Elif Ozbay GurlerHuseyin Hisil2025-10-0620242190-85082190-851610.1007/s13389-024-00351-7http://dx.doi.org/10.1007/s13389-024-00351-7https://gcris.yasar.edu.tr/handle/123456789/7298This manuscript provides complete inversion-free and explicit group law formulas in Jacobian coordinates for the genus 2 hyperelliptic curves of the form y 2 = x 5 + a 3 x 3 + a 2 x 2 + a 1 x + a 0 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y<^>2 = x<^>5+a_3 x<^>3+a_2 x<^>2+a_1 x+a_0$$\end{document} over a field K with char ( K ) not equal 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{char}(K) \ne 2$$\end{document} . The formulas do not require the use of polynomial arithmetic operations such as resultant mod or gcd computations but only operations in K.EnglishGroup law, Genus 2, Hyperelliptic curves, Explicit formulas, Jacobian coordinatesCURVESArticle