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On the Picard-Fuchs equation and the formal Brauer group of certain elliptic \(K3\)-surfaces. (English) Zbl 0539.14006

In this paper one studies elliptic pencils which can be put in Weierstrass form
\[ \text{(W)}\colon\quad y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6\quad\text{with } a_1,\dots,a_6\in\mathbb{Z}[t], \]
non-constant discriminant \(\Delta(t)\) and \(j\)-invariant \(J(t)\). It is shown:
(1) Assume (*): (W) reduces mod \(t\) to \(y^2+xy=x^3.\) Then the Picard-Fuchs equation of the pencil admits a unique solution \(f(t)=\sum f_nt^{n- 1}\) in \(\mathbb{Z}[[t]]\) with \(f_1=1\).
(2) In the situation of (1) let \(\ell(t)=\sum n^{-1}f_nt^n\) and let \(D\) be the gcd of the coefficients of \(\Delta(t)\). Assume (*) and also (**): \( \deg a_i(t)\leq 2i\) for \(1\leq i\leq 6\). Then the series \(G(t_1,t_2)\overset{\text{def}} = \ell^{-1}(\ell(t_1)+\ell(t_2))\) has coefficients in \(\mathbb{Z}[1/D]\); so \(G(t_1,t_2)\) is a 1-parameter formal group law over \(\mathbb{Z}[1/D]\).
(3) Let \(p\) be a prime number, \(p\nmid D\). Assume in addition to (*) and (**) also (***): equation (W) mod \(p\) is the Weierstrass form of an elliptic pencil on a \(K3\)-surface \(X_p\) over \(\mathbb{F}_ p\). Then \(G(t_1,t_2) \bmod p\) is a formal group law for the formal Brauer group of \(X_p\).
(4) Via Cartier-Dieudonné theory and crystalline cohomology the formal Brauer group of \(X_p\) is connected with the zeta-function of \(X_p/\mathbb{F}_p\). The resulting relation between \(f(t)\) and \(Z(X_p/\mathbb{F}_p,t)\) can be expressed as Atkin-Swinnerton-Dyer type congruences.
Besides the general theory sketched above the paper contains a number of concrete detailed examples in which \(f(t)\), \(Z(X_p/\mathbb{F}_p,t)\) and the resulting congruences are explicitly computed. Thus one shows, for instance:
Let \(u_n=0\) if \(n\) is odd, \(u_n=(-1)^m\sum_{k}\binom{m}{k}^ 2\binom{m+k}{k}\) if \(n=2m\) (famous numbers in Apéry’s irrationality proof for \(\zeta(2))\). For an odd prime \(p\) let \(\alpha_p=0\) if \(p\equiv 3\bmod 4\) resp. \(\alpha_p=2p-4a^2\) if \(p=a^2+4c^2,\) \(a,c\in\mathbb{Z}\). Then \(u_n+\alpha_pu_{n/p}+p^2u_{n/p^2}\equiv 0\bmod p^r\) if \(p^r\mid n\) (convention: \(u_q=0\) if \(q\not\in\mathbb{N})\).
Reviewer: Frits Beukers

MSC:

14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
14L05 Formal groups, \(p\)-divisible groups
14J25 Special surfaces
14F30 \(p\)-adic cohomology, crystalline cohomology
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11G35 Varieties over global fields
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