Decompositions of an Abelian surface and quadratic forms
[Décompositions d’une surface abélienne et formes quadratiques]
Annales de l'Institut Fourier, Tome 61 (2011) no. 2, pp. 717-743.

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.

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.

DOI : 10.5802/aif.2627
Classification : 14K02, 14H52, 11E16
Keywords: Abelian surface, elliptic curve, binary quadratic form
Mot clés : surface abélienne, courbe elliptique, forme quadratique
Ma, Shouhei 1

1 University of Tokyo Graduate School of Mathematical Sciences 3-8-1 Komaba, Meguro-ku Tokyo 153-8914 (Japan)
@article{AIF_2011__61_2_717_0,
     author = {Ma, Shouhei},
     title = {Decompositions of an {Abelian} surface and quadratic forms},
     journal = {Annales de l'Institut Fourier},
     pages = {717--743},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {61},
     number = {2},
     year = {2011},
     doi = {10.5802/aif.2627},
     zbl = {1231.14036},
     mrnumber = {2895071},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2627/}
}
TY  - JOUR
AU  - Ma, Shouhei
TI  - Decompositions of an Abelian surface and quadratic forms
JO  - Annales de l'Institut Fourier
PY  - 2011
SP  - 717
EP  - 743
VL  - 61
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2627/
DO  - 10.5802/aif.2627
LA  - en
ID  - AIF_2011__61_2_717_0
ER  - 
%0 Journal Article
%A Ma, Shouhei
%T Decompositions of an Abelian surface and quadratic forms
%J Annales de l'Institut Fourier
%D 2011
%P 717-743
%V 61
%N 2
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2627/
%R 10.5802/aif.2627
%G en
%F AIF_2011__61_2_717_0
Ma, Shouhei. Decompositions of an Abelian surface and quadratic forms. Annales de l'Institut Fourier, Tome 61 (2011) no. 2, pp. 717-743. doi : 10.5802/aif.2627. http://www.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

Cité par Sources :