## 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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