To paraphrase the definition given in Katz and Mazur's "Arithmetic moduli of elliptic curves", an elliptic curve over a base scheme $S$ is a pair of a group scheme $f: E \to S$ that is a smooth curve over $S$, together with a distinguised $S$-point/section: $$\infty: S \to E$$ such that all geometric fibres $E_s$, i.e. all pullbacks: $$E_s := E \times_{s, S, f} \mathrm{Spec} (k_s)$$ are curves of genus $1$ over $k_s$ (note that the arithmetic and geometric genera agree), wherein $k_s$ is the residue field at some point $s \in S$.
My question is then, why is the point $\infty$ needed to define an elliptic curve ? Alternatively, are there examples of genus 1 curves that are not elliptic ? Thank you.