A. [. Ballentine, E. L. Guillevic, C. García, M. Martindale, B. Massierer et al., Isogenies for point counting on genus two hyperelliptic curves with maximal real multiplication " . working paper or preprint
URL : https://hal.archives-ouvertes.fr/hal-01421031

]. T. Bec94 and . Becker, On Gröbner bases under specialization, Applicable Algebra in Engineering, Communication and Computing, vol.51, pp.1-8, 1994.

H. [. Birkenhake and . Lange, Complex abelian varieties Grundlehren der Mathematischen Wissenschaften, pp.27-35, 2003.

H. [. Birkenhake and . Wilhelm, Humbert surfaces and the Kummer plane, In: Transactions of the American Mathematical society, vol.3555, issue.13, pp.1819-1841, 2003.

K. [. Bröker and . Lauter, Modular Polynomials for Genus 2, LMS Journal of Computation and Mathematics, vol.7, pp.326-339, 2009.
DOI : 10.5802/jtnb.142

D. [. Bröker, K. Gruenewald, and . Lauter, Explicit CM theory for level 2-structures on abelian surfaces, Algebra & Number Theory, vol.44, issue.4, pp.495-528, 2011.
DOI : 10.1090/S0025-5718-02-01422-9

K. [. Bröker, A. Lauter, and . Sutherland, Modular polynomials via isogeny volcanoes, Mathematics of Computation, vol.81, issue.278, pp.1201-1231, 2012.
DOI : 10.1090/S0025-5718-2011-02508-1

J. H. Bruinier, Hilbert Modular Forms and Their Applications, pp.105-179, 2008.
DOI : 10.1007/978-3-540-74119-0_2
URL : http://arxiv.org/abs/math/0609763

K. [. Charles, E. Lauter, and . Goren, Cryptographic Hash Functions from Expander Graphs, Journal of Cryptology, vol.4, issue.2, pp.93-113, 2009.
DOI : 10.1007/978-1-4757-1920-8
URL : http://csrc.nist.gov/groups/ST/hash/documents/LAUTER_HashJuly27.pdf

J. Couveignes and R. Lercier, Elliptic periods for finite fields, Finite Fields and Their Applications, vol.15, issue.1, pp.1-22, 2009.
DOI : 10.1016/j.ffa.2008.07.004
URL : https://hal.archives-ouvertes.fr/hal-00630391

. J. Ddr, A. Dimitar, D. Dudeanu, and . Robert, Computing cyclic isogenies using in genus 2

T. [. Doche, D. Icart, and . Kohel, Efficient Scalar Multiplication by Isogeny Decompositions, Public Key Cryptography-PKC 2006, pp.191-206, 2006.
DOI : 10.1007/11745853_13
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.219.2150

]. A. Dud16 and . Dudeanu, Computational Aspects of Jacobians of Hyperelliptic Curves, 2016.

]. R. Dup06 and . Dupont, Moyenne arithmético-géométrique, suites de Borchardt et applications " . http://www.lix.polytechnique.fr/Labo/Regis, pp.2-4, 2006.

A. [. Elkies and . Kumar, K3 surfaces and equations for Hilbert modular surfaces, Algebra & Number Theory, vol.93, issue.10, pp.2297-2411, 2014.
DOI : 10.1007/BFb0097582
URL : http://arxiv.org/abs/1209.3527

]. A. Eng09 and . Enge, Computing modular polynomials in quasi-linear time, Math. Comp 78, pp.1809-1824, 2009.

A. [. Enge and . Sutherland, Class Invariants by the CRT Method, ANTS IX: Proceedings of the Algorithmic Number Theory 9th International Symposium, pp.142-156, 2010.
DOI : 10.1007/978-3-642-14518-6_14
URL : https://hal.archives-ouvertes.fr/inria-00448729

A. Enge and E. Thomé, Computing Class Polynomials for Abelian Surfaces, Experimental Mathematics, vol.23, issue.2, 2014.
DOI : 10.1090/S0025-5718-2013-02712-3
URL : https://hal.archives-ouvertes.fr/hal-00823745

F. [. Enge and . Morain, Comparing Invariants for Class Fields of Imaginary Quadratic Fields, Algorithmic number theory, pp.252-266, 2002.
DOI : 10.1007/3-540-45455-1_21

F. [. Fouquet and . Morain, Isogeny Volcanoes and the SEA Algorithm, Algorithmic number theory, pp.276-291, 2002.
DOI : 10.1007/3-540-45455-1_23
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.43.2649

]. E. Fre90 and . Freitag, Hilbert modular forms In: Hilbert Modular Forms, pp.5-71, 1990.

