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elliptic fibration
Context
Elliptic cohomology
Bundles
bundles

Context
Classes of bundles
Universal bundles
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Examples and Applications
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 ).

References
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