elliptic surface

Definition 1.

Let $k$ be afield and let $C/k$ be a smooth projective curve defined over thefield $k$ and has genus $g$ . The function field of $C/k$ will bedenoted by $K=k(C)$ . An elliptic surface $\\mathcal{E}$ over the curve $C$ is, by definition, a two-dimensional projective variety togetherwith:

1.
A morphism $\\pi:\\mathcal{E}\\to C$ such that for all butfinitely many points $t\\in C(\\overline{k})$ , the fiber $\\mathcal{E}_{t}=\\pi^{-1}(t)$ is a non-singular curve of genus $1$ ,
2.
A section to $\\pi$ (the zero section) $\\sigma_{0}:C\\to\\mathcal{E}$ .

With this definition, $\\mathcal{E}/K$ may be regarded as an elliptic curve over thefield $K$ .

Example 1.

The surface $y^{2}=x^{3}+t$ is an elliptic surface over the curve $\\mathbb{P}^{1}(\\mathbb{Q})$ . It may be regarded as an elliptic curve over the function field $\\mathbb{Q}(t)$ .

References

1 R. Miranda, The basic theory of elliptic surfaces,Dottorato di Ricerca in Matematica, Dipartimento di Mathematica dell’ UniversitÃÂ di Pisa, ETS Editrice Pisa, 1989.
2 J. Silverman, Advanced Topics in theArithmetic of Elliptic Curves, Graduate Texts in Mathematics 151,Springer-Verlag, New York.

单词	EllipticSurface
释义	elliptic surface Definition 1. Let $k$ be afield and let $C/k$ be a smooth projective curve defined over thefield $k$ and has genus $g$ . The function field of $C/k$ will bedenoted by $K=k(C)$ . An elliptic surface $\\mathcal{E}$ over the curve $C$ is, by definition, a two-dimensional projective variety togetherwith: 1. A morphism $\\pi:\\mathcal{E}\\to C$ such that for all butfinitely many points $t\\in C(\\overline{k})$ , the fiber $\\mathcal{E}_{t}=\\pi^{-1}(t)$ is a non-singular curve of genus $1$ , 2. A section to $\\pi$ (the zero section) $\\sigma_{0}:C\\to\\mathcal{E}$ . With this definition, $\\mathcal{E}/K$ may be regarded as an elliptic curve over thefield $K$ . Example 1. The surface $y^{2}=x^{3}+t$ is an elliptic surface over the curve $\\mathbb{P}^{1}(\\mathbb{Q})$ . It may be regarded as an elliptic curve over the function field $\\mathbb{Q}(t)$ . References 1 R. Miranda, The basic theory of elliptic surfaces,Dottorato di Ricerca in Matematica, Dipartimento di Mathematica dell’ UniversitÃÂ di Pisa, ETS Editrice Pisa, 1989. 2 J. Silverman, Advanced Topics in theArithmetic of Elliptic Curves, Graduate Texts in Mathematics 151,Springer-Verlag, New York.
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