G. Frey and E. Kani, Curves of genus 2 with elliptic differentials and associated Hurwitz spaces, Contemporary Mathematics, vol.14, p.33, 2009.
DOI : 10.1090/conm/487/09524

F. [. Galbraith, N. Hess, and . Smart, Extending the GHS Weil Descent Attack, Advances in Cryptology?EUROCRYPT 2002, pp.29-44, 2002.
DOI : 10.1007/3-540-46035-7_3
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.108.527

J. Zur-gathen and G. Jürgen, Modern Computer Algebra, pp.0-521, 1999.
DOI : 10.1017/CBO9781139856065

]. P. Gau00 and . Gaudry, Algorithmique des courbes hyperelliptiques et applications à la cryptologie, 2000.

]. P. Gau07 and . Gaudry, Fast genus 2 arithmetic based on Theta functions, Journal of Mathematical Cryptology, vol.13, pp.243-265, 2007.

T. [. Gaudry, D. Houtmann, C. Kohel, A. Ritzenthaler, and . Weng, The 2-Adic CM Method for Genus 2 Curves with Application to Cryptography, International Conference on the Theory and Application of Cryptology and Information Security, pp.114-129, 2006.
DOI : 10.1007/11935230_8
URL : https://hal.archives-ouvertes.fr/inria-00103435

]. E. Gor02 and . Goren, Lectures on Hilbert modular varieties and modular forms, American Mathematical Soc, 2002.

]. D. Gru08 and . Gruenewald, Explicit algorithms for Humbert surfaces, pp.17-24, 2008.

]. Gun63 and . Gundlach, Die Bestimmung der Funktionen zur Hilbertschen Modulgruppe des Zahlkörpers Q( ? 5), In: Math. Annalen, vol.152, pp.226-256, 1963.

F. Hirzebruch, D. Zagier, W. L. Baily, and T. Shioda, Classification of Hilbert Modular Surfaces, pp.43-78, 1977.
DOI : 10.1017/CBO9780511569197.005

G. Humbert, Sur les fonctions abéliennes singulières I, Journal de Mathématiques Pures et Appliquées, serie 5 V (1899), pp.233-350

G. Humbert, Sur les fonctions abéliennes singulières II, Journal de Mathématiques Pures et Appliquées, serie 5 VI, pp.279-386, 1900.

G. Humbert, Sur les fonctions abéliennes singulières III, Journal de Mathématiques Pures et Appliquées, pp.97-124, 1901.

J. Igusa, Arithmetic variety of moduli for genus 2, Annals of Mathematics, vol.723, 1960.

]. J. Igu62 and . Igusa, On Siegel modular forms of genus 2, 1962.

C. [. Ionica, D. Martindale, M. Robert, and . Streng, Isogeny graphs of ordinary abelian surfaces over a finite field, 2013.

E. [. Ionica and . Thomé, Isogeny graphs with maximal real multiplication, In: IACR Cryptology ePrint Archive, vol.2014, p.230, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00967742

M. Kalkbrener, On the stability of Gröbner bases under specializations, Journal of Symbolic Computation, vol.241, pp.51-58, 1997.

]. E. Kan and . Kani, Generalized Humbert Schemes and Intersections of Humbert Surfaces, p.47, 2013.

]. E. Kan94 and . Kani, Elliptic curves on abelian surfaces, Manuscripta Mathematica, vol.84, pp.199-223, 1994.

E. Kani, The moduli spaces of Jacobians isomorphic to a product of two elliptic curves, Collectanea Mathematica, vol.22, issue.3, pp.138-174, 2014.
DOI : 10.1112/plms/s3-48.1.175

]. O. Kin05 and . King, The subgroup structure of finite classical groups in terms of geometric configurations In: Surveys in Combinatorics, pp.29-56, 2005.

]. H. Lab16 and . Labrande, Explicit computation of the Abel-Jacobi map and its inverse, 2016.

