In a section of his 1876 treatise Theorie der Abelschen Functionen vom Geschlecht 3 Weber proved a formula that expresses the bitangents of a non-singular plane quartic in terms of Riemann theta constants (Thetanullwerte). The present note is devoted to a modern presentation of Weber's formula. In the end a connection with the universal bitangent matrix is also displayed.
The level moduli space A 4,8 g is mapped to the projective space by means of gradients of odd Theta functions, such a map turning out no to be injective in the genus 2 case. In this work a congruence subgroup Γ is located between Γ 2 (4, 8) and Γ 2 (2, 4) in such a way the map factors on the related level moduli space A Γ , the new map being injective on A Γ . Satake's compactification ProjA(Γ) and the desingularization ProjS(Γ) are also due to be investigated, since the map does not extend to the boundary of the compactification; to aim at this, an algebraic description is provided, by proving a structure theorem both for the ring of modular forms A(Γ) and the ideal of cusp forms S(Γ).
We use the gradients of theta functions at odd twotorsion points -thought of as vector-valued modular forms -to construct holomorphic differential forms on the moduli space of principally polarized abelian varieties, and to characterize the locus of decomposable abelian varieties in terms of the Gauss images of twotorsion points.
Aronhold's classical result states that a plane quartic can be recovered by the configuration of any Aronhold systems of bitangents, i.e. special $7$-tuples of bitangents such that the six points at which any sub-triple of bitangents touches the quartic do not lie on the same conic in the projective plane. Lehavi (cf. \cite{lh}) proved that a smooth plane quartic can be explicitly reconstructed from its $28$ bitangents; this result improved Aronhold's method of recovering the curve. In a 2011 paper \cite{PSV} Plaumann, Sturmfels and Vinzant introduced an $8 \times 8$\ud
symmetric matrix that parametrizes the bitangents of a nonsingular plane quartic. The starting point of their construction is \ud
Hesse's result for which every smooth quartic curve has exactly $36$ equivalence classes of\ud
linear symmetric determinantal representations. \ud
In this paper we tackle the inverse problem, i.e. the construction of the bitangent matrix starting from the 28 bitangents of the plane quartic, and we provide a Sage script intended for computing the bitangent matrix of a given curve
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