above type its period matrix can be reconstructed analytically from the data of hyperbolic capacities of a set of harmonic functions. The equation of the curve is then reconstructed from the period matrix via theta characteristics. The crucial point is to obtain for \(C\) a set of harmonic functions – with their boundary conditions – depending only on \(G\), without reference to the algebraic structure.

Let \(M\) be a domain in the hyperbolic plane, and let \(h\) be a harmonic function on \(M\). The capacity associated to it is \[ \int_M {||\nabla h||}^2 dM = \int_{\partial M} h \nu [h] d\mu \] where \(\nu [h]\) is the normal derivative and \(d\mu\) is the measure induced on the boundary.

A hyperelliptic curve \(C\) with the above real structure can be obtained by gluing together two spheres with \(g+1\) disks removed. Each sphere can in turn be obtained by gluing together two \(2g+2\)-gons. The authors show that starting from the data of the hyperelliptic curve in the form \(y^2=P(x)\), a set of hyperbolic capacities \(\{h_i\}\) can be obtained satisfying very simple boundary conditions on the \(2g+2\)-gons.

At this point the algorithm works in the following way. Given the lengths of a \(2g+2\) and the boundary conditions, the capacities \(\{h_i\}\) can be reconstructed numerically via harmonic polynomials. In turn this determines the period matrix, and then the equation of the curve via theta characteristics.

The second half of the paper is devoted to a series of cases where the correspondence is actually exact. For certain specific families of real hyperelliptic genus \(2\) curves, the equation can be obtained explicitly in exact form. In particular, the authors consider the following subspace of the real moduli space of genus \(2\) real hyperelliptic curves. If \(g=2\), the \(2g+2\)-gon is a hexagon characterized by three lengths \(l_1,l_2,l_3\). If two of them are equal, say \(l_2=l_3\), an involution is obtained acting on the corresponding curve. Since there is also the hyperelliptic involution, the group \({\mathbb Z}/2 \times {\mathbb Z}/2\) acts on these curves by automorphisms preserving the real structure. Conversely, if \({\mathbb Z}/2 \times {\mathbb Z}/2\) is contained in the group of automorphisms of \(C\) preserving the real structure, the resulting hexagon will have two lengths equal. For families in this subspace, the authors construct an action of the dihedral group \(D_5\) and show in several explicit examples how for the various fixed points of the \(D_5\) action it is possible to obtain explicit equations starting from the lengths \(l_1\), \(l_2\).

A final section is devoted to a compilation of tables and examples. Other interesting points addressed in the body of the paper, concern the application of the procedure to one-punctured tori, and to curves with half twists.

Reviewer: Ettore Aldrovandi (Tallahassee)

##### MSC:

30F10 | Compact Riemann surfaces and uniformization |

14P05 | Real algebraic sets |

32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |

32G20 | Period matrices, variation of Hodge structure; degenerations |

PDF
BibTeX
XML
Cite

\textit{P. Buser} and \textit{R. Silhol}, Duke Math. J. 108, No. 2, 211--250 (2001; Zbl 1019.30042)

Full Text:
DOI

##### References:

[1] | E. Bujalance, J. Etayo, M. Gamboa, and G. Gromadzki, Automorphism Groups of Compact Bordered Klein Surfaces: A Combinatorial Approach , Lecture Notes in Math. 1439 , Springer, Berlin, 1990. MR 92a:14018 · Zbl 0709.14021 · doi:10.1007/BFb0084977 |

[2] | P. Buser, Geometry and Spectra of Compact Riemann Surfaces , Progr. Math. 106 , Birkhäuser, Boston, 1992. MR 93g:58149 · Zbl 0770.53001 |

[3] | H. Farkas and I. Kra, Riemann Surfaces , Grad. Texts in Math. 71 , Springer, New York, 1980. MR 82c:30067 · Zbl 0475.30001 |

[4] | B. Gross and J. Harris, Real algebraic curves , Ann. Sci. École Norm. Sup. (4) 14 (1981), 157–182. MR 83a:14028 · Zbl 0533.14011 |

[5] | T. Kuusalo and M. Näätänen, Geometric uniformization in genus \(2\) , Ann. Acad. Sci. Fenn. Ser. A I Math. 20 (1995), 401–418. MR 96h:30083 |

[6] | S. M. Natanzon, Klein surfaces , Russian Math. Surveys 45 , no. 6 (1990), 53–108. MR 92i:14029 |

[7] | Z. Nehari, Conformal Mapping , McGraw-Hill, New York, 1952. MR 13:640h · Zbl 0048.31503 |

[8] | R. Rodrí guez and V. González-Aguilera, “Fermat’s quartic curve, Klein’s curve and the tetrahedron” in Extremal Riemann Surfaces (San Francisco, 1995), Contemp. Math. 201 , Amer. Math. Soc., Providence, 1997, 43–62. MR 97j:14033 · Zbl 0911.14021 |

[9] | M. Seppälä and R. Silhol, Moduli spaces for real algebraic curves and real abelian varieties , Math. Z. 201 (1989), 151–165. MR 90k:14043 · Zbl 0645.14012 · doi:10.1007/BF01160673 · eudml:174043 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.