Decompositions of an Abelian surface and quadratic forms
Annales de l'Institut Fourier, Volume 61 (2011) no. 2, pp. 717-743.

When a complex Abelian surface can be decomposed into a product of two elliptic curves, how many decompositions does the Abelian surface admit? We provide arithmetic formulae for the number of such decompositions.

Quand une surface abélienne complexe admet une décomposition en produit de deux courbes elliptiques, combien y a-t-il de telles décompositions possibles ? Nous donnons des formules arithmétiques pour le nombre de telles décompositions.

DOI: 10.5802/aif.2627
Classification: 14K02,  14H52,  11E16
Keywords: Abelian surface, elliptic curve, binary quadratic form
Ma, Shouhei 1

1 University of Tokyo Graduate School of Mathematical Sciences 3-8-1 Komaba, Meguro-ku Tokyo 153-8914 (Japan)
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Ma, Shouhei. Decompositions of an Abelian surface and quadratic forms. Annales de l'Institut Fourier, Volume 61 (2011) no. 2, pp. 717-743. doi : 10.5802/aif.2627. http://archive.numdam.org/articles/10.5802/aif.2627/

[1] Baily, W. L. Jr.; Borel, A. Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2), Volume 84 (1966), pp. 442-528 | DOI | MR | Zbl

[2] Birkenhake, C.; Lange, H. Complex abelian varieties. Second edition, Grundlehren der Mathematischen Wissenschaften, 302, Springer, 2004 | MR | Zbl

[3] Cassels, J. W. S. Rational quadratic forms. London Mathematical Society Monographs, 13, Academic Press, 1978 | MR | Zbl

[4] Cox, D. A. Primes of the form x 2 + n y 2 , Wiley-Interscience, 1989 | MR | Zbl

[5] Hayashida, T. A class number associated with a product of two elliptic curves, Natur. Sci. Rep. Ochanomizu Univ., Volume 16 (1965), pp. 9-19 | MR | Zbl

[6] Hosono, S.; Lian, B. H.; Oguiso, K.; Yau, S.-T. Fourier-Mukai number of a K3 surface, Algebraic structures and moduli spaces (CRM Proc. Lecture Notes), Volume 38, Amer. Math. Soc., Providence, 2004, pp. 177-192 | MR | Zbl

[7] Lange, H. Principal polarizations on products of elliptic curves, The geometry of Riemann surfaces and abelian varieties (Contemp. Math.), Volume 397, Amer. Math. Soc., Providence, 2006, pp. 153-162 | MR | Zbl

[8] Lehner, J.; Newman, M. Weierstrass points of Γ 0 (n), Ann. of Math. (2), Volume 79 (1964), pp. 360-368 | DOI | MR | Zbl

[9] Montgomery, H. L.; Weinberger, P. J. Notes on small class numbers, Acta Arith., Volume 24 (1973/74), pp. 529-542 | MR | Zbl

[10] Nikulin, V. V. Integral symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat., Volume 43 (1979) no. 1, pp. 111-177 | MR | Zbl

[11] Ruppert, W. M. When is an abelian surface isomorphic or isogeneous to a product of elliptic curves?, Math. Z., Volume 203 (1990) no. 2, pp. 293-299 | DOI | MR | Zbl

[12] Shioda, T. The period map of Abelian surfaces, J. Fac. Sci. Univ. Tokyo, Volume 25 (1978) no. 1, pp. 47-59 | MR | Zbl

[13] Shioda, T.; Mitani, N. Singular abelian surfaces and binary quadratic forms, Classification of algebraic varieties and compact complex manifolds (Lecture Notes in Math.), Volume 412, Springer, 1974, pp. 259-287 | MR | Zbl

[14] Stark, H. M. On complex quadratic fields with class-number two, Math. Comp., Volume 29 (1975), pp. 289-302 | MR | Zbl

[15] Taylor, D. E. The geometry of the classical groups, Sigma Series in Pure Mathematics, 9, Heldermann Verlag, 1992 | MR | Zbl

[16] Wall, C. T. C. Quadratic forms on finite groups, and related topics, Topology, Volume 2 (1963), pp. 281-298 | DOI | MR | Zbl

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