# nLab elliptic fibration

### Context

#### Bundles

bundles

fiber bundles in physics

## Context

• dependent type theory

• ## Classes of bundles

• covering space

• numerable bundle

• ## Presentations

• bundle gerbe

• groupal model for universal principal ∞-bundles

• microbundle

• ## Examples

• trivial vector bundle

• tautological line bundle

• basic line bundle on the 2-sphere?
• Hopf fibration

• canonical line bundle

• ## Constructions

• clutching construction

• inner product on vector bundles

• dual vector bundle

• projective bundle

• # Contents

## Idea

An elliptic fibration is a bundle of elliptic curves, possibly including some singular fibers.

An elliptic surface is an elliptic fibration over an algebraic curve.

## Properties

### Classification by local systems with modular group coefficients

Write $SL_2(\mathbb{Z})$ for the special linear group in dimension 2 with integer coefficients and write $SL_2(\mathbb{Z}) \to PSL_2(\mathbb{Z})$ for the projection to the corresponding projective linear group. Regarding this as the Möbius group it comes with its natural action on the upper half plane $\mathfrak{h}$. The homotopy quotient $\mathcal{M}_{ell}(\mathbb{C}) = \mathfrak{h}//SL_2(\mathbb{Z})$ is the moduli stack of elliptic curves over the complex numbers.

Accordingly, to any $SL_2(\mathbb{Z})$-principal bundle $P \to B$ (necessarily flat since $SL_2(\mathbb{Z})$ is a discrete group, hence a “local system”) is associated a $\mathfrak{h}$-fiber bundle such that a section of it defines a non-singular elliptic fibration.

One may turn this around: Given an elliptic fibration $E \to B$, then away from the points $S\subset B$ over which the fiber is singular, it is given by an $SL_2(\mathbb{Z})$-local system together with a section of the associated upper-half plane bundle on $B-S$.

With due technical care, this data uniquely characterizes the elliptic fibration (e.g. Miranda 88, prop. VI.3.3).

### Classification of the singular fibers

The remaining singular fibers follow an ADE classification (Kodaira 64, Néron 64, Kodaira 66)

• Wikipedia, elliptic surface

• Rick Miranda, The basic theory of elliptic surfaces, lecture notes 1988 (pdf)

• Viacheslav Nikulin, Elliptic fibrations on K3 surfaces (arXiv:1010.3904)

• Fedor Bogomolov, Yuri Tschinkel, Monodromy of elliptic surfaces (pdf)

• Takahiko Yoshida, Locally standard torus fibrations pdf

The ADE classification of the possible singular fibers is due to

• Kunihiko Kodaira, (1964). “On the structure of compact complex analytic surfaces. I”. Am. J. Math. 86: 751–798. doi:10.2307/2373157. Zbl 0137.17501.

• Kunihiko Kodaira, (1966). “On the structure of compact complex analytic surfaces. II”. Am. J. Math. 88: 682–721. doi:10.2307/2373150. Zbl 0193.37701.

and

• André Néron, (1964). “Modeles minimaux des variétés abeliennes sur les corps locaux et globaux”. Publications Mathématiques de l’IHÉS (in French) 21: 5–128.