Modular invariants for genus 3 hyperelliptic curves.

*(English)*Zbl 1420.14063Siegel modular forms of degree \(g\) are known to live on moduli spaces of \(g\)-dimensional principally polarized abelian varieties, and therefore also on moduli spaces of curves of genus \(g\). Many such modular forms, especially those that have meanings in terms of moduli questions, are constructed as products of theta constants. In the genus 3 case discussed in this paper, there are two such Siegel modular forms, \(\chi_{18}\) (which essentially characterizes the hyperelliptic locus) and \(\Sigma_{140}\) (which on the hyperelliptic locus locus distinguishes the Jacobians from the reducible varieties). In addition, the ring of Siegel modular forms of degree \(g\) admits an Igusa map \(\rho\) to the ring of invariants of binary forms of degree \(2g+2\). When \(g=3\), the ring of invariants of binary octics is generated by the Shioda invariants \(J_{l}\), \(2 \leq l\leq10\), of which 6 are algebraically independent.

The aim of this paper is to prove the algebraicity of Siegel modular functions of degree 3 that are regular outside the zeroes of \(\Sigma_{140}\), and to analyze the primes dividing the denominator. More explicitly, if \(f\) is a holomorphic Siegel modular form of degree 3, weight \(k\), and level 1, such that its image under the Igusa map is a polynomial in the Shioda invariants that has only integral coefficients, and consider the quotient \(j=\big(\frac{f^{140}}{\sigma_{140}^{k}}^{1/\gcd\{k,140\}}\). Then if \(Z\) is the period matrix associated with a hyperelliptic curve \(C\) of genus 3 that is defined over \(M\) then \(j(Z) \in M\), and moreover the primes in \(\mathcal{O}_{M}\) that divide the denominator of this algebraic numbers are only primes modulo which \(C\) has geometrically bad reduction. Several MAGMA calculations are also given, to illuminate the main result.

The paper is divided into 6 sections. Section 1 is the Introduction, including the main result. Section 3 introduces the basic ideas behind theta functions, the Igusa map, and the Shioda invariants. Section 3 presents the relations between the various incarnations of Siegel modular forms of degree \(g\) (holomorphic functions on the Siegel upper half-plane, sections on line bundles on the moduli stack of principally polarized abelian varieties, sections on line bundles on the moduli stack of curves) and the connection to invariants of binary forms of degree \(2g+2\) as well as of hyperelliptic curves of genus \(g\). Section 4 proves, using Thomae formulae, an identity involving, in genus 3, the restriction of \(\Sigma_{140}\) to the hyperelliptic locus and the discriminant of hyperelliptic curves, and uses it to deduce the main result. Section 5 contains the relevant MAGMA calculations, applied to 13 hyperelliptic curves of genus 3, including some with CM and some modular curves. Finally, Section 6 is a short conclusion.

The aim of this paper is to prove the algebraicity of Siegel modular functions of degree 3 that are regular outside the zeroes of \(\Sigma_{140}\), and to analyze the primes dividing the denominator. More explicitly, if \(f\) is a holomorphic Siegel modular form of degree 3, weight \(k\), and level 1, such that its image under the Igusa map is a polynomial in the Shioda invariants that has only integral coefficients, and consider the quotient \(j=\big(\frac{f^{140}}{\sigma_{140}^{k}}^{1/\gcd\{k,140\}}\). Then if \(Z\) is the period matrix associated with a hyperelliptic curve \(C\) of genus 3 that is defined over \(M\) then \(j(Z) \in M\), and moreover the primes in \(\mathcal{O}_{M}\) that divide the denominator of this algebraic numbers are only primes modulo which \(C\) has geometrically bad reduction. Several MAGMA calculations are also given, to illuminate the main result.

The paper is divided into 6 sections. Section 1 is the Introduction, including the main result. Section 3 introduces the basic ideas behind theta functions, the Igusa map, and the Shioda invariants. Section 3 presents the relations between the various incarnations of Siegel modular forms of degree \(g\) (holomorphic functions on the Siegel upper half-plane, sections on line bundles on the moduli stack of principally polarized abelian varieties, sections on line bundles on the moduli stack of curves) and the connection to invariants of binary forms of degree \(2g+2\) as well as of hyperelliptic curves of genus \(g\). Section 4 proves, using Thomae formulae, an identity involving, in genus 3, the restriction of \(\Sigma_{140}\) to the hyperelliptic locus and the discriminant of hyperelliptic curves, and uses it to deduce the main result. Section 5 contains the relevant MAGMA calculations, applied to 13 hyperelliptic curves of genus 3, including some with CM and some modular curves. Finally, Section 6 is a short conclusion.

Reviewer: Shaul Zemel (Jerusalem)

##### MSC:

14H42 | Theta functions and curves; Schottky problem |

11F46 | Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms |

11G15 | Complex multiplication and moduli of abelian varieties |

11E76 | Forms of degree higher than two |

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

14K25 | Theta functions and abelian varieties |

PDF
BibTeX
XML
Cite

\textit{S. Ionica} et al., Res. Number Theory 5, No. 1, Paper No. 9, 22 p. (2019; Zbl 1420.14063)

Full Text:
DOI

##### References:

[1] | Balakrishnan, JS; Ionica, S.; Lauter, K.; Vincent, C., Constructing genus-3 hyperelliptic Jacobians with CM, LMS J. Comput. Math., 19, 283-300, (2016) · Zbl 1404.11085 |

