Bayer, Pilar; Guàrdia, Jordi On equations defining fake elliptic curves. (English) Zbl 1093.11042 J. Théor. Nombres Bordx. 17, No. 1, 57-67 (2005). Given “fake elliptic curves” (i.e. principally polarized abelian surfaces with potential multiplication by quaternion algebras), it is a natural question to look for genus 2 algebraic curves whose Jacobians give rise to those abelian surfaces. The paper under review gives a method for computing such equations in the case where the fake elliptic curves also have complex multiplication. Those particular abelian surfaces are known to be isogenous to squares of elliptic curves with complex multiplication. As an illustration of the method, equations for curves whose Jacobians appear as special points on Shimura curves corresponding to quaternion algebra with discriminant 6, 10 and 15 are given. Reviewer: Pierre Parent (Talence) Cited in 1 ReviewCited in 1 Document MSC: 11G18 Arithmetic aspects of modular and Shimura varieties 11G15 Complex multiplication and moduli of abelian varieties Keywords:abelian sufaces; Shimura curves; explicit equations of curves PDFBibTeX XMLCite \textit{P. Bayer} and \textit{J. Guàrdia}, J. Théor. Nombres Bordx. 17, No. 1, 57--67 (2005; Zbl 1093.11042) Full Text: DOI Numdam EuDML Link References: [1] M. Alsina, Binary quadratic forms and Eichler orders. Journées Arithmétiques Graz 2003, in this volume. · Zbl 1079.11022 [2] M. Alsina, P. Bayer, Quaternion orders, quadratic forms and Shimura curves. CRM Monograph Series 22. AMS, 2004. · Zbl 1073.11040 [3] P. Bayer, Uniformization of certain Shimura curves. In Differential Galois Theory, T. Crespo and Z. Hajto (eds.), Banach Center Publications 58 (2002), 13-26. · Zbl 1036.11026 [4] K. Buzzard, Integral models of certain Shimura curves. Duke Math. J. 87 (1996), 591-612. · Zbl 0880.11048 [5] J. Guàrdia, Jacobian nullwerte and algebraic equations. Journal of Algebra 253 (2002), 112-132. · Zbl 1054.14041 [6] J. Guàrdia, Jacobian Nullwerte, periods and symmetric equations for hyperelliptic curves. In preparation. · Zbl 1177.11052 [7] M. Eichler, Zur Zahlentheorie der Quaternionen-Algebren. J. reine angew. Math. 195 (1955), 127-151. · Zbl 0068.03303 [8] K. Hashimoto, N. Murabayashi, Shimura curves as intersections of Humbert surfaces and defining equations of QM-curves of genus two. Tôhoku Math. J. 47 (1995), 271-296. · Zbl 0838.11044 [9] B.W. Jordan, On the Diophantine Arithmetic of Shimura Curves. Thesis. Harvard University, 1981. [10] J.S. Milne, Points on Shimura varieties mod p. Proceed. of Symposia in Pure Mathematics 33, part 2 (1979), 165-184. · Zbl 0418.14022 [11] A. Mori, Explicit Period Matrices for Abelian Surfaces with Quaternionic Multiplications. Bollettino U. M. I. (7), 6-A (1992), 197-208. · Zbl 0767.14018 [12] F. Rodríguez-Villegas, Explicit models of genus 2 curves with split CM. Algorithmic number theory (Leiden, 2000). Lecture Notes in Compt. Sci. 1838, 505-513. Springer, 2000. · Zbl 1032.11026 [13] V. Rotger, Abelian varieties with quaternionic multiplication and their moduli. Thesis. Universitat de Barcelona, 2002. [14] V. Rotger, Quaternions, polarizations and class numbers. J. reine angew. Math. 561 (2003), 177-197. · Zbl 1094.11022 [15] V. Rotger, Modular Shimura varieties and forgetful maps. Trans. Amer. Math. Soc. 356 (2004), 1535-1550. · Zbl 1049.11061 [16] G. Shimura , Construction of class fields and zeta functions of algebraic curves. Annals of Math. 85 (1967), 58-159. · Zbl 0204.07201 [17] G. Shimura , On the derivatives of theta functions and modular forms. Duke Math. J. 44 (1977), 365-387. · Zbl 0371.14023 [18] G. Shimura , Abelian varieties with complex multiplication and modular functions. Princeton Series, 46. Princeton University Press, 1998. · Zbl 0908.11023 [19] M.-F. Vignéras, Arithmétique des algèbres de quaternions. LNM 800. Springer, 1980. · Zbl 0422.12008 [20] A. Weil, Sur les périodes des intégrales abéliennes. Comm. on Pure and Applied Math. 29 (1976), 813-819. · Zbl 0342.14020 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.