Computing Riemann theta functions.

*(English)*Zbl 1092.33018This paper is based on Computer Algebra Systems (Java, Maple, with an excellent two-part appendix providing the necessary protocol for both implementations); the central problem is approximating values of theta functions (with characteristics) and their derivatives.

Theta functions are quasi-periodic analytic functions of \(g\) complex variables, where \(g\) is any positive integer (the genus, in the case when the periods are complete integrals over the first homology of a Riemann surface). The importance of this line of work is underscored by the rapid development, over the past three decades, of the area of “integrable equations”. The results obtained in this paper are very strong, will be useful, and are especially beautiful in their mathematical basis.

Much of algebraic geometry is being substantially advanced by computational methods (cf. systems like MACAULAY, SINGULAR or MAGMA). Two among the present authors already provided Maple-based techiniques to compute the genus and approximate the period lattice of plane curves [cf. B. Deconinck and M. van Hoeij, Phys. D 152–153, 28–46 (2001; Zbl 1054.14079)].

The key results on which this work is based are the Fourier expansion of theta functions and a Taylor-polynomial-type error estimate; an error estimate based on the length of the eigenvectors of the period matrix (“Fill Factor Error”); modular transformations that, without changing the theta function, bring the period matrix into ‘minimal’ form (in particular a theorem of C. L. Siegel, revisited and exploited in this setting); and hard estimates to obtain both pointwise and uniform approximations.

This is clever and strenuous work which will be valuable in applications to physics and PDE theory, as well as pure mathematics to study issues of special symmetry and reduction for period matrices. One aspect that is not treated and perhaps could be, within this work, is the (modular) dependence of theta on the period matrix, linked to the analytic variables by the heat equations; in that context, estimates for the size of the period lattice are available [P. Buser and P. Sarnak, Invent. Math. 117, No. 1, 27–56 (1994; Zbl 0814.14033)] and have great mathematical depth, being related to the spectrum of the Laplacian [A. Kokotov and D. Korotkin, Lett. Math. Phys. 71, No.3, 241–242 (2005; Zbl 1084.58505)].

Theta functions are quasi-periodic analytic functions of \(g\) complex variables, where \(g\) is any positive integer (the genus, in the case when the periods are complete integrals over the first homology of a Riemann surface). The importance of this line of work is underscored by the rapid development, over the past three decades, of the area of “integrable equations”. The results obtained in this paper are very strong, will be useful, and are especially beautiful in their mathematical basis.

Much of algebraic geometry is being substantially advanced by computational methods (cf. systems like MACAULAY, SINGULAR or MAGMA). Two among the present authors already provided Maple-based techiniques to compute the genus and approximate the period lattice of plane curves [cf. B. Deconinck and M. van Hoeij, Phys. D 152–153, 28–46 (2001; Zbl 1054.14079)].

The key results on which this work is based are the Fourier expansion of theta functions and a Taylor-polynomial-type error estimate; an error estimate based on the length of the eigenvectors of the period matrix (“Fill Factor Error”); modular transformations that, without changing the theta function, bring the period matrix into ‘minimal’ form (in particular a theorem of C. L. Siegel, revisited and exploited in this setting); and hard estimates to obtain both pointwise and uniform approximations.

This is clever and strenuous work which will be valuable in applications to physics and PDE theory, as well as pure mathematics to study issues of special symmetry and reduction for period matrices. One aspect that is not treated and perhaps could be, within this work, is the (modular) dependence of theta on the period matrix, linked to the analytic variables by the heat equations; in that context, estimates for the size of the period lattice are available [P. Buser and P. Sarnak, Invent. Math. 117, No. 1, 27–56 (1994; Zbl 0814.14033)] and have great mathematical depth, being related to the spectrum of the Laplacian [A. Kokotov and D. Korotkin, Lett. Math. Phys. 71, No.3, 241–242 (2005; Zbl 1084.58505)].

Reviewer: Emma Previato (Boston)

##### MSC:

33F05 | Numerical approximation and evaluation of special functions |

11F30 | Fourier coefficients of automorphic forms |

14K25 | Theta functions and abelian varieties |

65F15 | Numerical computation of eigenvalues and eigenvectors of matrices |

##### Keywords:

Theta functions; Pointwise/Uniform approximation; Fill factor; modular transformations; Black-box program (Maple; Java)##### References:

[1] | Handbook of mathematical functions, with formulas, graphs and mathematical tables, Edited by Milton Abramowitz and Irene A. Stegun. Fifth printing, with corrections. National Bureau of Standards Applied Mathematics Series, Vol. 55, National Bureau of Standards, Washington, D.C., (for sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 20402), 1966. |

