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 |

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

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