I am trying to understand a statement in Andrew Snowden's Notes regarding the representability of the functor $F_{\Gamma(3)}$ (Section 14.1).
Here, $F_{\Gamma(N)}(S)= \{isom \; classes \; of \; (E,(P,Q)) \; over \; S\}.$ In Section 14.1, the notes state:
" Riemann-Roch says we can pick a function $x$ on E w/ a pole of order 2 along the zero section, and no other poles. This will be unique upto $x \mapsto ax+b$. "
My question(1): What is the notion of a "function on E having a pole of order 2 along the zero section"?
Additionally: While reading Hida's book "Geometric modular forms and elliptic curves" (section 2.2.5), I encountered a similar concept where x is expressed as a power series.
Specific question(2): How can we write x as a power series and why does it have a pole of order 2?
Context: I have read the previous section in Hida's book (section 2.2.4) where he discusses the power series expansion of $\omega \in \Gamma(E, \underline{\omega}_{E/S})$, but I am unsure how this relates to writing x as a power series.