Explicit rational group law on hyperelliptic Jacobians of any genus

• Autorzy: Urbanik David

Identyfikatory

DOI

10.4064/ba230330-7-7

• Języki publikacji: EN

Abstrakty

EN

It is well-known that abelian varieties are projective, and so there exist explicit polynomial and rational functions which define both the variety and its group law. It is however difficult to find any explicit polynomial and rational functions describing these varieties or their group laws in dimensions greater than two. One exception can be found in Mumford’s classic “Lectures on Theta”, where he describes how to obtain an explicit model for hyperelliptic Jacobians as the union of several affine pieces described as the vanishing locus of explicit polynomial equations. In this article, we extend this work to give explicit equations for the group law on a dense open set. One can view these equations as generalizations of the usual chord-based group law on elliptic curves.

Słowa kluczowe

EN

abelian varieties; Jacobian; group law

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2023
• Tom: Vol. 71, no 1
• Strony: 35--52
• Opis fizyczny: Bibliogr. 14 poz.

Twórcy

autor

Urbanik David

• Institut des Hautes Études Scientifiques, 91440 Paris, France

Bibliografia

• [1] D. G. Cantor, Computing in the Jacobian of a hyperelliptic curve, Math. Comp. 48 (1987), 95-101.
• [2] C. Costello and K. Lauter, Group law computations on Jacobians of hyperelliptic curves, in: A. Miri and S. Vaudenay (eds.), Selected Areas in Cryptography, Springer, 2012, 92-117.
• [3] C. G. J. Jacobi, Über die Darstellung einer Reihe gegebner Werthe durch eine gebrochne rationale Function, J. Reine Angew. Math. 30 (1846), 127-156.
• [4] D. Grant, Formal groups in genus two, J. Reine Angew. Math. 411 (1990), 96-121.
• [5] F. Leitenberger, About the group law for the Jacobi variety of a hyperelliptic curve, Beiträge Algebra Geom. 46 (2005), 125-130.
• [6] J. S. Milne, Abelian varieties (v2.00), 2008; www.jmilne.org/math/.
• [7] D. Mumford, On the equations defining abelian varieties. I, Invent. Math. 1 (1966), 287-354.
• [8] D. Mumford, On the equations defining abelian varieties. II, Invent. Math. 3 (1967), 75-135.
• [9] D. Mumford, On the equations defining abelian varieties. III, Invent. Math. 3 (1967) 215-244.
• [10] D. Mumford, Tata Lectures on Theta II, Progress Math. 43, Birkhäuser, 1984.
• [11] J. H. Silverman, The Arithmetic of Elliptic Curves, Grad. Texts Math. 106, Springer, New York, 1986.
• [12] D. Urbanik, Hyperelliptic Jacobian arithmetic (source code), www.ihes.fr/~urbanik/work/jacarith_dburbani_August2018.zip, 2018.
• [13] E. V. Flynn, The group law on the Jacobian of a curve of genus 2, J. Reine Angew. Math. 439 (1993), 45-69.
• [14] E. V. Flynn, The Jacobian and formal group of a curve of genus 2 over an arbitrary ground field, Math. Proc. Cambridge Philos. Soc. 107 (1990), 425-441.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).