An elliptic surface is an elliptic fibration over an algebraic curve.
Write for the special linear group in dimension 2 with integer coefficients and write 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 . The homotopy quotient is the moduli stack of elliptic curves over the complex numbers.
Accordingly, to any -principal bundle (necessarily flat since is a discrete group, hence a “local system”) is associated a -fiber bundle such that a section of it defines a non-singular elliptic fibration.
One may turn this around: Given an elliptic fibration , then away from the points over which the fiber is singular, it is given by an -local system together with a section of the associated upper-half plane bundle on .
With due technical care, this data uniquely characterizes the elliptic fibration (e.g. Miranda 88, prop. VI.3.3).
Wikipedia, elliptic surface
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.