E. [. Labrande and . Thomé, Computing theta functions in quasi-linear time in genus??two and above, LMS Journal of Computation and Mathematics, vol.1, issue.A, pp.163-177, 2016.
DOI : 10.1017/CBO9780511619878
URL : https://www.cambridge.org/core/services/aop-cambridge-core/content/view/2EE73D9E67B049C98C369F681F28741B/S1461157016000309a.pdf/div-class-title-computing-theta-functions-in-quasi-linear-time-in-genus-two-and-above-div.pdf

K. Lauter, M. Naehrig, and T. Yang, Hilbert theta series and invariants of genus 2 curves, Journal of Number Theory, vol.161, 2015.
DOI : 10.1016/j.jnt.2015.02.020

K. Lauter and T. Yang, Computing genus 2 curves from invariants on the Hilbert moduli space, Journal of Number Theory, vol.131, issue.5, p.10, 2011.
DOI : 10.1016/j.jnt.2010.05.012
URL : http://doi.org/10.1016/j.jnt.2010.05.012

D. [. Lauter and . Robert, Improved CRT Algorithm for Class Polynomials in Genus 2 The Open Book Series, ANTS X ? Proceedings of the Tenth Algorithmic Number Theory Symposiumpdf. Slides: 2012-07-ANTS-SanDiego.pdf (30min, International Algorithmic Number Theory Symposium (ANTS-X), pp.437-461, 2012.

H. [. Lenstra, L. Lenstra, and . Lovász, Factoring polynomials with rational coefficients, Mathematische Annalen, vol.32, issue.4, pp.515-534, 1982.
DOI : 10.1007/BF01457454
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.310.318

]. R. Man94 and . Manni, Modular varieties with level 2 theta structure, American Journal of Mathematics, vol.116, pp.1489-1511, 1994.

]. C. Mar16 and . Martindale, In preparation, p.2016

]. E. Mil15 and . Milio, A quasi-linear time algorithm for computing modular polynomials in dimension 2, LMS Journal of Computation and Mathematics, vol.181, pp.603-632, 2015.

]. F. Mor95 and . Morain, Calcul du nombre de points sur une courbe elliptique dans un corps fini: aspects algorithmiques, J. Théor. Nombres Bordeaux, vol.7, pp.255-282, 1995.

]. D. Mum84 and . Mumford, Tata lectures on theta II, Progress in Mathematics. Birkhäuser, 1984.

]. S. Nag83 and . Nagaoka, On the ring of Hilbert modular forms over Z, Journal Math. Soc. Japan, vol.354, pp.589-608, 1983.

A. Novocin, D. Stehlé, and G. Villard, An LLL-reduction algorithm with quasilinear time complexity, Proceedings of the forty-third annual ACM symposium on Theory of computing, pp.403-412, 2011.
DOI : 10.1145/1993636.1993691
URL : https://hal.archives-ouvertes.fr/ensl-00534899

]. H. Res74 and . Resnikoff, On the Graded Ring of Hilbert Modular Forms Associated with Q( ? 5), In: Math. Ann, vol.208, pp.161-170, 1974.

]. D. Rob13 and . Robert, Computing cyclic isogenies using real multiplication " . (Notes) ANR Peace meeting, 2013.

]. D. Rob15, C. Journées, . Codage, L. Cryptographie, and . Londe-les-maures, Isogenies, Polarisations and Real Multiplication, 2015.

A. Rostovtsev and A. Stolbunov, Public-key cryptosystem based on isogenies In: International Association for Cryptologic Research, Cryptology ePrint Archive, p.145, 2006.

]. B. Run99 and . Runge, Endomorphism rings of abelian surfaces and projective models of their moduli spaces, Tohoku mathematical journal 51, pp.283-303, 1999.

]. Ser70 and . Serre, Le Probleme des Groupes de Congruence Pour SL 2, Annals of Mathematics, vol.923, pp.489-527, 1970.

]. N. Sma03 and . Smart, An analysis of Goubin's refined power analysis attack, Cryptographic Hardware and Embedded Systems-CHES 2003, pp.281-290, 2003.

]. B. Smi09 and . Smith, Isogenies and the Discrete Logarithm Problem in Jacobians of Genus 3 Hyperelliptic Curves. Feb, 2009.

M. Streng, Complex multiplication of abelian surfaces, 2010.

]. A. Sut11 and . Sutherland, Computing Hilbert class polynomials with the Chinese remainder theorem, In: Mathematics of Computation, vol.80273, pp.501-538, 2011.

J. Tate, Endomorphisms of abelian varieties over finite fields, Inventiones Mathematicae, vol.1, issue.No. 6, pp.133-144, 1966.
DOI : 10.1007/BF01404549

]. E. Tes06 and . Teske, An elliptic curve trapdoor system, Journal of cryptology, vol.191, pp.115-133, 2006.

]. G. Van82, . Van, and . Geer, On the geometry of a Siegel modular threefold

]. G. Van12, . Van-der, and . Geer, Hilbert modular surfaces, p.2012

A. Weng, Constructing hyperelliptic curves of genus 2 suitable for cryptography, Mathematics of Computation, vol.72, issue.241, pp.435-458, 2003.
DOI : 10.1090/S0025-5718-02-01422-9