Weber’s formula for the bitangents of a smooth plane quartic
Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 2 (2019), pp. 5-17.

In a section of his 1876 treatise Theorie der Abel’schen 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.

Dans une section de son traité Theorie der Abel’schen Functionen vom Geschlecht 3, paru en 1876, Weber a démontré une formule qui permet de déterminer les équations des bitangentes d’une quartique plane non singulière à partir des constantes theta de Riemann (Thetanullwerte). Le but de cette note est de présenter la formule de Weber en langage moderne. On aussi montre une connexion avec la matrice universelle des bitangentes.

Received:
Published online:
DOI: 10.5802/pmb.33
Classification: 14H42,  14H45,  14K25
Keywords: Plane quartics, theta functions, bitangents.
Alessio Fiorentino 1

1 Institut de recherche mathématique de Rennes - IRMAR, Université de Rennes 1
@article{PMB_2019___2_5_0,
     author = {Alessio Fiorentino},
     title = {Weber{\textquoteright}s formula for the bitangents of a smooth plane quartic},
     journal = {Publications math\'ematiques de Besan\c{c}on. Alg\`ebre et th\'eorie des nombres},
     pages = {5--17},
     publisher = {Presses universitaires de Franche-Comt\'e},
     number = {2},
     year = {2019},
     doi = {10.5802/pmb.33},
     language = {en},
     url = {https://pmb.centre-mersenne.org/articles/10.5802/pmb.33/}
}
TY  - JOUR
TI  - Weber’s formula for the bitangents of a smooth plane quartic
JO  - Publications mathématiques de Besançon. Algèbre et théorie des nombres
PY  - 2019
DA  - 2019///
SP  - 5
EP  - 17
IS  - 2
PB  - Presses universitaires de Franche-Comté
UR  - https://pmb.centre-mersenne.org/articles/10.5802/pmb.33/
UR  - https://doi.org/10.5802/pmb.33
DO  - 10.5802/pmb.33
LA  - en
ID  - PMB_2019___2_5_0
ER  - 
%0 Journal Article
%T Weber’s formula for the bitangents of a smooth plane quartic
%J Publications mathématiques de Besançon. Algèbre et théorie des nombres
%D 2019
%P 5-17
%N 2
%I Presses universitaires de Franche-Comté
%U https://doi.org/10.5802/pmb.33
%R 10.5802/pmb.33
%G en
%F PMB_2019___2_5_0
Alessio Fiorentino. Weber’s formula for the bitangents of a smooth plane quartic. Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 2 (2019), pp. 5-17. doi : 10.5802/pmb.33. https://pmb.centre-mersenne.org/articles/10.5802/pmb.33/

[1] Enrico Arbarello; Maurizio Cornalba; Phillip A. Griffiths; Joseph Harris Geometry of algebraic curves, Grundlehren der Mathematischen Wissenschaften, 267, Springer, 1985 | MR: 770932 | Zbl: 1235.14002

[2] Lucia Caporaso; Edoardo Sernesi Recovering plane curves from their bitangents, J. Algebr. Geom., Volume 12 (2003) no. 2, pp. 225-244 | Article | MR: 1949642 | Zbl: 1080.14523

[3] Francesco Dalla Piazza; Alessio Fiorentino; Riccardo Salvati Manni Plane quartics: the universal matrix of bitangents, Isr. J. Math., Volume 217 (2017), pp. 111-138 | Article | MR: 3625106 | Zbl: 1364.14027

[4] Igor V. Dolgachev Classical algebraic geometry: a modern view, Cambridge University Press, 2012 | Zbl: 1252.14001

[5] John Fay On the Riemann–Jacobi formula, Nachr. Akad. Wiss. Gött., Volume 1979 (1979), pp. 61-73 | MR: 568803 | Zbl: 0441.33001

[6] Georg Frobenius Über die constanten Factoren der Thetareihen, Kronecker J., Volume 98 (1885), pp. 244-263 | MR: 1580037 | Zbl: 17.0478.01

[7] Phillip A. Griffiths; Joseph Harris Principles of algebraic geometry, Pure and Applied Mathematics, John Wiley & Sons, 1978 | Zbl: 0408.14001

[8] Benedict H. Gross; Joseph Harris On some geometric constructions related to theta characteristics, Contributions to automorphic forms, geometry, and number theory, Johns Hopkins University Press, 2004, pp. 279-311 | Zbl: 1072.14032

[9] Jordi Guàrdia Jacobian Nullwerte and algebraic equations, J. Algebra, Volume 253 (2002) no. 1, pp. 112-132 | Article | MR: 1925010 | Zbl: 1054.14041

[10] Jordi Guàrdia On the Torelli problem and Jacobian Nullwerte in genus three, Mich. Math. J., Volume 60 (2011) no. 1, pp. 51-65 | Article | MR: 2785863 | Zbl: 1316.14021

[11] Jun-Ichi Igusa Theta functions, Grundlehren der Mathematischen Wissenschaften, 194, Springer, 1972 | Zbl: 0251.14016

[12] Jun-Ichi Igusa Multiplicity one theorem and problems related to Jacobi’s formula, Am. J. Math., Volume 105 (1983), pp. 157-187 | Article | MR: 692109 | Zbl: 0527.14037

[13] David Lehavi Any smooth plane quartic can be reconstructed from its bitangents, Isr. J. Math., Volume 146 (2005), pp. 371-379 | Article | MR: 2151609 | Zbl: 1076.14037

[14] Enric Nart; Christophe Ritzenthaler A new proof of a Thomae-like formula for non hyperelliptic genus 3 curves, Arithmetic, geometry, cryptography and coding theory (Contemporary Mathematics), Volume 686, American Mathematical Society, 2017, pp. 137-155 | Article | MR: 3630613 | Zbl: 1374.14026

[15] Harry E. Rauch; Hershel M. Farkas Theta functions with applications to Riemann surfaces, The Williams & Wilkins Company, 1974 | Zbl: 0292.30015

[16] Heinrich Weber Theorie der Abel’schen Functionen vom Geschlecht 3, Druck und Verlag von Georg Reimer, 1876 | Zbl: 08.0293.01

Cited by Sources: