Assume that $C$ is a projective curve and $X$ is an elliptic fibration over $C$.



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> What is the picard group of $X$? can we say something about it?
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I think it should be generated by multisections and (components of ) fibres. How can I see it formally?



Picard group of an elliptic fibration is generated by multisection and fibres
Do you know about the [Shioda-Tate formula](https://planetmath.org/shiodatateformula). (I'm surprised there isn't a Wikipedia article about it.) Anyway, it says that if the elliptic fibration does not split as a product and if you have at least one section, then the Neron-Severi group is generated by the following divisors:


1. One irreducible fiber.
2. For each reducible fiber, take all but one of
the components.
3. The zero-section.
4. Generators for the group of sections.


