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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.

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!

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1 Answer

I figure that there are better answers possible, but this might do for a stater:

  • Assuming that your $f$ and $g$ are minimimal (i.e. there is no $u$, with $\deg(u)>0$ s.t $u^4$ divides $f$ and $u^6$ divides $g$), then your Weierstrass equation defines a K3 surface if and only if $\deg(f)\leq 8$ and $\deg(g)\leq 12$ and at least one of $\deg(f)\geq 5, \deg(g)\geq 7$ holds. E.g. $y^2=x^3+z^{12}-1$ defines a smooth K3 surface.

  • This example also contradicts your claim about the fiber type. The above equation defines a smooth surface and has 12 II fibers. Smoothness of the Weierstrass equation implies only that each singular fiber is of type $I_1$ or $II$, not necessary of type $I_1$.

  • One of the things you seem to look for is the monodromy action on $H^1(E,\mathbb{Z})$ for $E$ a smooth fiber. You can find this in many places e.g. Barth-Hulek-Peters-Van de Ven Chapter V. Sec 7-13.

  • The rest of the problem can be reformulated in terms of the Picard-Fuchs differential equation for a family of elliptic curves. There is a book of Stiller studying this, but I have to admit that I never read that book. Texts explaining mirror symmetry to mathematicians contain often a description how to calculate PF equation. (E.g., the book of Cox and Katz.)

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Thanks for pointing out the error in the degree of f and g. I'm working on elliptically fibered CY3-folds over P2 as well and got mixed up there. (Your paper with Klaus Hulek on the Mordell-Weil rank of elliptic threefolds is quite useful in that work.) I also appreciate your pointing out that II fibers might be present. However, I would like to understand the case when we have 24 I1 fibers first. –  D. S. Park Mar 18 '11 at 18:08
    
You are also right to point out that this problem can be formulated in terms of Picard-Fuchs equations. Actually the initial problem I was interested in was finding a solution to a Picard-Fuchs equation. This is the reformulated version of that problem. –  D. S. Park Mar 18 '11 at 18:08
    
There are several further references for explicit calculation of Picard-Fuchs. E.g., there are papers by Peters, Beukers-Stienstra or Verrill. More recently explicit calculation of the (p-adic) PF equation is used by people in point count algorithm. There is a paper by Alan Lauder on average ranks of elliptic curves over function fields. This paper is based on algorithm that calculates solutions to the p-adic PF equation. –  Remke Kloosterman Mar 20 '11 at 13:09
    
Thanks for the kind response and references. I will be going through them. –  D. S. Park Mar 23 '11 at 15:58
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