Let $\tau $ be in the upper half-plane and $E_{\tau}$ be the associated elliptic curve. $E_{\tau}$ has an associated canonical principal polarization, which is equivalent to the datum of a skew-symmetric form on $H_1(E_{\tau})$ such that the associated quadratic form is positive-definite.
- What is the explicit formula, in terms of $\tau$, for this skew-symmetric form?
- What is the explicit formula, in terms of $\tau$, for the associated quadratic form? If the skew-symmetric form is denote $S(x,y)$ and the complex structure is given by $J$, then the associated quadratic form has bilinear form $S(Jx, y)$.)
I tried chasing through the definitions but have ended up with an answer that doesn't seem right, so I wonder if anyone knows off the top of their head or if this is written down somewhere.