Equations for the Jacobian

of a hyperelliptic curve

Author:
Paul van Wamelen

Journal:
Trans. Amer. Math. Soc. **350** (1998), 3083-3106

MSC (1991):
Primary 14H40; Secondary 14H42

DOI:
https://doi.org/10.1090/S0002-9947-98-02056-X

MathSciNet review:
1432144

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We give an explicit embedding of the Jacobian of a hyperelliptic curve, , into projective space such that the image is isomorphic to the Jacobian over the splitting field of . The embedding is a modification of the usual embedding by theta functions with half integer characteristics.

**[Fly90]**E. V. Flynn. The Jacobian and formal group of a curve of genus over an arbitrary ground field.*Math. Proc. Cambridge Philos. Soc.*, 107:425-441, 1990. MR**91b:14025****[GG93]**D. M. Gordon and D. R. Grant. Computing the Mordell-Weil rank of Jacobians of curves of genus .*Trans. Amer. Math. Soc.*, 337:807-824, 1993. MR**93h:11057****[Gra85]**D. R. Grant.*Theta Functions and Division Points on Abelian Varieties of Dimension Two*. PhD thesis, MIT, 1985.**[Gra90]**D. R. Grant. Formal groups in genus .*J. Reine Angew. Math.*, 411:96-121, 1990. MR**91m:14045****[Har77]**R. Hartshorne.*Algebraic Geometry*. Springer-Verlag, 1977. MR**57:3116****[Iit82]**Shigeru Iitaka.*Algebraic Geometry*. Springer-Verlag, 1982. MR**84j:14001****[Lan82]**S. Lang.*Introduction to Algebraic and Abelian Functions*. Springer-Verlag, 1982. MR**84m:14032****[LB92]**H. Lange and Ch. Birkenhake.*Complex Abelian Varieties*. Springer-Verlag, 1992. MR**94j:14001****[Mil86a]**J.S. Milne. Abelian varieties. In G. Cornell and J.H. Silverman, editors,*Arithmetic Geometry*. Springer-Verlag, 1986, pp. 103-150. MR**89b:14029****[Mil86b]**J.S. Milne. Jacobian varieties. In G. Cornell and J.H. Silverman, editors,*Arithmetic Geometry*. Springer-Verlag, 1986, pp. 167-212. MR**89b:14029****[Mum83]**D. Mumford.*Tata Lectures on Theta I*, volume 28 of*Progr. Math.*Birkhäuser, 1983. MR**85h:14026****[Mum84]**D. Mumford.*Tata Lectures on Theta II*, volume 43 of*Progr. Math.*Birkhäuser, 1984. MR**86b:14017****[Pil90]**J. Pila. Frobenius maps of abelian varieties and finding roots of unity in finite fields.*Math. Comp.*, 55(192):745-763, 1990. MR**91a:11071**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
14H40,
14H42

Retrieve articles in all journals with MSC (1991): 14H40, 14H42

Additional Information

**Paul van Wamelen**

Affiliation:
Department of Mathematics, University of California, San Diego, San Diego, California 92093

Address at time of publication:
Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803-4918

Email:
wamelen@math.lsu.edu

DOI:
https://doi.org/10.1090/S0002-9947-98-02056-X

Keywords:
Jacobian,
hyperelliptic curve,
theta function,
theta constant,
Thomae's identity

Received by editor(s):
December 5, 1995

Article copyright:
© Copyright 1998
American Mathematical Society