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 $12$ and $18$ respectively in $z$ in order for the manifold to be K3. 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.

1. 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)?

2. [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 $i$.

3. It would be great if someone can recommend some references where I can learn to deal with problems of this nature.

Thanks in advance!