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).

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.

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