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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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References:

[1] Artin, M.: SupersingularK3-surfaces. Ann. Sci. Ec. Norm. Super.7, 543-568 (1974) · Zbl 0322.14014
[2] Artin, M., Mazur, B.: Formal groups arising from algebraic varieties. Ann. Sci. Ec. Norn. Super.10, 87-132 (1977) · Zbl 0351.14023
[3] Barth, W., Peters, C., Van de Ven A.: Compact complex surfaces. Ergebnisse der Mathematik 3. Folge, Bd. 4, Berlin, Heidelberg, New York: Springer 1984
[4] Beauville, A.: Les familles stables de courbes elliptiques sur ?2 admettant quatre fibres singulières. C.R. Acad. Sci. Paris294, 657 (1982) · Zbl 0504.14016
[5] Berthelot, P., Ogus, A.: Notes on crystalline cohomology. Mathematical Notes 21. Princeton: Princeton University Press 1978 · Zbl 0383.14010
[6] Beukers, F.: Irrationality of ?2, periods of an elliptic curve and? 1(5). In: Approximations diophantiennes et nombres transcendants, Luminy 1982. Progress in Mathematics 31. Basel, Boston, Stuttgart: Birkhäuser 1983
[7] Beukers, F.: Arithmetical properties of Picard-Fuchs equations. Sém. de Theorie des Nombres, Paris 1982-1983, Progress in Mathematics 51. Basel, Boston, Stuttgart: Birkhäuser 1984
[8] Bombieri, E., Mumford, D.: Enriques classification of surfaces in char.p, II. In: Complex analysis and algebraic geometry (Bailey, Shioda, T., eds.) Cambridge: 1977 · Zbl 0348.14021
[9] Gauss, C.: Werke, Band II, pp. 87-91.
[10] Hazewinkel, M.: Formal groups and applications. New York: Academic Press 1978 · Zbl 0454.14020
[11] Honda, T.: On the theory of commutative formal groups. J. Math. Soc. Japan22, 213-246 (1970) · Zbl 0202.03101
[12] Shimura, G.: Introduction to the arithmetic theory of automorphic forms. Iwanami Shoten Publishers and Princeton: Princeton University Press 1971 · Zbl 0221.10029
[13] Honda, T., Miyawaki, I.: Zeta-functions of elliptic curves of 2-power conductor. J. Math. Soc. Japan26, 362-373 (1974) · Zbl 0273.14007
[14] Illusie, L.: Complexe de De Rham-Witt et cohomologie cristalline. Ann. Sci. Ecol. Norn. Super.12, 501-661 (1979) · Zbl 0436.14007
[15] Ince, E.: Ordinary differential equations. New York: Dover 1927 · JFM 53.0399.07
[16] Kas, A.: Weierstrass normal forms and invariants of elliptic surfaces. Trans. Am. Math. Soc.225, 259-266 (1977) · Zbl 0402.14014
[17] Katz, N.: Expansion coefficients as approximate solutions of differential equations. Cohomologiep-adique. Astérisque 119-120 (1984)
[18] Klein, F.: Vorlesungen über die Theorie der elliptischen Modulfunktionen, ausgearbeitet und vervollständigt von R. Fricke. Leipzig 1892 · JFM 22.0455.02
[19] Klein, F.: Vorlesungen über die hypergeometrische Funktion. Grundlehren der mathematischen Wissenschaften, Bd. 39. Berlin, Heidelberg, New York: Springer 1981
[20] Kodaira, K.: On compact analytic surfaces. II. Ann. Math.77, 563-626 (1963) · Zbl 0118.15802
[21] Lang, S.: Elliptic functions. New York: Addison-Wesley 1973 · Zbl 0316.14001
[22] Manin, Yu.: Rational points on algebraic curves over function fields. Izv. Akad. Nauk SSSR Ser. Mat.27, (1963) A.M.S. Transl. Ser. 2,50, 189-234 · Zbl 0178.55102
[23] Néron, A.: Modèles mininaux des variétés abéliennes sur les corps locaux et globaux, Publ. Math. I.H.E.S.21 (1964)
[24] Raynaud, M.: Modèles de Néron. C.R. Acad. Sci. Paris Sér. A262, 345 (1966)
[25] Schmickler-Hirzbruch, U.: Elliptische Flächen über ?1? mit drei Ausnahmefasern und die hypergeometrische Differentialgleichung. Diplomarbeit, Universität Bonn 1978
[26] Shioda, T.: On elliptic modular surfaces. J. Math. Soc. Japan24, 20-59 (1972) · Zbl 0226.14013
[27] Shioda, T., Inose, H.: On singularK3-surfaces. In: Complex analysis and algebraic geometry. (Baily, Shioda, T., eds.) Cambridge 1977 · Zbl 0374.14006
[28] Stienstra, J.: Cartier-Dieudonné theory for Chow groups. J. Reine Angew. Math.355, 1-166 (1985) · Zbl 0545.14013
[29] Tate, J.: The arithmetic of elliptic curves. Invent. Math.23, 179-206 (1974) · Zbl 0296.14018
[30] Van der Poorten, A.: A proof that Euler missed ... Apéry’s proof of the irrationality of ? (3). Math. Intell.1, 195-203 (1978) · Zbl 0409.10028
[31] Vinberg, E.: The two most algebraicK(3)-surfaces. Math. Ann.265, 1-21 (1983) · Zbl 0537.14025
[32] Beukers, F.: Une formule explicite dans la théorie des courbes elliptiques. Preprint (1984)
[33] Hartshorne, R.: Algebraic geometry. Graduate Text in Mathematics, vol. 52. Berlin, Heidelberg, New York: Springer 1977 · Zbl 0367.14001
[34] MacMahon: Combinatory analysis. New York: Chelsea 1066
[35] Cox, D., Parry, W.: Torsion in elliptic curves overk(t). Compositflo Math.41, 337-354 (1980). · Zbl 0442.14015
[36] Macdonald, I.: Affine root systems and Dedekind’s ?-function. Invent. Math.15, 91-143 (1972) · Zbl 0244.17005
[37] Oda, T.: Formal groups attached to elliptic modular forms. Invent. Math.61, 81-102 (1980) · Zbl 0433.14037
[38] Schoeneberg, B.: Über den Zusammenhang der Eisensteinschen Reihen und Thetareihen mit der Diskriminante der elliptischen Funktionen. Math. Ann.126, 177-184 (1953) · Zbl 0053.05403
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