[2] | Balakrishnan, J.S., Bianchi, F., Cantoral Farfán, V., Çiperiani, M., Etropolski, A.: Chabauty-Coleman experiments for genus 3 hyperelliptic curves. Preprint (2018) |

[3] | Basson, R.: Arithmétique des espaces de modules des courbes hyperelliptiques de genre \(3\) en caractéristique positive. Ph.D. thesis, Université de Rennes 1, Rennes (2015) |

[4] | Bouw, II; Wewers, S., Computing \(L\)-functions and semistable reduction of superelliptic curves, Glasg. Math. J., 59, 77-108, (2017) · Zbl 1430.11090 |

[5] | Fay, J.D.: Theta functions on Riemann surfaces. Lecture Notes in Mathematics, vol. 352. Springer, Berlin (1973) · Zbl 0281.30013 |

[6] | Galbraith, S.D.: Equations For modular curves. Ph.D. thesis, University of Oxford, Oxford (1996) |

[7] | Goren, EZ; Lauter, KE, Class invariants for quartic CM fields, Ann. Inst. Fourier (Grenoble), 57, 457-480, (2007) · Zbl 1172.11018 |

[8] | Ichikawa, T., On Teichmüller modular forms, Math. Ann., 299, 731-740, (1994) · Zbl 0803.30036 |

[9] | Ichikawa, T., Teichmüller modular forms of degree 3, Am. J. MAth., 117, 1057-1061, (1995) · Zbl 0856.11024 |

[10] | Ichikawa, T., Theta constants and Teichmüller modular forms, J. Number Theory, 61, 409-419, (1996) · Zbl 0920.11029 |

[11] | Ichikawa, T., Generalized Tate curve and integral Teichmüller modular forms, Am. J. Math., 122, 1139-1174, (2000) · Zbl 1062.14501 |

[12] | Igusa, J., Modular forms and projective invariants, Am. J. Math., 89, 817-855, (1967) · Zbl 0159.50401 |

[13] | Kılıçer, P., Labrande, H., Lercier, R., Ritzenthaler, C., Sijsling, J., Streng, M.: Plane quartics over \(\mathbb{Q}\) with complex multiplication. Preprint (2017) · Zbl 1409.14051 |

[14] | Kılıçer, P., Streng, M.: Rational CM points and class polynomials for genus three. In: preparation (2016) |

[15] | Kılıçer, P., Lauter, K.E.., Lorenzo García, E., Newton, R., Ozman, E., Streng, M.: A bound on the primes of bad reduction for CM curves of genus 3. https://arxiv.org/abs/1609.05826. (2016) |

[16] | Labrande, H.: Code for fast theta function evaluation. https://hlabrande.fr/math/research/. (2017). Accessed 11 Jan 2017 |

[17] | Lachaud, G.; Ritzenthaler, C.; Zykin, A., Jacobians among abelian threefolds: a formula of Klein and a question of Serre, Math. Res. Lett., 17, 323-333, (2010) · Zbl 1228.14028 |

[18] | Lercier, R.; Ritzenthaler, C., Hyperelliptic curves and their invariants: geometric, arithmetic and algorithmic aspects, J. Algebra, 372, 595-636, (2012) · Zbl 1276.14088 |

[19] | Lercier, R., Liu, Q., Lorenzo García, E., Ritzenthaler, C.: Reduction type of non-hyperelliptic genus 3 curves. Preprint (2018) |

[20] | Lockhart, P., On the discriminant of a hyperelliptic curve, Trans. Am. Math. Soc., 342, 729-752, (1994) · Zbl 0815.11031 |

[21] | Manni, RS, Slope of cusp forms and theta series, J Number Theory, 83, 282-296, (2000) · Zbl 0957.11025 |

[22] | Molin, P., Neurohr, C.: Hcperiods (2018) |

[23] | Mumford, D.: Tata lectures on theta. I. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, (2007). With the collaboration of C. Musili, M. Nori, E. Previato and M. Stillman, Reprint of the 1983 edition |

[24] | Mumford, D.: Tata lectures on theta. II. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, (2007). Jacobian theta functions and differential equations, With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman, and H. Umemura, Reprint of the 1984 original |

[25] | Ogg, AP, Hyperelliptic modular curves, Bull. Soc. Math. France, 102, 449-462, (1974) · Zbl 0314.10018 |

[26] | Poor, C., The hyperelliptic locus, Duke Math. J., 76, 809-884, (1994) · Zbl 0832.14020 |

[27] | Shimura, G.: Abelian varieties with complex multiplication and modular functions. Princeton Mathematical Series, vol. 46. Princeton University Press, Princeton (1998) · Zbl 0908.11023 |

[28] | Shioda, T., On the graded ring of invariants of binary octavics, Amer. J. Math., 89, 1022-1046, (1967) · Zbl 0188.53304 |

[29] | Tautz, W.; Top, J.; Verberkmoes, A., Explicit hyperelliptic curves with real multiplication and permutation polynomials, Canad. J. Math., 43, 1055-1064, (1991) · Zbl 0793.14022 |

[30] | Tsuyumine, S., On Siegel modular forms of degree 3, Am. J. Math., 108, 755-862, (1986) · Zbl 0602.10015 |

[31] | Tsuyumine, S., On the Siegel modular function field of degree three, Compositio Math., 63, 83-98, (1987) · Zbl 0632.10027 |

[32] | Weng, A., A class of hyperelliptic CM-curves of genus three, J. Ramanujan Math. Soc., 16, 339-372, (2001) · Zbl 1066.11028 |

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.