[2] | M. Babich and A. Bobenko, Willmore tori with umbilic lines and minimal surfaces in hyperbolic space, Duke Math. J. 72 (1993), no. 1, 151 – 185. · Zbl 0820.53005 · doi:10.1215/S0012-7094-93-07207-9 · doi.org |

[3] | E. D. Belokolos, A. I. Bobenko, V. Z. Enol’skii, A. R. Its, and V. B. Matveev, Algebro-geometric approach to nonlinear integrable problems, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1994. · Zbl 0809.35001 |

[4] | A. I. Bobenko, All constant mean curvature tori in \?³,\?³,\?³ in terms of theta-functions, Math. Ann. 290 (1991), no. 2, 209 – 245. · Zbl 0711.53007 · doi:10.1007/BF01459243 · doi.org |

[5] | A. I. Bobenko and L. A. Bordag, Periodic multiphase solutions of the Kadomsev-Petviashvili equation, J. Phys. A 22 (1989), no. 9, 1259 – 1274. · Zbl 0692.35082 |

[6] | Bernard Deconinck and Mark van Hoeij, Computing Riemann matrices of algebraic curves, Phys. D 152/153 (2001), 28 – 46. Advances in nonlinear mathematics and science. · Zbl 1054.14079 · doi:10.1016/S0167-2789(01)00156-7 · doi.org |

[7] | B. A. Dubrovin, Theta-functions and nonlinear equations, Uspekhi Mat. Nauk 36 (1981), no. 2(218), 11 – 80 (Russian). With an appendix by I. M. Krichever. · Zbl 0478.58038 |

[8] | B. A. Dubrovin, Ron Flickinger, and Harvey Segur, Three-phase solutions of the Kadomtsev-Petviashvili equation, Stud. Appl. Math. 99 (1997), no. 2, 137 – 203. · Zbl 0893.35112 · doi:10.1111/1467-9590.00059 · doi.org |

[9] | David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. · Zbl 1042.35002 |

[10] | M. Heil, Numerical tools for the study of finite gap solutions of integrable systems, Ph.D. thesis, Technischen Universität Berlin, 1995. · Zbl 0858.58044 |

[11] | Jun-ichi Igusa, Theta functions, Springer-Verlag, New York-Heidelberg, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 194. · Zbl 0251.14016 |

[12] | C. G. J Jacobi, Fundamenta Nova Theoriae Functionum Ellipticarum, Königsberg (1829). |

[13] | A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), no. 4, 515 – 534. · Zbl 0488.12001 · doi:10.1007/BF01457454 · doi.org |

[14] | David Mumford, Tata lectures on theta. I, Progress in Mathematics, vol. 28, Birkhäuser Boston, Inc., Boston, MA, 1983. With the assistance of C. Musili, M. Nori, E. Previato and M. Stillman. · Zbl 0509.14049 |

[15] | David Mumford, Tata lectures on theta. II, Progress in Mathematics, vol. 43, Birkhäuser Boston, Inc., Boston, MA, 1984. Jacobian theta functions and differential equations; With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman and H. Umemura. · Zbl 0549.14014 |

[16] | David Mumford, Tata lectures on theta. III, Progress in Mathematics, vol. 97, Birkhäuser Boston, Inc., Boston, MA, 1991. With the collaboration of Madhav Nori and Peter Norman. · Zbl 0744.14033 |

[17] | G. F. B. Riemann, Theorie der Abel’schen functionen, Journal für reine und angewandte Mathematik 54 (1857), 101-155. · ERAM 054.1427cj |

[18] | C. L. Siegel, Vorlesungen über ausgewählte Kapitel der Funktionentheorie. Teil III, Mathematisches Institut der Universität, Göttingen, 1966 (German). C. L. Siegel, Topics in complex function theory. Vol. III: Abelian functions and modular functions of several variables, Wiley-Interscience [A Division of John Wiley & Sons, Inc.], New York-London-Sydney, 1973. Translated from the original German by E. Gottschling and M. Tretkoff; Interscience Tracts in Pure and Applied Mathematics, No. 25. |

[19] | Brigitte Vallée, A central problem in the algorithmic geometry of numbers: lattice reduction, CWI Quarterly 3 (1990), no. 2, 95 – 120. · Zbl 0713.11089 |

[20] | E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge University Press, Cambridge, 1902. · JFM 45.0433.02 |

[21] | W. Wirtinger, Untersuchungen über thetafunctionen, B. G. Teubner, Leipzig, 1895. · JFM 26.0514.01 |

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