Suppose I have a smooth elliptically fibered K3 manifold over $\mathbb{P}^1$ defined by the Weierstrass equation,
\begin{equation} y^2=x^3+f(z)x+g(z) \end{equation}
where $x,y,z$ are local coordinates. $z$ is a local coordinate on the base. $f$ and $g$ should be of order $8$ and $12$ respectively in $z$ in order for the manifold to be K3 [I have wrongly asserted that $f$ and $g$ should be of order $12$ and $18$ initially. Thanks to Remke Kloosterman for pointing this out.] As the manifold is smooth, there would be 24 $I_1$ fibers at loci, $P_1, \cdots, P_{24}$.
It is clear that the period integrals, \begin{equation} A\equiv \oint_{\alpha} \lambda,\quad B\equiv \oint_{\beta} \lambda \end{equation} for a closed meromorphic 1-form $\lambda(z)$ for the two 1-cycles $\alpha,\beta$ of the torus are $SL(2,\mathbb{Z})$ doublets when viewed as `functions' of the base coordinate $z$. They are well defined in simply connected patches not containing the degeneration points. This pair would undergo monodromies when taken around the 24 degeneration points.
My questions are the following.
- Is there always a $\lambda$ that makes $(A,B)$ non-singular at all points $z$ of the base? I expect there to be a $\lambda$ where $(A,B)$ behave near all $P_i$ (up to some $SL(2,\mathbb{Z})$ transformation,)
\begin{equation} A\sim A_i (z-z_i)+…,\quad B\sim C_i + B_i (z-z_i) \ln (z-z_i)+… \end{equation}
for constants $A_i, B_i,C_i$ at leading order in $(z-z_i)$, so despite the monodromies their values are finite. In fact, I expect that $\lambda$ to be the meromorphic differential that satisfies,
\begin{equation} {d \lambda \over dz} = {dx \over y} \end{equation}
where $dx/y$ is the unique holomorphic differential on the torus fiber at given $z$. Is this true? Is such a $\lambda$ unique(up to an exact form)?
[Main Question] If there is such a lambda, I would like to know the values of $(A,B)$ at the points $P_i$. In other words, I would like to know the values of $C_i$ for each $i$.
It would be great if someone can recommend some references where I can learn to deal with problems of this nature.
Thanks in advance!